Katharina Eichinger

AND Manuel V Gnann

Christian Kuehn

MULTISCALE ANALYSIS FOR TRAVELING-PULSE SOLUTIONS TO THE STOCHASTIC FITZHUGH-NAGUMO EQUATIONS

Keywords: November 3, 2021. 2010 Mathematics Subject Classification. 35C07, 35K57, 35Q92, 35R60, 60H15 FitzHugh-Nagumo equations, stochastic reaction-diffusion equations, traveling waves, pulse, stability

We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh-Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat [Nonlinear Anal., 162:197-223, 2017] for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

 (1.1b)in which the independent variables t and x denote time and position on a neural axon, respectively. The dependent variable ũ denotes the electric potential and ṽ is a gating variable. The parameter ν ą 0 determines the strength of the diffusion B 2 x ũ, while the nonlinearity f pũq is a reaction term which typically has the form f pũq " χpũq ũ p1 ´ũq pũ ´aq with 0 ă a ă 1 and is suitably cut off by the factor χ. The parameter σ ą 0 determines the strength of the noise W . Here, we assume that W is an infinite-dimensional Wiener process taking values in a Hilbert space to be specified in what follows. The parameter ε ą 0 determines the strength of the coupling of the electric potential ũ to the gating variable ṽ and is assumed to be sufficiently small. The parameter γ ą 0 determines the decay of the gating variable ṽ.

The classical FitzHugh-Nagumo [START_REF] Fitzhugh | Mathematical models of threshold phenomena in the nerve membrane[END_REF][START_REF] Nagumo | An active pulse transmission line simulating nerve axon[END_REF] partial differential equations (PDEs) obtained for σ " 0 form a simplified, yet qualitatively very similar, model for the Hodgkin-Huxley equations [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF], which was a key part of Hodgkin's and Huxley's Nobel prize awarded in 1963. By now, the FitzHugh-Nagumo system is a standard model for the generation and transmission of electrical signals in neuroscience [START_REF] Ermentrout | Mathematical Foundations of Neuroscience[END_REF][START_REF] Izhikevich | Dynamical Systems in Neuroscience[END_REF]. The literature on the PDE version of the FitzHugh-Nagumo system (σ " 0) is very large, see for example the recent papers [START_REF] Carter | Fast pulses with oscillatory tails in the fitzhugh-nagumo system[END_REF][START_REF] Guckenheimer | Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit[END_REF] and detailed references therein. For the ODE version (σ " 0, ν " 0), the literature is vast [START_REF] Rocsoreanu | The FitzHugh-Nagumo Model -Bifurcation and Dynamics[END_REF], mostly due to the crucial role played by bistable nonlinearities in all areas of nonlinear science and the commonly found multiple time scale structure of the FitzHugh-Nagumo system [START_REF] Kuehn | Multiple Time Scale Dynamics[END_REF]. Also the SODE variant for ν " 0 is quite well-studied, mainly due to a flurry of activity since the mid 1990s; see e.g. [START_REF] Bashkirtseva | Analysis of excitability for the fitzhugh-nagumo model via a stochastic sensitivity function technique[END_REF][START_REF] Berglund | From random Poincaré maps to stochastic mixed-mode-oscillation patterns[END_REF][START_REF] Berglund | Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model[END_REF][START_REF] Lindner | Effects of noise in excitable systems[END_REF][START_REF] Lindner | Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance[END_REF][START_REF] Muratov | Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle[END_REF]]. Yet, the full SPDE variant (1.1) has only attracted major attention quite recently, including a large number of numerical studies [START_REF] Lord | An Introduction to Computational Stochastic PDEs[END_REF][START_REF] Sauer | Analysis and approximation of stochastic nerve axon equations[END_REF][START_REF] Sauer | Reliability of signal transmission in stochastic nerve axon equations[END_REF][START_REF] Shardlow | Numerical simulation of stochastic PDEs for excitable media[END_REF][START_REF] Tuckwell | Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron[END_REF][START_REF] Tuckwell | Stochastic partial differential equations in neurobiology: linear and nonlinear models for spiking neurons[END_REF][START_REF] Tuckwell | Analytical and simulation results for stochastic Fitzhugh-Nagumo neurons and neural networks[END_REF] as well as analytical studies regarding existence, regularity, invariant measures and attractors [START_REF] Bates | Random attractors for stochastic reaction-diffusion equations on unbounded domains[END_REF][START_REF] Berglund | Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions[END_REF][START_REF] Bonaccorsi | Analysis of the stochastic Fitzhugh-Nagumo system[END_REF][START_REF] Kuehn | Random attractors for stochastic partly dissipative systems[END_REF][START_REF] Li | A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations[END_REF][START_REF] Wang | Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains[END_REF]. The question regarding stochastic stability of pulses for additive noise is far less studied. We refer to [START_REF] Hamster | Stability of travelling waves for reaction-diffusion equations with multiplicative noise[END_REF] for multiplicative noise with a regularized equation (diffusion in the second variable) and to the review [START_REF] Kuehn | Travelling waves in monostable and bistable stochastic partial differential equations[END_REF] for the stochastic Nagumo case (ε " 0). For further biophysical motivation regarding various noise terms in the FitzHugh-Nagumo equation we refer to the review [START_REF] Lindner | Effects of noise in excitable systems[END_REF].

Here we contribute to a more detailed understanding of stochastic pulse stability of the FitzHugh-Nagumo equations (1.1) exploiting the multiscale nature of the problem. More precisely, our aim is to understand the dynamics of (1.1) near a deterministically stable pulse in the regime, where the parameters σ ą 0 (strength of the noise) and ε ą 0 (coupling to the gating variable) are small. The subsequent analysis generalizes the recent analysis of Krüger and Stannat [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF] for corresponding scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. We remark that the idea of tracking small noise fluctuations for SPDEs around traveling waves via a multiscale SODE approximation goes back at least to the early 1980s and works by Ebeling, Mikhailov, and Schimansky-Geier [START_REF] Mikhailov | Effect of fluctuation on plane front propagation in bistable nonequilibrium systems[END_REF][START_REF] Mikhailov | Stochastic motion of the propagating front in bistable media[END_REF]. From the viewpoint of applications the extension from Nagumo to FitzHugh-Nagumo is a crucial generalization as the FitzHugh-Nagumo model (1.1) is far more realistic than the one studied in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF] since only (1.1) allows for deterministically stable traveling-pulse solutions. In fact, deterministically stable localized pulses model far better the real action potentials generated in neurons in comparison to the deterministically stable traveling fronts appearing in the Nagumo equation. From a mathematical viewpoint, our generalization is important as it does extend beyond the setting of treating the equation in a Hilbert space in which the linearization of the traveling wave is self-adjoint (this is the case in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF]) or generates an analytic semigroup and thereby makes the methods of multiscale approximation available to a broad class of stochastic SPDE-ODE reaction-diffusion systems.

1.2. Traveling-pulse solutions and their stability. We recall some of the well-known results on existence of traveling-pulse solutions to the deterministic version of (1.1) where σ " 0. These solutions have the form ũpt, xq " ûpξq and ṽpt, xq " vpξq, where ξ " x `st and s P R is the velocity of the traveling wave. The tuple pû, vq therefore fulfills the set of equations dû dξ " û1 for ξ P R,

(1.2a)

dû 1
dξ "

1 ν `sû 1 ´f pûq `v ˘for ξ P R, (1.2b)
dv dξ " ε s pû ´γvq for ξ P R.

(1.2c)

Traveling-pulse solutions to (1.2) are solutions ps, û, vq t such that pû, û1 , vq t Ñ pb 1 , b 2 , b 3 q t as ξ Ñ ˘8 where pb 1 , b 2 , b 3 q t P R 3 is a stationary solution to (1.2) fulfilling b 2 " 0, f pb 1 q " b 3 , and b 1 " γb 3 , i.e., they are homoclinic orbits of the dynamical system (1.2). In what follows we will also have the convention that we mean non-trivial traveling pulses. Furthermore, we are only interested in solutions pû, vq t such that pû, vq t Ñ p0, 0q t as ξ Ñ ˘8. Indeed, this situation will occur if the equilibrium point pb 1 , b 2 , b 3 q t " p0, 0, 0q t is unique, which is the case e.g. if γ ě 0 is sufficiently small [START_REF] Rocsoreanu | The FitzHugh-Nagumo Model -Bifurcation and Dynamics[END_REF]. Further note that the velocity s of the traveling wave is not a parameter but a functional of f and ε.

Next, we give a very brief overview of the existing literature on the existence of homoclinic orbits of (1.2) corresponding to traveling-pulse solutions. For a wave speed s " 0 and ε " 0, we recover the planar ODE associated to traveling waves of the Nagumo equation. Using the resulting Hamiltonian structure of the ODE, it is easy to see that a homoclinic orbit exists for s " 0 and ε " 0. A singular perturbation argument in combination with Melnikov's method [START_REF] Hastings | On the existence of homoclinic and periodic orbits for the Fitzhugh-Nagumo equations[END_REF][START_REF] Kuehn | Multiple Time Scale Dynamics[END_REF][START_REF] Szmolyan | Transversal heteroclinic and homoclinic orbits in singular perturbation problems[END_REF] yields the existence of a slow pulse with wave speed s « 0. Yet, an application of Sturm-Liouville theory [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF][START_REF] Kuehn | PDE Dynamics: An Introduction[END_REF] shows that the slow pulse is unstable. As it is deterministically already unstable, considering this pulse under the influence of noise is not expected to be biophysically relevant as the noisy small perturbations will be amplified exponentially near the slow pulse. Yet, there is also a fast pulse corresponding to much higher wave speeds s " spf, εq. Carpenter [START_REF] Carpenter | Traveling-wave solutions of nerve impulse equations[END_REF] and Conley [START_REF] Conley | On traveling wave solutions of nonlinear diffusion equations[END_REF] constructed these homoclinic orbits to (1.2) employing the method of isolating blocks of the fast and slow subsystems [START_REF] Conley | Isolated invariant sets and isolating blocks[END_REF]. See also [START_REF] Gardner | The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index[END_REF], where the methods developed in [START_REF] Conley | On traveling wave solutions of nonlinear diffusion equations[END_REF] have been further improved. Then a generalization of the pulse construction to large classes of FitzHugh-Nagumo-like models has been provided in [START_REF] Evans | Double impulse solutions in nerve axon equations[END_REF]. Later, a fully geometric construction of the fast pulse via the Exchange Lemma [START_REF] Jones | Tracking invariant manifolds up to exponentially small errors[END_REF][START_REF] Jones | Tracking invariant manifolds with differential forms in singularly perturbed systems[END_REF][START_REF] Kuehn | Multiple Time Scale Dynamics[END_REF] was proved by Jones, Kopell and Langer [START_REF] Jones | Construction of the FitzHugh-Nagumo pulse using differential forms[END_REF] using differential forms. The connection in parameter space between slow and fast pulses has been proved in [START_REF] Krupa | Fast and slow waves in the FitzHugh-Nagumo equation[END_REF]. Then the parametric bifurcation structure has been analyzed in a more refined way in [START_REF] Carter | Fast pulses with oscillatory tails in the fitzhugh-nagumo system[END_REF][START_REF] Champneys | When Shil'nikov meets Hopf in excitable systems[END_REF][START_REF] Guckenheimer | Homoclinic orbits of the FitzHugh-Nagumo equation: The singular limit[END_REF][START_REF] Guckenheimer | Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system[END_REF]. A non-perturbative approach to construct traveling-pulse solutions of (1.2) has been carried out by Arioli and Koch [START_REF] Arioli | Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation[END_REF] for the value ε " 0.01 using computer-assisted proofs.

Stability of the fast traveling-pulse solutions to the deterministic version of (1.1) with σ " 0 in the space of bounded uniformly continuous functions has been obtained by Jones [START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF] for the case of a cubic polynomial, in the sense that solutions starting sufficiently close to a wave profile decay to a translate of pûp¨`stq, vp¨`stqq t .

The proof relies on analysis developed by Evans [START_REF] Evans | Nerve axon equations. I. Linear approximations[END_REF][START_REF] Evans | Nerve axon equations. II. Stability at rest[END_REF][START_REF] Evans | Nerve axon equations. III. Stability of the nerve impulse[END_REF][START_REF] Evans | Nerve axon equations. IV. The stable and the unstable impulse[END_REF][START_REF] Evans | Errata: "Nerve axon equations. III. Stability of the nerve impulse[END_REF][START_REF] Evans | Erratum: "Nerve axon equations. II. Stability at rest[END_REF], where it is proved that in the space of bounded uniformly continuous functions that linear stability implies nonlinear stability and that the point spectrum of the linearization around the traveling pulse is determined by the roots of a function Dpλq, the Evans function. Jones proves that instability can only occur due to eigenvalues of the linear operator near λ " 0. By calculating the winding number of Dpλq for a small circle around λ " 0, it is shown that only two eigenvalues lie in it, one of which is λ " 0 (related to the translation invariance of the problem) and the other is negative because dD dλ p0q ą 0. For a more recent stream-lined stability analysis allowing for more general reaction terms f but still restricted to the space of bounded uniformly continuous functions, we refer to [START_REF] Arioli | Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation[END_REF][START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF] while stability in L 2 pR; R 2 q and H 1 pR; R 2 q is proved in [START_REF] Ghazaryan | Stability of traveling waves for degenerate systems of reaction diffusion equations[END_REF][START_REF] Rottmann-Matthes | Computation and Stability of Patterns in Hyperbolic-Parabolic Systems[END_REF][START_REF] Yurov | Stability estimates for semigroups and partly parabolic reaction diffusion equations[END_REF] (see §2.2 for further details). Nonlinear stability of the fast FitzHugh-Nagumo pulse with oscillatory tails (instead of monotone tails as considered in this paper) has been proved by Carter, de Rijk, and Sandstede in [START_REF] Carter | Stability of traveling pulses with oscillatory tails in the FitzHugh-Nagumo system[END_REF]. For general introductions into the subject, we refer to [START_REF] Chicone | Evolution Semigroups in Dynamical Systems and Differential Equations[END_REF][START_REF] Sandstede | Stability of travelling waves[END_REF].

1.

3. An approach for computing the velocity correction. Our aim is to investigate the following decomposition ũpt, xq " û px `st `ϕptqq `uϕ pt, xq,

(1.3a) ṽpt, xq " v px `st `ϕptqq `vϕ pt, xq (1.3b)
of solutions to (1.1), where the function ϕptq is a random correction to the position of the wave front, and u ϕ pt, xq and v ϕ pt, xq denote lower-order fluctuations that are uniquely defined through (1.3) for any choice of ϕ " ϕptq. Ideally, we would like to choose ϕ to minimize the distance in the direction of the traveling wave between the solution X :" pũ, ṽq t of the FitzHugh-Nagumo SPDEs (1.1) and the suitably translated traveling wave X " pû, vq . However, as the minimization problem (1.4) is not necessarily convex, uniqueness of a minimizer is not ensured. We follow the approach in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF] and replace (1.4) by the weaker condition for a critical point of finding ϕ " ϕptq such that

0 " ´Π0 st`ϕ ´Xpt, ¨q ´Xp¨`st `ϕq ¯, d X dξ p¨`st `ϕq ¯H , (1.5)
where p¨, ¨qH denotes the inner product of H. This approach has been employed by Inglis and MacLaurin in [START_REF] Inglis | A general framework for stochastic traveling waves and patterns, with application to neural field equations[END_REF] for more general classes of SPDE systems with the drawback that results only hold up to the first stopping time when the local minimum turns into a saddle. Here, we follow the work around proposed in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF] in the sense that ϕptq is approximated by a process ϕ m ptq, which in our case fulfills the random ordinary differential equation (RODE)

dϕ m dt ptq " m ´Π0 st`ϕ m ptq ´Xpt, ¨q ´Xp¨`st `ϕm ptqq ¯, d X dξ p¨`st `ϕm ptqq ¯H (1.6)
for given initial condition and a relaxation parameter m ą 0 that is chosen sufficiently large. Notably, approximating ϕ with ϕ m through (1.6) implies differentiability while this is in general not true for (1.5). By further analyzing (1.6) we will construct ϕ as a solution of an SODE and pu ϕ , v ϕ q t as the solution of a system of an SPDE coupled to an SODE in such a way that pu ϕ , v ϕ q t fulfill estimates in suitable function spaces, making precise what the notion 'lower order' means.

In [55, §3.4], it has been pointed out that with the strategy described above the first-order fluctuations around the traveling wave are orthogonal in their suitably chosen inner product. In fact, in the case of of second-order bistable reaction-diffusion equations, the spatial linearization around the traveling wave, the frozen-wave operator, is of Sturm-Liouville type. Hence, one can always find a weighted L 2 -inner product in which it is self-adjoint, so that one can replace the projection operator used in (1.4) with an orthogonal projection induced by this inner product. One cannot expect a similar approach to be applicable in our situation (1.1) of an SPDE coupled to an ODE or other more complicated systems of SPDEs. Instead, we will build up our analysis on Riesz spectral projections of the frozen-wave operator that do not require a self-adjoint structure and yield a partition into two subspaces invariant under the linearized flow. Note that non-orthogonal Riesz spectral projections have also been employed by Hamster and Hupkes in [START_REF] Hamster | Stability of traveling waves for reaction-diffusion equations with multiplicative noise[END_REF][START_REF] Hamster | Stability of travelling waves for reaction-diffusion equations with multiplicative noise[END_REF][START_REF] Hamster | Stability of traveling waves for systems of reaction-diffusion equations with multiplicative noise[END_REF][START_REF] Hamster | Travelling waves for reaction-diffusion equations forced by translation invariant noise[END_REF] by projecting onto the eigenvector of the adjoint of the frozen-wave operator. Their SODE to determine ϕ is more involved compared to (1.6). Their method applies to a variety of (systems) of reaction-diffusion equations with special forms of multiplicative noise (partially only including a one-dimensional Wiener process). However, in all situations treated there, the frozen-wave operator is sectorial with spectral angle larger than π 2 and therefore generates an analytic semigroup. Specifically, in [41, (1.2)], [42, (1.3)], [43, (1.17)] the second component of the FitzHugh-Nagumo system is regularized by adding cB 2

x ṽ with some c ą 0 to the right-hand side of (1.1b) (see [START_REF] Chen | Traveling pulse solutions to FitzHugh-Nagumo equations[END_REF][START_REF] Cornwell | On the existence and stability of fast traveling waves in a doubly diffusive FitzHugh-Nagumo system[END_REF] for existence and stability of the pulse in this case), which is different from the only partly parabolic FitzHugh-Nagumo system without diffusion in the second component as treated for instance in [START_REF] Arioli | Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation[END_REF][START_REF] Carpenter | Traveling-wave solutions of nerve impulse equations[END_REF][START_REF] Conley | On traveling wave solutions of nonlinear diffusion equations[END_REF][START_REF] Gardner | The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index[END_REF][START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF][START_REF] Jones | Construction of the FitzHugh-Nagumo pulse using differential forms[END_REF][START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF]. In our setting, the frozen-wave operator has essential spectrum parallel to the imaginary axis (see §2.2 and §A.2 below) and therefore is so far not covered by this approach.

Historically, one can track back stability and fluctuation analysis of bistable scalar SPDEs, such as the Nagumo SPDE, at least to early works by Ebeling, Mikhailov, and Schimansky-Geier in [START_REF] Mikhailov | Effect of fluctuation on plane front propagation in bistable nonequilibrium systems[END_REF][START_REF] Mikhailov | Stochastic motion of the propagating front in bistable media[END_REF], where it was recognized that the deterministic reference wave speed should be corrected by a stochastic term, which in turn satisfies an SODE. Many further works followed, e.g., using a more rigid/frozen stochastic frame in combination with numerical simulations by Lord and Thümmler in [START_REF] Lord | Computing stochastic traveling waves[END_REF], employing functional inequalities to establish stability bounds by Stannat in [START_REF] Stannat | Stability of travelling waves in stochastic bistable reaction-diffusion equations[END_REF], the adaptation of a rigorous multiscale expansion by Krüger and Stannat in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF] originally motivated by work on the related problem of stochastic traveling waves for bistable neural field equations by the same authors in [START_REF] Krüger | Front propagation in stochastic neural fields: a rigorous mathematical framework[END_REF] and by Inglis and MacLaurin in [START_REF] Inglis | A general framework for stochastic traveling waves and patterns, with application to neural field equations[END_REF], and work on long-time stochastic stability tracking for suitable multiplicative noise by Hamster and Hupkes in [START_REF] Hamster | Stability of travelling waves for reaction-diffusion equations with multiplicative noise[END_REF] via stochastic convolution estimates by the same authors in [START_REF] Hamster | Stability of traveling waves for reaction-diffusion equations with multiplicative noise[END_REF]. We also remark that in case of stochastic dispersive PDEs, the random modulation of soliton solutions (i.e., standing or traveling wave packages) has been investigated by de Bouard and Debussche for the stochastic Korteweg-de Vries equation in [START_REF] De Bouard | Random modulation of solitons for the stochastic Korteweg-de Vries equation[END_REF] and by de Bouard and Fukuizumi for the Gross-Pitaevskii equation in [START_REF] De Bouard | Modulation analysis for a stochastic NLS equation arising in Bose-Einstein condensation[END_REF]. Furthermore, multiscale expansions also frequently appear in the context of SPDE amplitude equations at bifurcation points as treated for instance by Blömker in [START_REF] Blömker | Amplitude Equations for Stochastic Partial Differential Equations[END_REF] and by Blömker, Hairer, and Pavliotis in [START_REF] Blömker | Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities[END_REF].

1.4. Outline. We continue with the setting and auxiliary results in §2, followed by the main results in §3. The proofs of auxiliary results are contained in Appendix A while the proofs of the main results can be found in §4. In §2.1 and §A.1 we construct solutions to (1.1) using the variational approach for equations with locally monotone coefficients [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF][START_REF] Liu | Stochastic partial differential equations: an introduction[END_REF]. In §2.2 and §A.2, we then formulate the SPDE of perturbations around the traveling wave which to leading-order is governed by the linearization around the deterministically-stable fast FHN pulse. Results on the deterministic linearized evolution of perturbations around the traveling wave are provided in Proposition 2.6 and Proposition 2.8 below. Afterwards, in §3.1 and §4.1, we derive an SODE approximating the correction of the wave velocity (cf. Proposition 3.2 below). The leading-order part of this SODE is an Ornstein-Uhlenbeck-type process with a linear damping in the drift due to the relaxation method of the frame and additive stochastic fluctuations obtained from projecting the infinite-dimensional noise onto the deterministic translation-invariant mode. Practically, this entails that the deterministic reference wave has phase diffusion along the translation direction. Subsequently, in §3.2 and §4.2, we prove a multiscale expansion in terms of the linearized evolution (cf. Theorem 3.3 below), which is further investigated in §3.3 and §4.3 in the limit as m Ñ 8 of immediate relaxation (cf. Theorem 3.4 and Proposition 3.5 below). In particular, our results yield bounds on ' the first exit time where the multiscale decomposition cannot be guaranteed to hold anymore and ' on the second moment of fluctuations transverse to the traveling wave mode after correcting the wave velocity. Concluding remarks and an outlook on future research can be found in §5.

Setting and auxiliary results

For what follows, we fix a stochastic basis, that is, a complete filtered probability space `Ω, F, pF t q tPr0,T s , P ˘, with a complete and right-continuous filtration pF t q tPr0,T s , where T P p0, 8q is arbitrary.

2.1. Existence and uniqueness of solutions using the variational approach. In line with the classical findings for the deterministic FitzHugh-Nagumo PDE discussed above, we make the following assumption on (1.1) or (1.2), respectively. In particular, we are only going to study stochastic perturbations to the deterministically stable fast pulse solution:

Global Assumption 2.1. In a right-neighborhood of ε " 0 the system (1.2) has a non-trivial homoclinic orbit of the point p0, 0, 0q t in phase space. The corresponding traveling-pulse solution pû, vq t of the FitzHugh-Nagumo equations (1.1) with σ " 0 is locally asymptotically stable up to translation.

We will assume certain properties on the reaction term f pwq that are fulfilled for instance by the choice f pwq " χpwq wp1 ´wqpw ´aq, where a P p0, 1q and χ P C 8 pRq meets χ |r´c 1 ,8q " 1 and χpwq "

c 2 2
w 2 for w P p´8, ´c2 s, where 1 ă c 1 ă c 2 sufficiently large (cut-off at ´8), for which existence of a traveling-pulse solution (cf. Global Assumption 2.1) is guaranteed provided ε ą 0 is small. Here, we directly make the usual abstract bi-stability assumptions [START_REF] Chen | Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations[END_REF][START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF][START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF] for f so that the nonlinearity effectively behaves like the classical cubic nonlinearity chosen for the Nagumo and FitzHugh-Nagumo equations.

Global Assumptions 2.2. We have f P C 3 pRq and there exist a P p0, 1q and d ą 0 with f p0q " f paq " f p1q " 0, (2.1a)

f pwq ă 0 for w P p0, aq Y p1, 8q, (2.1b)
f pwq ą 0 for w P p´8, 0q Y pa, 1q, (2.1c)

f 1 p0q ă 0, f 1 paq ą 0, f 1 p1q ă 0, (2.1d)
f pwq ´w d ‰ 0 for w P Rzt0u, (2.1e)
ˆ1 0 f pwq dw ą 0, (2.1f
)

η 1 :" sup wPR f 1 pwq ă 8, (2.1g
)

f pw 1 `w2 q ´f pw 1 q ´f 1 pw 1 qw 2 ď η 2 p1 `|w 1 | `|w 2 |q |w 2 | 2 for w 1 , w 2 P R, (2.1h
)

f 1 pwq ď η 3 ´1 `|w| 2 ¯for w P R, (2.1i
)

|f pw 1 q ´f pw 2 q| ď η 4 |w 1 ´w2 | ´1 `|w 1 | 2 `|w 2 | 2 ¯for w 1 , w 2 P R, (2.1j
)

f 1 pw 1 `w2 q ´f 1 pw 1 q ´f 2 pw 1 qw 2 ď η 5 |w 2 | 2 for w 1 , w 2 P R, (2.1k
)

f 1 pw 1 q ´f 1 pw 2 q ď η 6 |w 1 ´w2 | p1 `|w 1 | `|w 2 |q for w 1 , w 2 P R, (2.1l) f 2 pw 1 q ´f 2 pw 2 q ď η 7 |w 1 ´w2 | for w 1 , w 2 P R, (2.1m)
where η 2 , η 3 , η 4 , η 5 , η 6 , η 7 ă 8.

Note that the linearization of (1.2) in p0, 0, 0q t only depends on f 1 p0q and because of (2.1d) this point is hyperbolic and hence d j û dξ j for j P t0, 1, 2, 3, 4, 5u and d j v dξ j for j P t0, 1, 2, 3, 4u are exponentially decaying as |ξ| Ñ ˘8. In particular, d j û dξ j L 8 pRq ă 8 and

d j û dξ j L 2 pRq
ă 8 for all j P t0, 1, 2, 3, 4, 5u, as well as

d j v dξ j L 8 pRq
ă 8 and

d j v dξ j L 2 pRq
ă 8 for all j P t0, 1, 2, 3, 4u.

For ϕ " 0, we may use (1.3) as a definition of pu, vq :" pu 0 , v 0 q and employing (1.2), we obtain the system dupt, xq " `νB

and introduce the rigged space triple (Gelfand triple)

V :" H 1 pRq ? ε kL 2 pRq ãÑ H :" L 2 pRq ? ε kL 2 pRq " H ˚ãÑ V ˚" H ´1pRq ? ε kL 2 pRq, (2.5)
where

pY 1 , Y 2 q H :" Z ˆR pεw 1 w 2 `q1 q 2 q dx, (2.6a
)

pY 1 , Y 2 q V :" Z ˆR pεw 1 w 2 `εpB x w 1 qpB x w 2 q `q1 q 2 q dx, (2.6b)
with

Y j :" ˆwj q j ˙and Z :" ˆˆR ˆε ´dû dξ ¯2 `´dv dξ ¯2˙d ξ ˙´1 , (2.7)
and where

X :" ˆu v ˙, (2.8a)
A : r0, T s ˆV ˆΩ Ñ V ˚,
pt, X, ωq Þ Ñ Apt, Xq :"

ˆνB 2 x u `f pu `ûp¨`stqq ´f pûp¨`stqq ´v ε pu ´γvq ˙, (2.8b)
B : r0, T s ˆV ˆΩ Ñ L 2 pU ; Hq , pt, X, ωq Þ Ñ ˆU Ñ H, N Þ Ñ Bpt, XqN :" ˆσ? QN 0 ˙˙. (2.8c)
Here, we assume that U :" L 2 pRq, Q P LpU q is a symmetric nonnegative-definite bounded linear operator in U with finite trace, L 2 pU ; Hq denotes the space of Hilbert-Schmidt operators U Ñ H, and W U is a U -valued pF t q tPr0,T s -adapted cylindrical id U -Wiener process. Note that W " ? QW U is in fact a U -valued pF t q tPr0,T s -adapted Q-Wiener process. We remark that the assumption of Q having finite trace leads to a noise term in (2.4) which is not translation invariant. However, since we study stability of a pulse with exponential tails traveling with finite speed on an axon (which in reality has finite length), this assumption appears to be acceptable regarding the relevance of our results in terms of applications. Further note that the scaling of the first component with ? ε in (2.6) changes the geometry of the orthogonal sums indicated by the symbol ? ε k rather than k in (2.5). On the other hand, the normalization with Z merely implies the convenient property that d X dξ H " 1 but leaves all angles between vectors unchanged.

In order to show existence of solutions to (2.4), one could use the concept of mild solutions as employed for the FitzHugh-Nagumo SPDE with additive noise in [START_REF] Kuehn | Random attractors for stochastic partly dissipative systems[END_REF]. The approach presented in [START_REF] Krüger | A multiscale-analysis of stochastic bistable reaction-diffusion equations[END_REF], which we build upon, uses variational solutions [START_REF] Krylov | Stochastic evolution equations[END_REF][START_REF] Liu | Stochastic partial differential equations: an introduction[END_REF][START_REF] Prévot | A Concise Course on Stochastic Partial Differential Equations[END_REF] for equations with locally monotone coefficients [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF][START_REF] Liu | Stochastic partial differential equations: an introduction[END_REF]. As we will show in Proposition 2.5 below, the variational solution (constructed in Proposition 2.4 below) also turns out to be a mild solution. We remark that the authors of [START_REF] Hamster | Stability of travelling waves for reaction-diffusion equations with multiplicative noise[END_REF][START_REF] Hamster | Stability of traveling waves for systems of reaction-diffusion equations with multiplicative noise[END_REF][START_REF] Hamster | Travelling waves for reaction-diffusion equations forced by translation invariant noise[END_REF] use [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF] to construct solutions, too, but as previously mentioned, their system is regularized by adding cB 2

x u with c ą 0 to the second component in (2.8b), which also changes the Gelfand triple to H 1 pR; R 2 q ãÑ L 2 pR; R 2 q ãÑ H ´1pR; R 2 q. Since the variational approach has not been directly worked out for the FitzHugh-Nagumo SPDE with additive noise and no regularization of the second component, we use this opportunity to fill this gap within this work as an auxiliary step. Therefore, we use the concept of variational solutions (cf. [65, Definition 1.1] and [66, Definition 5.1.2]): Definition 2.3. A variational solution to (2.4) is a continuous H-valued pF t q tě0 -adapted process pXpt, ¨qq tPr0,T s such that the dt b P-equivalence class X meets X P L 2 pr0, T s ˆΩ, dt b P; V q and such that the solution formula Xpt, ¨q " Xp0, ¨q `ˆt

0 A `t1 , Xpt 1 , ¨q˘d t 1 `ˆt 0 B `t1 , Xpt 1 , ¨q˘d W U pt 1
, ¨q for t P r0, ts (2.9)

is satisfied, P-almost surely, where X denotes any V -valued progressively measurable dt b Pversion of X.

Under the given hypothesis, we can prove existence of solutions and regularity in space under additional assumptions:

Proposition 2.4. For p P r6, 8q and T ą 0, the stochastic evolution equation (2.4) has for any Xp0, ¨q " pup0, ¨q, vp0, ¨qq t " X p0q " `up0q , v p0q ˘t P L p pΩ, F 0 , P; Hq a unique variational solution.

The solution further satisfies X P L p `Ω, F, P; C 0 pr0, T s; Hq ˘.

For the subsequent result, we define the stronger space

V :" H 1 pRq ? ε kH 1 pRq (2.10a)
with inner product pY 1 , Y 2 q V :" Z ˆR pεw 1 w 2 `εpB x w 1 qpB x w 2 q `q1 q 2 `pB x q 1 qpB x q 2 qq dx

(2.6a) " pY 1 , Y 2 q H `pB x Y 1 , B x Y 2 q H , (2.10b)
where Y j and Z are as in (2.7).

Proposition 2.5. In the situation of Proposition 2.4, the solution X is also a mild solution in the sense of [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF] with linear operator

ˆνB 2 x 0 0 ´εγ ˙: H Ě H 2 pRq ? ε kL 2 pRq Ñ H.
If additionally ? Q P L 2 pU ; H 1 pRqq, u p0q P L 2 pRq, and v p0q P H 1 pRq, then tu, v P C 0 `r0, T s; H 1 pRq ˘, P-almost surely.

If further u p0q P H 1 pRq, i.e., X p0q P V, then X P C 0 pr0, T s; Vq, P-almost surely.

We We have the following result, which is proved in §A.2.1.

Proposition 2.6 (linearized evolution). The family of operators pL st q tě0 , with

L st : D pL st q " H 3 pRq ? ε kH 1 pRq Ñ V (2.10)
" H 1 pRq ? ε kH 1 pRq, generates an evolution family `Pst,st 1 ˘tět 1 ě0 in V with P st,st 1 LpVq ď e βpt´t 1 q , (2.16a)

where β :" f 1 pûq ´f 1 p0q W 1,8 pRq ´min ´f 1 p0q, εγ ( , (2.16b)
and we have used the norm in the ξ " x `st coordinate in the last expression.

We also write

L # Y :" ˆνB 2 ξ w `f 1 pûq w ´q ´sB ξ w ε pw ´γqq ´sB ξ q ˙" ˆνB 2 ξ `f 1 pûq ´sB ξ ´1 ε ´εγ ´sB ξ ˙Y (2.17)
for the linearized evolution in the moving frame, i.e., with respect to ξ " x `st, and call L # the frozen-wave operator. Defining the translation operator T c by (c) the point spectrum σ p pBq as the set of all λ P C such that the index indpλ id E ´Bq of λ id E ´B satisfies indpλ id E ´Bq " 0 but λ id E ´B is not invertible; (d) the essential spectrum σ ess pBq as the set of all λ P C such that λ id E ´B is not a Fredholm operator with indpλ id E ´Bq " 0. (e) for a C-Hilbert space E with inner product p¨, ¨qE the numerical range

T c Y :" Y p¨`cq for any c P R, (2.18
R E pBq :" tpBY, Y q E : Y P DpBq, Y E " 1u .
Next, we are going to list (spectral) properties of the frozen-wave operator in the statements below, which make use of a complex Hilbert space

H C :" L 2 pR; Cq ? ε kL 2 pR; Cq, (2.21a)
endowed with the inner product

pY 1 , Y 2 q H C :" Z ˆR pεw 1 w 2 `q1 q 2 q dξ, Y j :" ˆwj q j ˙, Z (2.7)
"

ˆˆR ˆε ´dû dξ ¯2 `´dv dξ ¯2˙d ξ ˙´1 .
(2.21b) As mentioned in §1.2 already, the spectral properties of L # have in fact been studied by Jones [START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF] (based on the framework developed in [START_REF] Evans | Nerve axon equations. I. Linear approximations[END_REF][START_REF] Evans | Nerve axon equations. II. Stability at rest[END_REF][START_REF] Evans | Nerve axon equations. III. Stability of the nerve impulse[END_REF][START_REF] Evans | Nerve axon equations. IV. The stable and the unstable impulse[END_REF][START_REF] Evans | Errata: "Nerve axon equations. III. Stability of the nerve impulse[END_REF][START_REF] Evans | Erratum: "Nerve axon equations. II. Stability at rest[END_REF]) with shorter proofs given by Yanagida in [START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF] and by Arioli and Koch in [2, §3] in the space of bounded uniformly continuous functions. Furthermore, in [2, §4.1], bounds on the eigenvalues for square integrable functions on the torus have been derived by a scaling of the components analogous to the one used in (2.6) or (2.21b). Here, we adapt these approaches in order to give corresponding results in H C in §A.2.2, a choice being compatible with the Hilbert-space setting of §2.1. Directly applying the deterministic results [START_REF] Arioli | Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation[END_REF][START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF][START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF] would require to adapt our stochastic framework to Banach spaces, which we briefly comment on in §5.

We emphasize that the following stability result, Proposition 2.8, in the space of square-integrable functions is (up to a scaling of the components of Y ) in more generality proved by Ghazaryan, Latushkin, and Schecter in [START_REF] Ghazaryan | Stability of traveling waves for degenerate systems of reaction diffusion equations[END_REF] and by Yurov in [START_REF] Yurov | Stability estimates for semigroups and partly parabolic reaction diffusion equations[END_REF]. These proofs mainly rely on applying the Gearhart-Prüss theorem (see e.g. [START_REF] Kapitula | Spectral and dynamical stability of nonlinear waves[END_REF]Theorem 4.1.5]) and Palmer's theorem [START_REF] Ben-Artzi | Dichotomy of systems and invertibility of linear ordinary differential operators[END_REF] and are more complicated than our relatively elementary arguments provided in §A.2.2. See also [START_REF] Rottmann-Matthes | Computation and Stability of Patterns in Hyperbolic-Parabolic Systems[END_REF] for an alternative proof using the Laplace transform by Rottmann-Matthes and [START_REF] Rottmann-Matthes | Linear stability of traveling waves in first-order hyperbolic PDEs[END_REF] by the same author, where the method has been applied to first-order hyperbolic PDEs. Proposition 2.8 (properties of the frozen-wave operator L # ). For the frozen-wave operator

L # : DpL # q :" H 2 pR; Cq ? ε kH 1 pR; Cq Ñ H C (2.21a) " L 2 pR; Cq ? ε kL 2 pR; Cq it holds (a) L # [L # | DpL # qXH ] generates a C 0 -semigroup ´P # st ¯tě0 [ ´P # st | H ¯tě0 ] in H C [H].
(b) We have P st,st 1 " T st P # spt´t 1 q | V T ´st 1 with P st,st 1 as in Proposition 2.6. (c) The spectrum σpL # q " σ ess pL # q 9

Yσ p pL # q consists of (i) essential spectrum σ ess pL # q Ď tλ P C : Re λ ď ´κu with κ :" min ´f 1 p0q, εγ (

and (ii) point spectrum σ p pL # q consisting of an isolated eigenvalue 0 of multiplicity 1 with eigenvector d X dξ such that σ p pL # qzt0u Ď tλ P C : Re λ ď λ ˚pεqu for an eigenvalue λ ˚pεq ă 0, which we will call the Jones eigenvalue [START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF] (see [START_REF] Yanagida | Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations[END_REF] for more general reaction terms f as treated here).

(d) Defining the Riesz spectral projections using Dunford calculus as

Π #,0 :" 1 2πi ‰ |λ|"r ´λ id H C ´L# ¯´1 dλ and Π # :" id H C ´Π#,0 , (2.23)
where r :" 1 2 min t|λ ˚pεq| , κu, it follows that for any ϑ ă mintκ, ´λ˚p εqu there exists C ϑ P r0, 8q independent of t P r0, 8q such that

P # st Π # LpH C q ď C ϑ e ´ϑt , (2.24)
where ¨ LpH C q denotes the operator norm of bounded linear operators

L 2 pR; Cq ? ε kL 2 pR; Cq Ñ L 2 pR; Cq ? ε kL 2 pR; Cq.
(e) The operators Π #,0 ˇˇH , Π # ˇˇH : H Ñ H are well-defined and bounded, too, with

Π #,0 LpHq ď Π #,0 LpH C q , Π # LpHq ď Π # LpH C q
, and (2.24) holds true with H C replaced by H.

Main results

For the following results, we can choose T P r0, 8q arbitrarily.

3.1. Correction of the wave velocity. In accordance with (1.6) we define the translated spectral Riesz projections

Π 0 st`ϕ :" T st`ϕ Π #,0 T ´st´ϕ , Π st`ϕ :" T st`ϕ Π # T ´st´ϕ , (3.1)
and postulate the RODE as an approximation for the noise-induced wave velocity change

9 ϕ m ptq " dϕ m dt ptq " m ´Π0 st`ϕ m ptq X m pt, ¨q, d X dξ p¨`st `ϕm ptqq ¯H , (3.2a)
where m ą 0 and

X m pt, xq :" X ϕ m pt, xq (2.12)
" Xpt, xq ´Xpx `st `ϕm ptqq " L 2 pRq ? ε kL 2 pRq, for X p0q P V we obtain the necessary condition for a minimum to be

0 " B ϕ Π 0 st`ϕ X ϕ pt, ¨q 2 H (2.12) " B ϕ Π 0 st`ϕ ´Xpt, ¨q ´Xp¨`st `ϕq ¯ 2 H (3.1) " B ϕ Π #,0 ´Xpt, ¨´st ´ϕq ´X ¯ 2 H " ´2 ´Π#,0 ´X pt, ¨´st ´ϕq ´X ¯, Π #,0 ´´B x X¯p t, ¨´st ´ϕq ¯¯H " ´ˆR B ξ diag `?ε, 1 ˘¨´Π #,0 ´X pt, ¨´st ´ϕq ´X ¯¯pξq 2 dξ loooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooon "0 ´2 ´Π#,0 ´X pt, ¨´st ´ϕq ´X ¯, d X dξ ¯H " ´2 ´Π0 st`ϕ ´Xpt, ¨q ´Xp¨`st `ϕq ¯, d X dξ p¨`st `ϕq ¯H (2.12) " ´2 ´Π0 st`ϕ X ϕ pt, ¨q, d X dξ p¨`st `ϕq ¯H ,
so that (1.5) holds true, while (1.6) or (3.2a) correspond to a relaxation of this condition with numerical parameter m.

By choosing ϕ m from (3.2a) as the velocity adaption, the SPDE (2.13) becomes

dX m pt, ¨q " ´Lst`ϕ m ptq X m pt, ¨q `Rm pt, X m pt, ¨q, ¨q ´9 ϕ m ptq d X dξ p¨`st `ϕm ptqq ¯dt `ˆσ 0 ˙dW pt, ¨q, (3.2b)
where

R m pt, Y, ¨q :" R ϕ m pt, Y, ¨q (2.15)
" ˆf pw `ûp¨`st `ϕm qq ´f pûp¨`st `ϕm qq ´f 1 pûp¨`st `ϕm qq w 0 ˙,

(3.3a) Y " ˆw q ˙. (3.3b)
The proof of the following proposition is contained in §4.1.

Proposition 3.2. Suppose ? Q P L 2 pL 2 pRq; H 1 pRqq and u p0q , v p0q P H 1 pRq. (a) P-almost surely, there exists a unique pF t q tě0 -adapted solution ϕ m P C 1 pr0, T sq to the pathwise ODE (3.2a) subject to ϕ m p0q " 0. (b) The correction 9 ϕ m to the velocity s of the traveling pulse satisfies the SODE

d 9 ϕ m ptq " ´m 9 ϕ m ptq ´1 `´Π 0 st`ϕ m ptq B x X m pt, ¨q, d X dξ p¨`st `ϕm ptqq ¯H ¯dt `σm ´Π0 st`ϕ m ptq p1, 0q t dW pt, ¨q, d X dξ p¨`st `ϕm ptqq ¯H `m ´Π0 st`ϕ m ptq R m pt, X m pt, ¨q, ¨q, d X dξ p¨`st `ϕm ptqq ¯H dt. (3.4)
3.2. Reduced stochastic dynamics and multiscale analysis. Since for small values of σ, X m and 9 ϕ m are expected to be small, we may also consider the reduced set of equations (in which by linearity σ can be scaled out)

d 9 ϕ m 0 ptq " ´m 9 ϕ m 0 ptq dt `m ´Π0 st p1, 0q t dW pt, ¨q, d X dξ p¨`stq ¯H , (3.5a
)

dX m 0 pt, ¨q " ´Lst X m 0 pt, ¨q ´9 ϕ m 0 ptq d X dξ p¨`stq ¯dt `p1, 0q t dW pt, ¨q. (3.5b)
Note that the SPDE (3.5b) describes the stochastic fluctuations around the stable fast pulse with stochastically adapted wave velocity. Due to the construction (cf. the definition of L st in (2.14)), the aforementioned SPDE arises from a multiscale argument and represents a leading-order approximation to track the fluctuations around the deterministically stable fast pulse.

The following statements establish existence of solutions to (3.5) and yield a multiscale expansion of the solution X " pũ, ṽq t to the original equation (1.1). Their proof is contained in §4.2.

Theorem 3.3. For a Q P L 2 pL 2 pRq; H 1 pRqq, X m 0 p0, ¨q " X p0q 0 P V, 9 ϕ m 0 p0q " m ´Π#,0 X p0q , d X dξ ¯H , ϕ m 0 p0q " 0,
ϕ m 0 ptq " `1 ´e´mt ˘´Π #,0 X p0q , d X dξ ¯H `ˆt 0 ´1 ´e´mpt´t 1 q ¯´Π 0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H , (3.6a)
X m 0 pt, ¨q " P st,0 X p0q 0 ´ϕm 0 ptq d X dξ p¨`stq `ˆt 0 P st,st 1 p1, 0q t dW pt 1 , ¨q, (3.6b)
P-almost surely, where `Pst,st 1 ˘tět 1 ě0 is given by Proposition 2.6.

(b) Let X be defined as in (2.12) using Proposition 2.4 and Proposition 2.5. Define the stopping times

τ q,σ :" inf ` t P r0, T s : Xpt, ¨q V ě σ 1´q ( Y tT u ˘, (3.7a)
τ m q,σ :" inf

` t P r0, T s : |ϕ m 0 ptq| ě σ ´q( Y tT u ˘, (3.7b)
where q P " 0, 1 2 ˘. Then, on min τ q,σ , τ m q,σ ( " T ( we have Xpt, ¨q ": X p¨`st `σϕ m 0 ptqq `σX m 0 pt, ¨q `σS m pt, ¨q,

where the remainder S m meets the estimate

S m pt, ¨q V ď Cσ 1´2q `1 `σ1´q ˘, P-almost surely, (3.9)
with C ă 8 being independent of σ. (c) For q P " 0, 1 2 ˘it holds P

" min τ q,σ , τ m q,σ ( " T ‰ ě 1 ´Cσ 2q Ñ 1 as σ OE 0 for a constant C ă 8.
Note that by scaling out σ beforehand, we have X p0q 0

:" σ ´1X p0q , where X p0q is the initial condition for the unscaled equations (2.4) describing the fluctuations around the traveling wave.

3.3. Immediate relaxation. In the limit as m Ñ 8 (immediate relaxation), the system (3.5) further simplifies and we expect the solution pX m 0 , ϕ m 0 q to (3.5) to converge to

ϕ 8 0 ptq :" ´Π#,0 X p0q 0 , d X dξ ¯H `ˆt 0 ´Π0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H , (3.10a)
X 8 0 pt, ¨q :" P st,0 Π # X p0q 0 `ˆt 0 P st,st 1 Π st 1 p1, 0q t dW pt 1 , ¨q, (3.10b)
where `Pst,st 1 ˘tět 1 ě0 is given by Proposition 2.6.

The proof of the following statement is contained in §4.3.

Theorem 3.4. Suppose ? Q P L 2 pL 2 pRq; H 1 pRqq and X p0q 0 P V. Let ϕ m be given as in Proposition 3.2 (a), X m :" X ϕ m be defined as in (2.12) using Proposition 2.4 and Proposition 2.5, and ϕ m 0 and X m 0 be given by Theorem 3.3. (a) We have Π 0 st X 8 0 pt, ¨q " 0 or equivalently Π st X 8 0 pt, ¨q " X 8 0 pt, ¨q for all t P r0, T s, P-almost surely. (b) For any δ ą 0, we have

E « sup tPrδ,T s |ϕ m 0 ptq ´ϕ8 0 ptq| ff Ñ 0 as m Ñ 8, (3.11a)
E « sup tPrδ,T s X m 0 pt, ¨q ´X8 0 pt, ¨q V ff Ñ 0 as m Ñ 8. (3.11b)
(c) Suppose q P " 0, 1 2 ˘, let τ q,σ be defined as in (3.7a), and τ 8 q,σ :" inf

` t P r0, T s : |ϕ 8 0 ptq| ě σ ´q( Y tT u ˘. (3.12)
On min τ q,σ , τ where C ă 8 is independent of σ. (d) For q P " 0, 1 2 ˘it holds P " min τ q,σ , τ 8 q,σ ( " T ‰ ě 1 ´Cσ 2q Ñ 1 as σ OE 0 for some C ă 8.

(e) On min τ q,σ , τ 8 q,σ ( " T ( , where q P " 0, 1 2 ˘, the function

R Q ϕ Þ Ñ Π 0 st ´Xpt, ¨q ´X p¨`st `σϕq ¯ 2 H P R
is for 0 ď t ď T fixed, P-almost surely, locally approximately minimal at ϕ " ϕ 8 0 ptq in the sense that

B ϕ Π 0 st ´Xpt, ¨q ´X p¨`st `σϕq ¯ 2 H ˇˇˇϕ "ϕ 8 0 ptq " O `σ3´2q ˘" o `σ2 ˘, (3.15a)
and

B 2 ϕ Π 0 st ´Xpt, ¨q ´Xp¨`st `σϕq ¯ 2 H ˇˇˇϕ "ϕ 8 0 ptq " σ 2 ˆ2 d X dξ
E " X 8 0 pt, ¨q 2 H ı ď 2C 2 ϑ e ´2ϑt X p0q 0 2 H `C2 ϑ 1 ´e´2ϑt ϑ Π # 2 LpHq εZ a Q 2 L 2 pL 2 pRqq (3.16)
with ϑ ă min tκ, ´λ˚p εqu, where κ and the Jones eigenvalue λ ˚pεq have been introduced in Proposition 2.8, and the constant C ϑ P r0, 8q only depends on ϑ (and the parameters γ, ε, f , and ν of the deterministic FitzHugh-Nagumo system (1.1) with σ " 0) but is independent of t, T , and σ.

The inequality (3.16) is crucial for theoretical and application purposes as it bounds the expected size of fluctuations/deviations from the deterministic pulse. It shows that the second moment of the remaining deviations from the pulse, after correcting the wave velocity, are bounded by a sum of two contributions. The first term is simply due to the initial data X p0q 0 and is exponentially decaying as in (2.24) of Proposition 2.8 (d). The second term is due to noise around the traveling wave, where once more the decay constant of (2.24) enters. From a theoretical viewpoint the multiscale estimate for the second moment can then be useful in Doob/Markov-type inequalities to control the probabilities of individual sample paths [START_REF] Antonopoulou | Motion of a droplet for the stochastic mass-conserving Allen-Cahn equation[END_REF][START_REF] Berglund | Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond[END_REF][START_REF] Gnann | Towards sample path estimates for fast-slow SPDEs[END_REF].

Note that the noise contribution is of lower order compared to the second moment of fluctuations around the corresponding deterministic traveling wave (i.e., without stochastic velocity adaptation) because we have shifted appropriately to minimize the deviations in direction of the derivative of the traveling wave (which corresponds to the eigenvector with eigenvalue 0 of the frozen-wave operator). This is made more precise in the following proposition, in which we compute the second moment of the fluctuations in direction of the derivative of the traveling wave on the event min τ q,σ , τ 8 q,σ ( " T ( where the multiscale decomposition holds true. The proof can be found in §4.3. Proposition 3.6. There exists a sequence pQ N q N PN with ? Q N P L 2 pL 2 pRq; H 1 pRqq such that for the deviations without adapting the wave velocity X :" X 0 of (2.12) (i.e., with ϕ " 0) with respect to Q N in the definition of the noise (2.8c), the second moment of deviations in direction of the traveling wave satisfies

E " ´Π0 st Xpt, ¨q, d X dξ p¨`stq ¯2 H 1 tmintτq,σ,τ 8 q,σ u"T u  " σ 2 ´Π#,0 X p0q 0 , d X dξ ¯2 H `σ2 εZ ˆt › › ›p1, 0q t ´Π#,0 ¯˚d X dξ › › › 2 H `opN 0 q ˙`opσ 2 q, where › › ›p1, 0q t `Π#,0 ˘˚d X dξ › › › 2 H
ą 0, opN 0 q can depend on t but not on σ, and opσ 2 q can depend on t and N . In particular, asymptotically the in σ leading contribution of this moment grows linearly in time.

Proofs of main results

4.1. Correction of the wave velocity. In this section, we prove Propositions 3.2.

Proof of Proposition 3.2 (a). We use the path-wise definition

F : r0, T s ˆR Þ Ñ R, pt, ϕq Þ Ñ m ´Π0 st`ϕ ´Xpt, ¨q ´Xp¨`st `ϕq ¯, d X dξ p¨`st `ϕq

¯H

and note that by employing translation invariance of integrals, we have

F pt, ϕq (3.1)
" m ´Π#,0 ´Xpt, ¨´st ´ϕq ´X ¯, d X dξ ¯H for pt, ϕq P r0, T s ˆR. Then, we can compute for pt 1 , ϕ 1 q, pt 2 , ϕ 2 q P r0, T s ˆR,

|F pt 1 , ϕ 1 q ´F pt 2 , ϕ 2 q| ď m Π #,0 LpHq Xpt 1 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 2 ´ϕ2 q H d X dξ H
and on noting that

Xpt 1 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 2 ´ϕ2 q H ď Xpt 1 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 1 ´ϕ1 q H ` Xpt 2 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 2 ´ϕ2 q H , with Xpt 1 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 1 ´ϕ1 q H (2.11) ď Xpt 1 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 1 ´ϕ1 q H ` Xp¨´ϕ 1 q ´X p¨`spt 2 ´t1 q ´ϕ1 q H " Xpt 1 , ¨q ´Xpt 2 , ¨q H ` ˆϕ1 spt 1 ´t2 q`ϕ 1 d X dξ p¨´ξqdξ H ď Xpt 1 , ¨q ´Xpt 2 , ¨q H `|s| |t 1 ´t2 | d X dξ H and Xpt 2 , ¨´st 1 ´ϕ1 q ´Xpt 2 , ¨´st 2 ´ϕ2 q H " ˆst 2 `ϕ2 st 1 `ϕ1 B ξ Xpt 2 , ¨´ξqdξ H ď ˆst 2 `ϕ2 st 1 `ϕ1 B ξ Xpt 2 , ¨´ξq H dξ ď p|s| |t 1 ´t2 | `|ϕ 1 ´ϕ2 |q sup tPr0,T s B ξ Xpt, ¨q H (2.11) ď p|s| |t 1 ´t2 | `|ϕ 1 ´ϕ2 |q ˜sup tPr0,T s B ξ Xpt, ¨q H ` d X dξ H ¸,
and applying Proposition 2.5, we obtain sup tPr0,T s B ξ Xpt, ¨q H ă 8, P-almost surely, so that F is, P-almost surely, continuous and globally Lipschitz continuous in the second component. Making use of the Picard-Lindelöf theorem finishes the proof.

Proof of Proposition 3.2 (b). We can rewrite (3.2a) according to Taking the differential yields

d 9 ϕ m ptq " m ´Π#,0 pdX m q pt, ¨´st ´ϕm ptqq , d X dξ ¯H ´ps `9 ϕ m ptqq m ´Π#,0 B ξ X m pt, ¨´st ´ϕm ptqq , d X dξ ¯H dt (3.1) " m ´Π0 st`ϕ m ptq dX m pt, ¨q , d X dξ p¨`st `ϕm ptqq ¯H dt ´ps `9 ϕ m ptqq m ´Π0 st`ϕ m ptq B ξ X m pt, ¨q , d X dξ p¨`st `ϕm ptqq ¯H dt (3.2b) " m ´Π0 st`ϕ m ptq L st`ϕ m ptq X m pt, ¨q , d X dξ p¨`st `ϕm ptqq ¯H dt `m ´Π0 st`ϕ m ptq R m pt, X m pt,
Π 0 st`ϕ m ptq `Lst`ϕ m ptq ´sB x ˘" T st`ϕ m ptq Π #,0 L # (2.23) " T st`ϕ m ptq L # Π #,0
and therefore

´Π0 st`ϕ m ptq `Lst`ϕ m ptq ´sB x ˘Xm pt, ¨q , d X dξ p¨`st `ϕm ptqq ¯H " ´L# Π #,0 X m pt, ¨´st ´ϕm ptqq , d X dξ ¯H (2.6a),(2.7) " ´L# d X dξ , d X dξ ¯H ´Π#,0 X m pt, ¨´st ´ϕm ptqq , d X dξ ¯H (2.20)
" 0, P-almost surely, so that we end up with (3.4).

4.2.

Reduced stochastic dynamics and multiscale analysis. In this section, we prove Theorem 3.3. We note that the existence and uniqueness of mild solutions for all non-autonomous linear SPDEs appearing in the next proof is guaranteed by standard results, see for instance [START_REF] Seidler | Da Prato-Zabczyk's maximal inequality revisited I[END_REF]Theorem 1.3] or the very general approach in [START_REF] Veraar | Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations[END_REF].

Proof of Theorem 3.3 (a). Since equation (3.5a) decouples from (3.5b), it forms a linear SDE for which we have a unique mild solution given by 9 ϕ m 0 ptq " me ´mt ´Π#,0 X p0q 0 , d X dξ ¯H `m ˆt 0 e ´mpt´t 1 q ´Π0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H , P-almost surely. Another integration using ϕ m 0 p0q " 0 yields

ϕ m 0 ptq " `1 ´e´mt ˘´Π #,0 X p0q 0 , d X dξ ¯H `m ˆt 0 ˆt1 0 e ´mpt 1 ´t2 q ´Π0 st 2 p1, 0q t dW pt 2 , ¨q, d X dξ p¨`st 2 q ¯H dt 1 , P-almost surely. Utilizing m ˆt 0 ˆt1 0 e ´mpt 1 ´t2 q ´Π0 st 2 p1, 0q t dW pt 2 , ¨q, d X dξ p¨`st 2 q ¯H dt 1 " ˆt 0 ˆt´t 2 0 me ´mt 3 dt 3 ´Π0 st 2 p1, 0q t dW pt 2 , ¨q, d X dξ p¨`st 2 q ¯H " ˆt 0 ´1 ´e´mpt´t 2 q ¯´Π 0 st 2 p1, 0q t dW pt 2 , ¨q, d X dξ p¨`st 2 q ¯H ,
we end up with (3.6a).

Since ϕ m 0 is already uniquely defined, we may uniquely solve (3.5b) with the mild-solution formula Since pL st q tě0 generates an evolution family `Pst,st 1 ˘tět 1 ě0 (cf. Proposition 2.6), we find the mild-solution representation S m pt, ¨q "

X m 0 pt, ¨q " P st,0 X p0q 0 ´ˆt 0 9 ϕ m 0 pt 1 qP st,st 1 d X dξ p¨`st 1 q dt 1 `ˆt 0 P st,st
ˆt 0 P st,st 1 `Sm 1 pt 1 , ¨q `Sm 2 pt 1 , ¨q `Sm 3 pt 1 , ¨q˘d t 1 , P-almost surely. (4.3)
Note that (4.3) can be rigorously justified by following the arguments detailed at the beginning of the proof of Proposition 2.5. We continue by treating the terms ´t 0 P st,st 1 S m j pt 1 , ¨q dt 1 for j P t1, 2, 3u separately.

First observe that we have the point-wise estimate ´`f 1 pû `ũ m 0 q ´f 1 pûq ´f 2 pûq ũm 0 ˘dû dξ ``f 1 pû `ũ m 0 q ´f 1 pûq ˘Bx ũm 0 ¯p1, 0q t . Now, we note that `f 1 pû `ũ m 0 q ´f 1 pûq ´f ¯.

|S m 1 pt, xq| (2.
The combination with (4.4) yields

S m 1 pt, ¨q V (2.10b) ď σ ´1 C 1 ũm 0 pt, ¨q 2 H 1 pRq ´1 ` ũm 0 pt, ¨q H 1 pRq ¯, (4.5)
where C 1 ă 8 is independent of σ.

Next, we estimate S m 2 . Notice that on one hand we have

f
where C 2 ă 8 is independent of σ.

For estimating ´t 0 P st,st

σ ˆt 0 P st,st 1 S m 3 pt 1 , ¨qdt 1 " ´ˆt 0 P st,st 1 pB t 1 ´sB x q ´X `¨`st 1 `σϕ m 0 pt 1 q ˘´X p¨`st 1 q ´d X dξ p¨`st 1 qσϕ m 0 pt 1 q ¯dt 1 " ´X p¨`st `σϕ m 0 ptqq `Xp¨`stq `d X dξ p¨`stqσϕ m 0 ptq `ˆt 0 `pB t 1 P st,st 1 q `Pst,st 1 sB x ˘´X `¨`st 1 `σϕ m 0 pt 1 q ˘´X p¨`st 1 q ´d X dξ p¨`st 1 qσϕ m 0 pt 1 q ¯dt 1
" ´X p¨`st `σϕ m 0 ptqq `Xp¨`stq `d X dξ p¨`stqσϕ m 0 ptq ´ˆt 0 P st,st 1 pL st 1 ´sB x q ´X `¨`st 1 `σϕ m 0 pt 1 q ˘´X p¨`st 1 q ´d X dξ p¨`st 1 qσϕ m 0 pt 1 q ¯dt 1 , P-almost surely, and therefore by boundedness of pL st ´sB x q| H 3 pRq ? ε kH 2 pRq : U :" H 3 pRq ? ε kH 2 pRq Ñ V (cf. (2.14)) and employing Proposition 2.6, we have

σ ˆt 0 P st,st 1 S m 3 pt 1 , ¨q dt 1 V (2.16a) ď X p¨`st `σϕ m 0 ptqq ´Xp¨`stq ´d X dξ p¨`stqσϕ m 0 ptq V `C ˆt 0 e βpt´t 1 q X `¨`st 1 `σϕ m 0 pt 1 q ˘´X p¨`st 1 q ´d X dξ p¨`st 1 qσϕ m 0 pt 1 q U dt
¯ L 2 pRq ď 1 2 d j`2 v dξ j`2 L 2 pRq
pσϕ m 0 ptqq 2 for j P t0, 1, 2u. As a result, we obtain

ˆt 0 P st,st 1 S m 3 pt 1 , ¨q dt 1 V ď σ ´1 C 3 ˆpσϕ m 0 ptqq 2 `ˆt 0 e βpt´t 1 q `σϕ m 0 pt 1 q ˘2 dt 1 ˙ ď σ ´1 C 3 e βt `β ´1 β max t 1 Pr0,ts `σϕ m 0 pt 1 q ˘2 , (4.8)
P-almost surely, where C 3 ă 8 is independent of σ.

We collect (4.5), (4.7), and (4.8), and obtain because of (2.16) of Proposition 2.6 and (4.3),

S m pt, ¨q V ď σ ´1 C 1 ˆt 0 e βpt´t 1 q
´ ũm 0 pt 1 , ¨q Proof of Theorem 3.3 (c). We fix an element ω P Ω. If for σ P p0, 1s we have t :" τ q,σ pωq ă T , then

H 1 pRq ` ũm 0 pt 1 , ¨q 2
H 1 pRq ¯dt 1 `σ´1 C 2 ˆt 0 e βpt´t 1 q σ ϕ m 0 pt 1 q ũm 0 pt 1 , ¨q H 1 pRq dt 1 `σ´1 C 3 e βt `β ´1 β max t 1 Pr0,ts `σϕ m 0 pt 1 q ˘2 ď σ ´1 C 4 max t 1 Pr0,ts ˆ´ ũm 0 pt 1 , ¨q H 1 pRq `σ ϕ m 0 pt 1 q ¯2 ` ũm 0 pt 1 , ¨q 3
H 1 pRq ˙, 3
σ 1´q (3.7a) " Xpt, ¨q V (2.11)
" Xpt, ¨q ´Xp¨`stq

V (3.8) ď
σ S m pt, ¨q V ` X p¨`st `σϕ m 0 ptqq ´Xp¨`stq `σX m 0 pt, ¨q

V (3.9) ď
Cσ 2´2q ` X p¨`st `σϕ m 0 ptqq ´Xp¨`stq `σX m 0 pt, ¨q

V ,
where C ă 8 is independent of σ P p0, 1s. This gives with help of Markov's inequality

P rτ q,σ ă T s ď P " max tPr0,T s Xpt, ¨q V ě σ 1´q  ď P " max tPr0,T s X p¨`st `σϕ m 0 ptqq ´Xp¨`stq `σX m 0 pt 0 , ¨q V ě σ 1´q `1 ´Cσ 1´q ˘ ď 2σ 2q´2 p1 ´Cσ 1´q q 2 E " max tPr0,T s X p¨`st `σϕ m 0 ptqq ´Xp¨`stq 2 V  `2σ 2q p1 ´Cσ 1´q q 2 E " max tPr0,T s X m 0 pt, ¨q 2 V  ď Cσ 2q p1 ´Cσ 1´q q 2 ˆE " max tPr0,T s |ϕ m 0 ptq| 2  `E " max tPr0,T s X m 0 pt, ¨q 2 V ˙,
where C ă 8 is independent of σ P p0, 1s X ´0, C ´1 1´q ¯. Hence, in order to conclude that P rτ q,σ ă T s ď Cσ 2q Ñ 0 as σ OE 0 for some C ă 8, it suffices to prove that

E " max tPr0,T s |ϕ m 0 ptq| 2  ă 8 and E " max tPr0,T s X m 0 pt, ¨q 2 V  ă 8.
This follows from the representations (3.6) and the Burkholder-Davis-Gundy inequality [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]Theorem 3.28] or [START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF]Lemma 7.7] to estimate the martingale. Indeed, we have

E " max tPr0,T s |ϕ m 0 ptq| 2  ď 2 ´Π#,0 X p0q 0 , d X dξ ¯H 2 `2E max tPr0,T s ˆt 0 ´1 ´e´mpt´t 1 q ¯´Π 0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H 2 (3.1) ď 2 Π #,0 2 LpHq X p0q 0 2 H d X dξ 2 H `2C 1 T εZ Π #,0 2 LpHq d X dξ 2 H a Q 2 L 2 pL 2 pRqq ă 8,
as well as

E " max tPr0,T s X m 0 pt, ¨q 2 V  ď 3 max tPr0,T s P st,0 X p0q 0 2 V `3 d X dξ 2 V E max tPr0,T s |ϕ m 0 ptq| 2 `3E max tPr0,T s ˆt 0 P st,st 1 p1, 0q t dW pt 1 , ¨q 2 V (2.16) ď 3e 2βT X p0q 0 2 V `3 d X dξ 2 V E max tPr0,T s |ϕ m 0 ptq| 2 `3C 2 e 2βT ´1 2β εZ a Q 2 L 2 pL 2 pRqq ă 8,
where C 1 , C 2 ă 8 are independent of σ and Proposition 2.6 has been employed.

In the same way, we obtain

P " τ m q,σ ă T ‰ (3.7b) ď P " max tPr0,T s |ϕ m 0 ptq| ě σ ´q ď σ 2q E " max tPr0,T s |ϕ m 0 ptq| 2  Ñ 0 as σ OE 0.
Π 0 st X 8 0 pt, ¨q (3.10b) " Π 0 st P st,0 Π # X p0q 0 `ˆt 0 Π 0 st P st,st 1 Π st 1 p1, 0q t dW pt 1 , ¨q (3.1)
"

T st Π #,0 P # st Π # X p0q 0 `ˆt 0 T st Π #,0 P # spt´t 1 q Π # T ´st 1 p1, 0q t dW pt 1 , ¨q (2.23)
"

T st P # st Π #,0 Π # looomooon (2.23) " 0 X p0q 0 `ˆt 0 T st P # spt´t 1 q Π #,0 Π # looomooon (2.23) " 0
T ´st 1 p1, 0q t dW pt 1 , ¨q " 0, P-almost surely, so that because of (2.23) and (3.1) the claim follows.

Proof of Theorem 3.4 (b). Using Theorem 3.3 (a), we obtain for the difference of ϕ m 0 and ϕ 8 0 the following form

pϕ 8 0 ´ϕm 0 q ptq (3.6a),(3.10a) " e ´mt ´Π#,0 X p0q 0 , d X dξ ¯H `ˆt 0 e ´mpt´t 1 q ´Π0 st 1 p1, 0q t dW pt 1 ,

¯H

" 0 for any δ ą 0. For the remaining term we apply the simplification

ˆt 0 e ´mpt´t 1 q ´Π0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H (2.18), (3.1)
" ˆt 0 e ´mpt´t 1 q ´Π#,0

T ´st 1 p1, 0q t dW pt 1 , ¨q, d X dξ ¯H (2.18) " ´Π#,0 p1, 0q t W pt, ¨´stq, d X dξ ¯H ´m ˆt 0 e ´mpt´t 1 q ´Π#,0 p1, 0q t W pt 1 , ¨´st 1 q, d X dξ ¯H dt 1
`s ˆt 0 e ´mpt´t 1 q ´Π#,0 p1, 0q t pB x W qpt 1 , ¨´st 1 q, d X dξ ¯H dt 1 (4.12) " e ´mt ´Π#,0 p1, 0q t W pt, ¨´stq, d X dξ ¯H `m ˆt 0 e ´mpt´t 1 q ´Π#,0 p1, 0q t `W pt, ¨´stq ´W pt 1 , ¨´st 1 q ˘, d X dξ ¯H dt 1 `s ˆt 0 e ´mpt´t 1 q ´Π#,0 p1, 0q t pB x W qpt 1 , ¨´st 1 q, d X dξ ¯H dt Proof of Theorem 3.4 (c). We introduce the auxiliary stopping time τ 8 q,σ,c :" inf

` t P r0, T s : |ϕ 8 0 ptq| ě σ ´q ´c( Y tT u ˘,
where c P p0, σ ´qs, and start by proving that we can use the multiscale decomposition (3.8) of Theorem 3.3 (b) on min τ q,σ , τ 8 q,σ,c

( " T ( X E p , P-almost surely, where E p P F is an event with P rE p s ě 1 ´p, and where p P p0, 1s is arbitrary. On min τ q,σ , τ 8 q,σ,c ( " T ( we have

ϕ 8 0 p0q `c 2 (3.10a) " ´Π#,0 X p0q 0 , d X dξ ¯H `c 2 ă σ ´q
Now, note that for any m ą 0 we have

ϕ m 0 ptq (3.6a) " `1 ´e´mt ˘´Π #,0 X p0q 0 , d X dξ ¯H `ˆt 0 ´1 ´e´mpt´t 1 q ¯´Π 0 st 1 p1, 0q t dW pt 1 , ¨q, d X dξ p¨`st 1 q ¯H (2.18),(3.1) " `1 ´e´mt ˘´Π #,0 X p0q 0 , d X dξ ¯H `ˆt 0 ´1 ´e´mpt´t 1 q ¯´Π #,0 T ´st 1 p1, 0q t dW pt 1 , ¨q, d X dξ ¯H (2.18) " `1 ´e´mt ˘´Π #,0 X p0q 0 , d X dξ ¯H ´m ˆt 0 e ´mpt´t 1 q ´Π#,0 p1, 0q t W pt 1 , ¨´st 1 q, d X dξ ¯H dt 1 `s ˆt 0 ´1 ´e´mpt´t 1 q ¯´Π #,0 p1, 0q t pB x W qpt 1 , ¨´st 1 q, d X dξ ¯H dt

"

Xpt, ¨q ´Xp¨`st `σϕ 0 ptqq ´σX 8 0 pt, ¨q

V (3.8) ď σ S m pt, ¨q V `σ X 8 0 pt, ¨q ´Xm 0 pt, ¨q V ` X p¨`st `σϕ 8 0 ptqq ´X p¨`st `σϕ m 0 ptqq V (3.9) ď Cσ 2´2q `1 `σ1´q ˘`σ X 8 0 pt, ¨q ´Xm 0 ptq V `bσ |ϕ 8 0 ptq ´ϕm 0 ptq| ? Z c 2ε dû dξ W 1,8 pRq û W 1,1 pRq `2 dv dξ W 1,8 pRq v W 1,1 pRq
Ñ Cσ 2´2q `1 `σ1´q ˘as m Ñ 8, P-almost surely, i.e., (3.14) holds true on min τ q,σ , τ 8 q,σ,c

( " T ( X E p , where C ă 8 is independent of σ, p, and c. Setting E :" Ť pPp0,1s E p , it follows P rEs " 1 and we infer that the point-wise estimate (3.14) holds true on min τ q,σ , τ 8 q,σ,c ( " T ( , too. Since min τ q,σ , τ 8 q,σ ( " T ( " ď cPp0,σ ´q s min τ q,σ , τ 8 q,σ,c ( " T ( , we infer that (3.14) is also valid on min τ q,σ , τ 8 q,σ ( " T ( .

Proof of Theorem 3.4 (d). Because of Theorem 3.3 (c), it suffices to prove P " τ 8 q,σ ă T ‰ ď Cσ 2q Ñ 0 as σ OE 0 for some C ă 8. Indeed, by Markov's inequality, we obtain

P " τ 8 q,σ ă T ‰ (3.
´Π#,0 d 2 X dξ 2 p¨`σϕq , d X dξ ¯H dϕ `2σ 3 ˆϕ8 0 ptq 0 ´d X dξ , d 2 X dξ 2 p¨`σϕq ¯H dϕ `2σ 4 ˆϕ8 0 ptq 0 ˆϕ8 0 ptq 0 ´Π#,0 d 2 X d 2 ξ p¨`σϕq , d 2 X dξ 2 `¨`σϕ 1 ˘¯H dϕ dϕ 1 ,
we can estimate

ˆϕ8 0 ptq 0 ´Π#,0 d 2 X dξ 2 p¨`σϕq , d X dξ ¯H dϕ ď σ ´q Π #,0 LpHq d 2 X dξ 2 H d X dξ H , ˆϕ8 0 ptq 0 ´d X dξ , d 2 X dξ 2 p¨`σϕq ¯H dϕ ď σ ´q d X dξ H d 2 X dξ 2 H , and
ˆϕ8 0 ptq 0 ˆϕ8 0 ptq 0 ´Π#,0 d 2 X d 2 ξ p¨`σϕq , d 2 X dξ 2 `¨`σϕ 1 ˘¯H dϕ dϕ 1 ď σ ´2q Π #,0 LpHq d 2 X
X 8 0 pt, ¨q H ď P # st Π # X p0q 0 H ` ˆt 0 P # spt´t 1 q Π # T ´st 1 p1, 0q t dW pt 1 , ¨q H (2.24) ď C ϑ e ´ϑt X p0q 0 H ` ˆt 0 P # spt´t 1 q Π # T ´st 1 p1, 0q t dW pt 1 , ¨q H ,
P-almost surely, where ϑ has been introduced in (2.22). Therefore, using Itô's isometry, the second moment can be bounded as follows

E " X 8 0 pt, ¨q 2 H ı ď 2C 2 ϑ e ´2ϑt X p0q 0 2 H `2 ˆt 0 P # spt´t 1 q Π # T ´st 1 p1, 0q t a Q 2 L 2 pL 2 pRq;Hq dt 1 .
For an orthonormal basis pe k q kPN of L 2 pRq we can write

P # spt´t 1 q Π # T ´st 1 p1, 0q t a Q 2 L 2 pL 2 pRq;Hq " 8 ÿ k"1 P # spt´t 1 q Π # T ´st 1 p1, 0q t a Qe k 2 H ď C 2 ϑ e ´2ϑpt´t 1 q Π # 2 LpHq εZ 8 ÿ k"1 a Qe k 2 L 2 pRq ď C 2 ϑ e ´2ϑpt´t 1 q Π # 2 LpHq εZ a Q 2 L 2 pL 2 pRqq
, so that eventually we arrive at (3.16).

Proof of Proposition 3.6. Suppose Q " Q N P L 2 pL 2 pRq; H 1 pRqq to be specified further below. With (2.12)-(2.15), the deviations around the traveling wave without stochastic velocity adaption (ϕ " 0) satisfy the following mild-solution formula Xpt, ¨q " σP st,0 X p0q 0 `ˆt 0 P st,st 1 R 0 pt 1 , Xpt 1 , ¨q, ¨qdt 1 `σ ˆt 0 P st,st 1 p1, 0q t dW pt 1 , ¨q, which can be justified with arguments analogous to those given at the beginning of the proof of Proposition 2.5. Thanks to Proposition 2.8 (b) and (2.18), in the moving frame this corresponds to

Xpt, ¨´stq " σP # st X p0q 0 `ˆt 0 P # spt´t 1 q R 0 pt 1 , Xpt 1 , ¨´st 1 q, ¨´st 1 qdt 1 `σ ˆt 0 P # spt´t 1 q p1, 0q t dW pt 1 , ¨´st 1 q.
We are interested in the deviations in direction of the derivative of the traveling wave which is given by the projection Π #,0 defined in (2.23). Note that by (2.20) we have Π #,0 P # st " P # st Π #,0 " Π #,0 , so that

Π #,0 Xpt, ¨´stq " σΠ #,0 X p0q 0 `ˆt 0 Π #,0 R 0 pt 1 , Xpt 1 , ¨´st 1 q, ¨´st 1 qdt 1 `σ ˆt 0 Π #,0 p1, 0q t dW pt 1 , ¨´st 1 q.
Hence,

´Π#,0 Xpt, ¨´stq, d X dξ ¯H "σ ´Π#,0 X p0q 0 , d X dξ ¯H `ˆt 0 ´Π#,0 R 0 pt 1 , Xpt 1 , ¨´st 1 q, ¨´st 1 q, d X dξ ¯H dt 1 loooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooon ":I 1 `σ ˆt 0 ´Π#,0 p1, 0q t dW pt 1 , ¨´st 1 q, d X dξ ¯H looooooooooooooooooooooooomooooooooooooooooooooooooon ":I 2 .
In order to compute E " ´Π#,0 Xpt, ¨´stq, d X dξ ¯2 H 1 tmintτq,σ,τ 8 q,σ u"T u  we develop the square and consider the terms separately. For the first term note that

E " σ 2 ´Π#,0 X p0q 0 , d X dξ ¯2 H 1 tmintτq,σ,τ 8 q,σ u"T u  " σ 2 ´Π#,0 X p0q 0 , d X dξ ¯2 H ´P `min τ q,σ , τ 8 q,σ ( ă T ˘σ2 ´Π#,0 X p0q 0 , d X dξ ¯2 H " σ 2 ´Π#,0 X p0q 0 , d X dξ ¯2 H `opσ 2 q,
since by Theorem 3.4 (d) we have P `min τ q,σ , τ 8 q,σ ( ă T ˘ď Cσ 2q for some C ă 8 depending on t and N but independent of σ. Now note that using (2.15), with the same argumentation as in the estimate of S m 1 (cf. (4.2a)) in the proof of Theorem 3.3 (b) it holds

E " I 2 1 1 tmintτq,σ,τ 8 q,σ u"T u ı ď C 2 σ 2 σ 2´4q p1 `σ1´q q 2
for a constant C ă 8 depending on t and N but independent of σ. For the third term we write again

E " I 2 2 1 tmintτq,σ,τ 8 q,σ u"T u ı " E " I 2 2 ‰ ´E " I 2 2 1 tmintτq,σ,τ 8 q,σ uăT u ı ,
we will recognize that E " I 2 2 1 tmintτq,σ,τ 8 q,σ uăT u ı " opσ 2 q depending on t and N by dominated convergence together with Theorem 3.4 (d) and the fact that σ ´2I 2 2 1 tmintτq,σ,τ 8 q,σ uăT u ď σ ´2I 2 2 , the latter is a random variable independent of σ whose expectation is finite by the computations that follow. Take an orthonormal basis pe k q kPN of L 2 pRq with e k P C 8 c pRq for each k P N (existence of such a basis follows by applying the Gram-Schmidt algorithm to a countable dense subset of L 2 pRq in C 8 c pRq). With Itô's isometry we obtain

E " I 2 2 ‰ " σ 2 ˆt 0 8 ÿ k"1 ´Π#,0 T ´st 1 p1, 0q t a Q N e k , d X dξ ¯2 H dt 1 .
Now take the sequence of Hilbert Schmidt operators pQ N q N PN with

a Q N e k " " e k : k ď N, 0 : k ą N,
and compute the with N increasing second moment of I 2 for each

N P N E " I 2 2 ‰ " σ 2 ˆt 0 N ÿ k"1 ´Π#,0 p1, 0q t e k p¨´st 1 q, d X dξ ¯2 H dt 1 (2.6a) " σ 2 ε 2 Z 2 ˆt 0 N ÿ k"1 ´ek p¨´st 1 q, ´´Π #,0 ¯˚d X dξ ¯1¯2 L 2 pRq dt 1 ,
where ´`Π #,0 ˘˚d X dξ ¯1 denotes the first component of `Π#,0 ˘˚d X dξ . Taking the limit N Ñ 8, we obtain with Parseval's identity

E " I 2 2 ‰ Ñ σ 2 ε 2 Z 2 ˆt 0 › › › ´´Π #,0 ¯˚d X dξ ¯1› › › 2 L 2 pRq dt 1 " σ 2 εZt › › ›p1, 0q t ´Π#,0 ¯˚d X dξ › › › 2 H .
By the previous estimates, the mixed terms E

" 2σ ´Π#,0 X p0q 0 , d X dξ ¯H I 1 1 tmintτq,σ,τ 8
q,σ u"T u ı and Er2I 1 I 2 1 tmintτq,σ,τ 8 q,σ u"T u s are of order opσ 2 q depending on t and N by the Cauchy-Schwarz inequality. For the third mixed term, we write again

E " 2σ ´Π#,0 X p0q 0 , d X dξ ¯H I 2 1 tmintτq,σ,τ 8 q,σ u"T u ı " E " 2σ ´Π#,0 X p0q 0 , d X dξ ¯H I 2 ı ´E " 2σ ´Π#,0 X p0q 0 , d X dξ ¯H I 2 1 tmintτq,σ,τ 8 q,σ uăT u ı " opσ 2 q
depending on t and N since the first term in the second line is equal to 0 because I 2 is a martingale (and equal to 0 for t " 0) and the second term in the second line is of order opσ 2 q by the Cauchy-Schwarz inequality in conjunction with dominated convergence as σ OE 0.

In order to ensure linear growth in time, we prove that p1, 0q t `Π#,0 ˘˚d X dξ ‰ 0 by contradiction. Assume that p1, 0q t `Π#,0 ˘˚d X dξ " 0. Then for φ P L 2 pRq we have

0 " εZ ´´´Π #,0 ¯˚d X dξ ¯1 , φ ¯L2 pRq " ´d X dξ , Π #,0 p1, 0q t φ ¯H .
Since Π #,0 is a projection on the linear subspace generated by d X dξ (see (2.23)), we obtain #,0 p1, 0q t φ " 0. Hence, for any Y P H it holds

0 " ´Y, Π #,0 p1, 0q t φ ¯H " εZ ´´´Π #,0 ¯˚Y ¯1 , φ ¯L2 pRq .
The above equality is in particular valid for a non-trivial eigenfunction Y " pw, qq t ‰ 0 of `L# ˘ẘith eigenvalue 0, that is, ´`Π #,0 ˘˚Y ¯1 " 0. Note that such an eigenfunction exists because of (2.20) or Proposition 2.8 (cii), respectively. Indeed, we have indpL # q " ind ´`L # ˘˚¯" 0 by Definition 2.7 (c) and because for any Ỹ P H C we have

0 " ´Ỹ , L # d X dξ ¯HC " ´´L # ¯˚Ỹ , d X dξ ¯HC ,
the range of `L# ˘˚is orthogonal to d X dξ , so that the dimension of the kernel of `L# ˘˚is at least 1. Interchanging the roles of L # and `L# ˘˚shows that the range of L # is orthogonal to the kernel of `L# ˘˚. Since the range of L # has codimension 1 we conclude that the dimension of the kernel of `L# ˘˚is 1.

For such Y we additionally have

´Π#,0 ¯˚Y (2.23) " ˜´1 2πi ‰ |λ|"r ´λ id H C ´´L # ¯˚¯´1 dλ ¸Y " ˜1 2πi ‰ |λ|"r ´λ id H C ´´L # ¯˚¯´1 dλ ¸Y " Y.
With ´`Π #,0 ˘˚Y ¯1 " 0 this implies immediately that w " 0. In order to determine the second component, we compute `L# ˘˚Y explicitly. Let Y j " pw j , q j q t P C 8 c pR; Cq 2 for j " 1, 2, then by (2.17) we obtain through integration by parts and regrouping terms

´L# Y 1 , Y 2 ¯HC " εZν ˆRpB 2 ξ w 1 qw 2 dξ `εZ ˆR f 1 pûqw 1 w 2 dξ ´εZ ˆR q 1 w 2 dξ
´εZs ˆRpB ξ w 1 qw 2 dξ `εZ ˆR w 1 q 2 dξ ´εZγ ˆR q 1 q 2 dξ ´Zs ˆRpB ξ q 1 qq 2 dξ " ˆY1 ,

ˆνB 2 ξ `f 1 pûq `sB ξ 1 ´ε ´εγ `sB ξ ˙Y2 ˙HC " ´Y1 , ´L# ¯˚Y 2 ¯HC .
Now ´`Π #,0 ˘˚Y ¯1 " 0 is equivalent to νB 2 ξ w `f 1 pûqw `sB ξ w `εq " 0. With w " 0 this implies q " 0, a contradiction to being non-trivial.

Conclusions & Outlook

In this work, we have shown how to derive a multiscale decomposition near a deterministically stable traveling wave for the FitzHugh-Nagumo SPDE-ODE system. This decomposition into a component along the translation invariant family and into its complement exploits the small noise and small time scale separation parameters to derive leading-order dynamics. More precisely, the stochastically adjusted wave speed is given by an SODE to account for stochastic phase dynamics along the deterministically neutral translation mode. The fluctuations in the infinitely many remaining modes are captured by an SPDE system. Locally, near the wave, these two equations can be linearized to provide a relatively explicit solution representation which approximates well the deviations in the corresponding modes. In particular, our approach does not require an analytic semigroup generated by the linearization and it applies to much wider classes of SPDE systems as well as other patterns.

Natural generalizations of our work could be to treat the case of a cylindrical Wiener process allowing for translation-invariant (white) noise or to allow for multiplicative noise. Furthermore, it would be a desirable goal to obtain optimal estimates on the relevant stopping times, where the approach breaks down which has been addressed for the FitzHugh-Nagumo system with a regularizing Laplacian in the second component in [START_REF] Hamster | Travelling waves for reaction-diffusion equations forced by translation invariant noise[END_REF] and for stochastic neural field equations in [START_REF] Maclaurin | Wandering bumps in a stochastic neural field: a variational approach[END_REF]. These could then be compared with high-accuracy numerical simulations; cf. the references in numerics in the introduction. Of course, other classes of traveling wave patterns and effects could also be tackled via multiscale decomposition, e.g., periodic wave trains, deterministically chaotic waves, stochastic pulse splitting effects induced by large deviations or propagation failure, just to name a few. Many of these effects are somewhat understood for special examples of scalar SPDEs (see [START_REF] Kuehn | Travelling waves in monostable and bistable stochastic partial differential equations[END_REF] and the references therein) but it is clear that the dynamics can be far more complicated for SPDE systems. From a technical viewpoint, we have already pointed out that applying stochastic slow manifold methods and related fast-slow sample path estimates would be natural directions for future research. Furthermore, a generalization from the Hilbert-space setting to Banach spaces appears possible as the variational approach also works if V and V ˚are Banach spaces (cf. Assumptions A.1), we do not require any orthogonality, and Riesz spectral projections are available in Banach spaces, too. Notably, related techniques in the deterministic setting, such as Lyapunov-Schmidt or center-manifold reductions work in Banach spaces as well. In summary, it seems evident that rigorous multiscale methods for pattern formation in SPDEs have already been very successful but still need additional development.

Appendix A. Proofs of auxiliary results

A.1. Existence and uniqueness of solutions using the variational approach. In this section, we prove Proposition 2.4 and Proposition 2.5. Note that the formulation (2.4) for general operators A : r0, T s ˆV ˆΩ Ñ V ˚and B : r0, T s ˆV ˆΩ Ñ L 2 pU ; Hq can be found in [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF] and existence of variational solutions has been established under the following conditions:

Assumptions A.1. The Hilbert space H is separable, V is a reflexive Banach space which is continuously and densly embedded into H, and pW U ptq, t ě 0q is a cylindrical Wiener process on a separable Hilbert space U with respect to a complete filtered probability space ´Ω, F, pF t q tPr0,T s , P with a complete and right-continuous filtration pF t q tPr0,T s . Furthermore, there exist α ą 1, β ě 0, θ ą 0, C 1 , C 2 ă 8, a positive pF t q tPr0,T s -adapted process F P L p 2 pr0, T s ˆΩ; dt ˆPq with p ě β `2, and g : V Ñ r0, 8q measurable and locally bounded, such that the following conditions (LR1)-(LR6) hold for all Y, Y 1 , Y 2 P V and pt, ωq P r0, T s ˆΩ:

(LR1) Hemicontinuity: the map R Q τ Þ Ñ V ˚xApt, Y 1 `τ Y 2 q, Y y V P R is continuous. (LR2) Local monotonicity: 2 V ˚xApt, Y 1 q ´Apt, Y 2 q, Y 1 ´Y2 y V ` Bpt, Y 1 q ´Bpt, Y 2 q 2 L 2 pU ;Hq ď pC 1 `gpY 2 qq Y 1 ´Y2 2 H . (LR3) Coercivity: 2 V ˚xApt, Y q, Y y V ` Bpt, Y q 2 L 2 pU ;Hq `θ Y α V ď F ptq `C1 Y 2 H . (LR4) Growth: Apt, Y q α α´1 V ˚ď pF ptq `C1 Y α V q ´1 ` Y β H ¯. (LR5) Bpt, Y q 2 L 2 pU ;Hq ď C 2 ´F ptq ` Y 2 H ¯. (LR6) gpY q ď C 2 p1 ` Y α V q ´1 ` Y β H ¯.
Note that conditions (LR1) and (LR3) are the same as the classical ones from [START_REF] Krylov | Stochastic evolution equations[END_REF], while conditions (LR2) and (LR4) are weaker. A main theorem presented in [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF] reads as follows:

Theorem A.2 (Liu and Röckner [START_REF] Liu | SPDE in Hilbert space with locally monotone coefficients[END_REF]). Under Assumptions A.

F ptq :" max " 1 `´2 ? ε `η4 ´1 `3 û 2 L 8 pRq ¯¯4 , εZσ a Q 2 L 2 pU q * , (A.1g) p ě 6. (A.1h)
Proof of (LR1). We have for Y " pw, qq t , Y 1 " pw 1 , q 1 q t , Y 2 " pw 2 , q 2 q t P V ,

V ˚xApt, Y 1 `τ Y 2 q, Y y V (2.8b) " εZ H ´1pRq @ νB 2 x pw 1 `τ w 2 q , w D H 1 pRq `εZ H ´1pRq xf
`3 `τ 2 1 `τ 2 2 ˘ w 2 2 L 8 pRq ¯ w 2 L 2 pRq |τ 1 ´τ2 | w L 2 pRq (2.6b) ď η 4 ˆ1 `6 û 2 L 8 pRq `6 εZ Y 1 2 V `3pτ 2 1 `τ 2 2 q εZ Y 2 2 V ˙ Y 2 H Y H |τ 1 ´τ2 | ,
where the Sobolev embedding theorem in form of w j L 8 pRq ď w j H 1 pRq has been applied.

Proof of (LR2). For proving local monotonicity, observe that for Y 1 " pw 1 , q 1 q t , Y 2 " pw 2 , q 2 q t P V ,

2 V ˚xApt, Y 1 q ´Apt, Y 2 q, Y 1 ´Y2 y V (2.8b) " ´2εZν B x pw 1 ´w2 q 2 L 2 pRq
`2εZ pf pw 1 `û p¨`stqq ´f pw 2 `û p¨`stqq , w 1 ´w2 q L 2 pRq ´2εZγ q 1 ´q2 2 L 2 pRq .

On noting that 2εZ pf pw 1 `û p¨`stqq ´f pw 2 `û p¨`stqq , w 1 ´w2 q L 2 pRq

(2.1g)

ď 2εZη 1 w 1 ´w2 2 L 2 pRq (2.6b) ď 2η 1 Y 1 ´Y2 2 H ,
and B pt, Y 1 q ´B pt, Y 2 q

(2.8c) " 0, this results in

2 V ˚xApt, Y 1 q ´Apt, Y 2 q, Y 1 ´Y2 y V ` B pt, Y 1 q ´B pt, Y 2 q 2 L 2 pU ;Hq ď 2η 1 Y 1 ´Y2 2 H (A.1d) ď C 1 Y 1 ´Y2 2
H . Note that we have verified the classical monotonicity assumption in [START_REF] Krylov | Stochastic evolution equations[END_REF], i.e., local monotonicity is not needed here.

Proof of (LR3). For proving coercivity, we note that for Y " pw, qq t P V ,

V ˚xApt, Y q, Y y V (2.8b) "
´εZν B x w 2 L 2 pRq `εZ pf pw `û p¨`stqq ´f pû p¨`stqq , wq L 2 pRq ´εZγ q 2 L 2 pRq . Using εZ pf pw `û p¨`stqq ´f pû p¨`stqq , wq L 2 pRq (2.1g)

ď εZη 1 w 2 L 2 pRq (2.6a) ď η 1 Y 2 H and Bpt, Y q 2 L 2 pU ;Hq (2.6a),(2.8c) " εZσ ? Q 2 L 2 pU q
, we obtain the estimate

V ˚xApt, Y q, Y y V ` Bpt, Y q 2 L 2 pU ;Hq ď εZσ a Q 2 L 2 pU q ´εZν B x w 2 L 2 pRq `η1 Y 2 H (2.6) " εZσ a Q 2 L 2 pU q `pη 1 `νq Y 2 H ´ν Y 2 V (A.1) ď F ptq `C1 Y 2 H ´θ Y 2 V , which yields (LR3).
Proof of (LR4). We have for Y " pw, qq t , Ỹ " p w, qq t P V ,

V ˚AApt, Y q, Ỹ E V (2.
ď εZν B x w L 2 pRq B x w L 2 pRq `εZη 4 ´1 `3 û 2 L 8 pRq `2 w 2 L 8 pRq ¯ w L 2 pRq w L 2 pRq
`εZ q L 2 pRq w L 2 pRq `εZ w L 2 pRq q L 2 pRq `εZγ q L 2 pRq q L 2 pRq

(2.6) ď ˆ Y V ˆν `4η 4 εZ Y 2 H ˙` Y H ´2? ε `εγ `η4 ´1 `3 û 2 L 8 pRq ¯¯˙ Ỹ V ,
where we have used that

w 2 L 8 pRq ď 2 w L 2 pRq dw dx L 2 pRq (2.6) ď 2 εZ Y H Y V .

This implies

Apt, Y q

α α´1 V (A.1a) " Apt, Y q 2 V ď 2 Y 2 V ˆν `4η 4 εZ Y 2 H ˙2 `2 ´2? ε `εγ `η4 ´1 `3 û 2 L 8 pRq ¯¯2 Y 2 H ď ˆ1 `´2 ? ε `εγ `η4 ´1 `3 û 2 L 8 pRq ¯¯4 `4 max " ν 2 , 16η 2 4 ε 2 Z 2 * Y 2 V ˙´1 ` Y 4 H (A.1) ď pF ptq `C1 Y α V q ´1 ` Y β H ¯,
i.e., (LR4). Note that here we have not obtained the classical growth condition in [START_REF] Krylov | Stochastic evolution equations[END_REF].

Proof of (LR5). We have Bpt, Y q 2 L 2 pU ;Hq (2.6a),(2.8c)

" εZσ ? Q 2 L 2 pU q ď C 2 ´F ptq ` Y 2 H ¯.
Proof of (LR6). This trivially holds because g " 0.

Proof of Proposition 2.5. Denote by X " pu, vq t the solution from Proposition 2.4. We first apply [66, Proposition G.0.5 (i)] and verify the conditions there to conclude that X meets a mild-solution representation. Next, we prove additional regularity provided ? Q P L 2 pU ; H 1 pRqq, u p0q P L 2 pRq, and v p0q P H 1 pRq. From the first component of (2.4) or (2.9) we derive the mild-solution formula tupt, ¨q " ? 4πνt denotes the heat kernel generated by νB 2

ˆt 0 K t´t 1 ˚`f `upt 1 , ¨q `ûp¨`st 1 q ˘´f `ûp¨`st 1 q ˘´vpt 1 , ¨q˘t 1 dt 1 `ˆt 0 K t´t 1 ˚upt 1 , ¨q dt 1 `ˆt 0 K t´t 1 ˚t1 dW pt 1 ,
x and ˚the convolution on the real line. For the second component of (2.4) or (2.9) we get analogously vpt, ¨q " e ´εγt v p0q `ε ˆt 0 e ´εγpt´t 1 q upt 1 , ¨q dt 1 , P-almost surely.

Differentiation in space yields tpB x uqpt, ¨q " ˆt 0 K t´t 1 ˚´f 1 pupt 1 , ¨q `ûp¨`st 1 qq ´Bx upt 1 , ¨q `dû dξ p¨`st 1 q ¯¯t 1 dt

1 ´ˆt 0 K t´t 1 ˚´f 1 `ûp¨`st 1 q ˘dû dξ p¨`st 1 q `Bx vpt 1 , ¨q¯t 1 dt 1 `ˆt 0 K t´t 1 ˚Bx upt 1 , ¨q dt 1 `ˆt 0 K t´t 1 ˚`t 1 dpB x W qpt 1 , ¨q˘, (A.3)
P-almost surely, and B x vpt, ¨q " e ´εγt ´Bx v p0q ¯`ε ˆt 0 e ´εγpt´t 1 q pB x uqpt 1 , ¨q dt 1 , P-almost surely. (A.4)

We estimate the three lines on the right-hand side of (A.3) separately:

Using K t´t 1 L 1 pRq " 1, we obtain for the first term on the right-hand side of (A.3) with Young's convolution inequality

ˆt 0 K t´t 1 ˚´f 1 pupt 1 , ¨q `ûp¨`st 1 qq ´Bx upt 1 , ¨q `dû dξ p¨`st 1 q ¯¯t 1 dt 1 L 2 pRq (2.1i) ď η 3 ˆt 0 ´1 `2 û 2 L 8 pRq `2 upt 1 , ¨q 2 H 1 pRq ¯ˆ B x upt 1 , ¨q L 2 pRq ` dû dξ L 2 pRq ˙t1 dt 1 ď η 3 T 2 ´1 `2 û 2 L 8 pRq ¯˜T dû dξ L 2 pRq `2T 1
2 ? 3 B x u L 2 pr0,T sˆRq 2η 3 T dû dξ L 2 pRq u 2 L 2 pr0,T s;H 1 pRqq `2η 3 T 3 2 ? 3 u 2 C 0 pr0,T s;L 2 pRqq B x u L 2 pr0,T sˆRq `2η 3 ˆt 0 B x upt 1 , ¨q 2 L 2 pRq t 1 B x upt 1 , ¨q L 2 pRq dt 1 ,
where the Sobolev embedding on the real line in form of upt 1 , ¨q L 8 pRq ď upt 1 , ¨q H 1 pRq has been applied.

For the second line of (A.3), we obtain similarly

ˆt 0 K t´t 1 ˚´f 1 `ûp¨`st 1 q ˘dû dξ p¨`st 1 q `Bx vpt 1 , ¨q¯t 1 dt 1 L 2 pRq (2.1i) ď η 3 T 2 2 ˆ1 ` dû dξ 2 L 8 pRq ˙ dû dξ L 2 pRq `T 3 2 ? 3 B x v L 2 pr0,T sˆRq ,
where we may use

B x v L 2 pr0,T sˆRq (A.4) ď 1 ? 2εγ B x v p0q L 2 pRq `1 γ B x u L 2 pr0,T sˆRq , P-almost surely. (A.5)
Finally, the first integral on third line of (A.3) yields

ˆt 0 K t´t 1 ˚Bx upt 1 , ¨q dt 1 L 2 pRq ď ? T B x u L 2 pr0,T sˆRq ,
while the second integral gives with help of Itô's isometry

E « ˆt 0 K t´t 1 ˚`t 1 dpB x W qpt 1 , ¨q˘ 2 L 2 pRq ff ď ˆt 0 pt 1 q 2 pK t´t 1 ˚q B x a Q 2 L 2 pU q dt 1 ď T 3 3 a Q 2 L 2 pU ;H 1 pRqq .
Using X " pu, vq t P L 2 pr0, T s; V q X C 0 pr0, T s; For proving continuity, observe that for T ě t 2 ě t 1 ą 0 we have

t 2 B x upt 2 , ¨q ´t1 B x upt 1 , ¨q " pt 2 ´t1 q B x upt 2 , ¨q `t1 pB x upt 2 , ¨q ´Bx upt 1 , ¨qq .
Then, it follows pt 2 ´t1 q B x upt 2 , ¨q Ñ 0 as t 2 Ñ t 1 in L 2 pRq by the reasoning before if one translates in time and uses uniqueness. For the remaining term, we derive once more from the first component of (2.4) or (2.9) A.2. Linearization around the traveling wave. In this section, we give the proofs of Proposition 2.6 and Proposition 2.8. A.2.1. Fixed frame. We focus on investigating the linearized evolution generated by the family of operators pL st q tě0 defined in (2.14).

B x upt 2 , ¨q ´Bx upt 1 , ¨q " K t 2 ´t1 ˚pB x uqpt 1 , ¨q ´Bx upt 1 , ¨q `ˆt 2 t 1 K t 2 ´t1 ˚´f 1 `upt 1 , ¨q `ûp¨`st 1 q ˘´B x upt 1 , ¨q `dû dξ p¨`st 1 q ¯¯dt 1 ´ˆt 2 t 1 K t 2 ´t1 ˚´f 1 `ûp¨`st 1 q ˘dû dξ p¨`st 1 q ´Bx vpt 1 , ¨q¯d t 1 `ˆt 2 t 1 K t 2 ´t1 ˚dpB x W qpt 1 ,
Proof of Proposition 2.6. Note that in view of (2.14) we may write

L st " L ˘8 `Rst with $ ' ' & ' ' % L ˘8 :" ˆνB 2 x `f 1 p0q ´1 ε ´εγ ˙, R st :" ˆf 1 pûp¨`stqq ´f 1 p0q 0 0 0 ˙. (A.7)
For Y " pw, qq t P pC 8 c pRqq 2 we have

pL ˘8Y, Y q V " εZ ´ν d 2 w dx 2 `f 1 p0qw ´q, w ¯H1 pRq `εZ pw ´γq, qq H 1 pRq " εZ ˆ´ν d 2 w dx 2 2 L 2 pRq ´`ν ´f 1 p0q ˘ dw dx 2 L 2 pRq `f 1 p0q w 2 L 2 pRq εZγ q 2 H 1 pRq ď ´min ´f 1 p0q, εγ ( Y 2 V ´εZν d 2 w dx 2 2 L 2 pRq
.

By density of pC 8 c pRqq 2 in DpL ˘8q " H 3 pRq ? ε kH 1 pRq, we infer that L ˘8 `κ id V with κ :" min ´f 1 p0q, εγ

( (2.1d) ě 0 is dissipative.
In order to prove that L ˘8 `pκ ´1q id V : D pL ˘8q Ñ V is a bijection, we define M pY 1 , Y 2 q :" ´εZν ´dw 1 dx , dw 2 dx ¯H1 pRq `εZ ``f 1 p0q `κ ´1˘w 1 ´q1 , w 2 ˘H1 pRq `εZ pw 1 `pκ ´1 ´γqq 1 , q 2 q H 1 pRq for Y j :" pw j , w j q t P H 2 pRq ? ε kH 1 pRq and we recognize that M : `H2 pRq ? ε kH 1 pRq ˘2 Ñ V is bilinear, continuous, and ´M is coercive, so that by the Lax-Milgram theorem for any 2 P V there exists Y 1 P H 2 pRq ? ε kH 1 pRq such that M ´Y1 , Ỹ ¯" ´Y2 , Ỹ ¯V for all Ỹ P H 2 pRq ? ε kH 1 pRq. This implies L ˘8Y 1 `pκ ´1qY LpVq ď e ´κt . Now, since û and dû dξ are bounded and f P C 1 pRq, the family pR st q tě0 is uniformly bounded Ñ V with R st LpVq ď f 1 pûq ´f 1 p0q W 1,8 pRq ă 8. By [74, Chapter 5, Theorem 2.3] the family pL st q tě0 of linear operators generates an evolution family `Pst,st 1 ˘tět 1 ě0 of bounded linear operators V Ñ V meeting estimate (2.16), i.e., P st,st 1 LpVq ď e βpt´t 1 q with β :" f 1 pûq ´f " T ´st pB t ´Lst q P st,st 1 T st 1 " 0, so that with In what follows, we provide a more detailed spectral analysis of L # , leading to the proofs of Proposition 2.8 (c) and (d).

T ´st P st,st 1 T st 1 ˇˇt"t 1 " T ´st 1 id V T st 1 " T ´st 1 T st 1 | V " id V the claim
Lemma A. " L # ˘8 `R# : H C Ą DpL # q " H 2 pR; Cq ? ε kH 1 pR; Cq Ñ H C is a relatively compact perturbation of L # ˘8 : H C Ą DpL # q Ñ H C . Proof. Though the proof is standard, for the sake of a self-contained presentations we provide all necessary details. Suppose that pY n q n P `H1 pR; Cq ? ε kL 2 pR; Cq ˘N with Y n " pw n , q n q t and sup nPN w n H 1 pR;Cq ă 8. Because of

R # Y n (A.8b) "
ˆpf 1 pûq ´f 1 p0qq w n 0 ˙, we first prove that ppf 1 pûq ´f 1 p0qq w n q n has a convergent subsequence in L 2 pR; Cq. Therefore, note that f 2 pûpξqq dû dξ and |f 1 pûpξqq ´f 1 p0q| ď sup wPr0,ûpξqsYrûpξq,0s |f 2 pwq| |ûpξq| decay exponentially as |ξ| Ñ ˘8. Now, setting y :" arctan ξ and φ n pyq :" a 1 `ξ2 pf 1 pûpξqq ´f 1 p0qq w n pξq, we may compute that dy " for an n-independent constant C ă 8. By the Rellich-Kondrachov theorem, pφ n q n has a convergent subsequence in L 2 ``´π 2 , π 2 ˘; C ˘and because φ n ´φm L 2 pp´π 2 , π 2 q;Cq " `f 1 pûpξqq ´f 1 p0q ˘pw n ´wm q L 2 pR;Cq , we infer that ppf 1 pûpξqq ´f 1 p0qq w n q n has a convergent subsequence in L 2 pR; Cq.

Suppose that pY n q n P `DpL # q ˘N meets sup nPN " L 2 pR; Cq ? ε kL 2 pR; Cq are closed (which follows by an interpolation argument as in the proof of Lemma A.5) and L # is a compact pertubation of # by Lemma A.5.

Proof of Proposition 2.8 (cii). It suffices to prove that for λ P σ p pL # q any corresponding eigenvector Y " pw, qq t P DpL # q is bounded and has infinitely many bounded derivatives to conclude that in H C there are no additional eigenvalues compared to the ones obtained by Yanagida [91, §5.1] (cf. [START_REF] Jones | Stability of the travelling wave solution of the FitzHugh-Nagumo system[END_REF][START_REF] Antonopoulou | Motion of a droplet for the stochastic mass-conserving Allen-Cahn equation[END_REF]Theorem] in the case of the cubic polynomial without cut off).

Indeed, for Y as above satisfying L # Y " λY , we have

ν d 2 w dξ 2
(2.17)

" λw ´f 1 pûqw `s dw dξ `q loooooooooooooomoooooooooooooon PH 1 pR;Cq , s dq dξ (2.17) " ´λq `εw ´εγq looooooooomooooooooon PH 1 pR;Cq , which immediately yields w P H 3 pR; Cq and q P H 2 pR; Cq. Inductively, we obtain that w, q P H k pR; Cq for any k P N, giving smoothness and boundedness of w, q, and all derivatives.

With the stability analysis in [91, §5.1] we conclude that, except for the simple eigenvalue 0, all eigenvalues are to the left of the imaginary axis with real part bounded by λ ˚pεq. We further remark that in (2.20) it has already been noted that d X dξ is an eigenvector to the eigenvalue 0.

In what follows, we need a regularizing effect of the semigroup `P ˘8 st ˘tě0 of Lemma A.3 in the first component.

Lemma A.6. For Y p0q " `wp0q , q p0q ˘t P H C we write Y ˘8pt, ¨q :" pw ˘8pt, ¨q, q ˘8pt, ¨qq t :" P ˘8 st Y p0q . Then, it holds w ˘8, q ˘8, ? tB ξ w ˘8 From the equation λX λ ´L# X λ " Y, where |λ| " r, and because the coefficients of L # are real (cf. (2.17)), it follows that λ X λ ´L# X λ " Y.

Because r ą 0 us chosen sufficiently small we have λ P ρpL # q (cf. Definition 2.7 (a)) and it follows due to uniqueness X λ " X λ . Hence,

Π #,0 Y " r 2π

"

  Xpt, xq `Xpx `stq ´Xpx `st `ϕm ptqq. Remark 3.1. Indeed, from (1.4) with the choices (3.1) and H (2.5)

  we have: (a) The system (3.5) has a unique mild solution [82, Definition 1.1] given by

2 H`

 2 O `σ1´2q ˘˙ą 0 (3.15b) as σ OE 0, P-almost surely. Note that Theorem 3.4 (a) is a generalization of the orthogonality property in [55, §3.3]. Theorem 3.4 (c) and Theorem 3.4 (d) yield a multiscale expansion in the immediate-relaxation limit. Finally, Theorem 3.4 (e) relates to our original motivation in (1.4) to obtain the correction of the translation of the wave through a minimization argument. The following result is proved in in §4.3. Proposition 3.5. Suppose that ? Q P L 2 pL 2 pRq; L 2 pRqq and X p0q 0 P H. Then the second moment of X 8 0 satisfies the bound

  9 ϕ m ptq (3.2a) " m ´Π0 st`ϕ m ptq X m pt, ¨q, d X dξ p¨`st `ϕm ptqq ¯H (3.1) " m ´Π#,0 X m pt, ¨´st ´ϕm ptqq , d X dξ ¯H , P-almost surely.

4. 3 . 5 .

 35 Immediate relaxation. Here, we give the proofs of Theorem 3.4 and Proposition 3.Proof of Theorem 3.4 (a). Utilizing Proposition 2.8 (b) and (d) yields

  follows by uniqueness of the evolution family.Note that (A.11) just provides a rough estimate on the action of the semigroup ´P # st ¯tě0

  3.Lemma A.4 can be lifted to obtain the essential spectrum of L # using the following compactness argument:Lemma A.5 (compactness). The operator (cf. (A.8b)) R # : H C Ą H 1 pR; Cq ? ε kL 2 pR; Cq Ñ H C is compact. In particular, L # (A.8)

1 .

 1 Introduction 1.1. The stochastic FitzHugh-Nagumo equations. We consider the stochastic FitzHugh-Nagumo equations dũpt, xq " `νB 2 x ũpt, xq `f pũpt, xqq ´ṽpt, xq ˘dt `σ dW pt, xq for pt, xq P R `ˆR, (1.1a) dṽpt, xq " ε pũpt, xq ´γṽpt, xqq dt for pt, xq P R `ˆR,

  , we use the following Hilbert space setting. Suppose that L 2 pRq is the standard L 2 -space of R-valued functions on the real line,H 1 pRq :" u P L 2 pRq : B x u P L 2 pRq ( ,where B x u denotes the distributional derivative, and H ´1pRq denotes the topological dual of H 1 pRq relative to L 2 pRq. The Laplacian B 2x u for u P H 1 pRq can be defined via its bilinear form as

	2 x upt, xq `f pupt, xq `ûpx `stqq ´f pûpx `stqq ´vpt, xq ˘dt `σ dW pt, xq, dvpt, xq " ε pupt, xq ´γvpt, xqq dt. In order to analyze (2.2)H 1 pRq @ w, B 2 x u D H ´1pRq :" ´ˆR pB x wq pB x uq dx.	(2.2a) (2.2b) (2.3)
	Then, we may recast the system (2.2) in form of the abstract stochastic evolution equation
	dXpt, ¨q " Apt, Xpt, ¨qq dt `Bpt, Xpt, ¨qq dW U pt, ¨q

  will give the proof of Proposition 2.4 and Proposition 2.5 in §A.1.

	2.2. Linearization around the traveling wave. We recall the global Assumptions 2.1 and
	write X :" pû, vq t for the deterministically stable fast traveling pulse. Then we define Xpt, xq :" pũpt, xq, ṽpt, xqq t " Xpt, xq `Xpx `stq, and in line with (1.3) set	(2.11)
	X ϕ pt, xq :" pu ϕ pt, xq, v ϕ pt, xqq t :" Xpt, xq ´Xpx `st `ϕq " Xpt, xq `Xpx `stq ´Xpx `st `ϕq, where ϕ " ϕptq is yet to be determined. We split the stochastic evolution equation (2.4) around (2.12) the traveling-wave profile X into a linear and nonlinear part using (1.2). This yields
	dX ϕ pt, ¨q " ´Lst`ϕptq X ϕ pt, ¨q `Rϕptq pt, X ϕ pt, ¨q, ¨q ´9 ϕptq d dξ p¨`st `ϕptqq ¯dt X	`ˆσ 0	˙dW pt,	¨q
									(2.13)
	where and	L st`ϕ Y :"	ˆνB 2 x w `f 1 pûp¨`st `ϕqq w ε pw ´γqq	´q	˙, Y :"	ˆw q	˙,	(2.14)
	R ϕ pt, Y, ¨q :"	ˆf pw `ûp¨`st `ϕqq ´f pûp¨`st `ϕqq ´f 1 pûp¨`st `ϕqq w 0	˙.	(2.15)

  p¨`st `σϕ m 0 ptqqq ´f 1 pû p¨`stqq û p¨`st `σϕ m

						0	0 ptqq ˙dt
						`σ 9 ϕ m 0 ptq d dξ p¨`st `σϕ m X 0 ptqq dt
	and			dX m 0 pt,	¨q (3.5b) " L st X m 0 pt, ¨q dt ´9 ϕ m 0 ptq d dξ p¨`stq dt X	`ˆ1 0 ˙dW pt, ¨q,
	we get with
	that					ũm 0 pt, ¨q :" upt, ¨q `û p¨`stq ´û p¨`st `σϕ m 0 ptqq	(4.1)
	where				dS m pt, ¨q ´Lst S m pt, ¨q dt " S m 1 pt, ¨q `Sm 2 pt, ¨q `Sm 3 pt, ¨q,
	S m 1 pt, ¨q :" σ	´1 ˆf pû p¨`st `σϕ m 0 ptqq `ũ m 0 pt, ¨qq ´f pû p¨`st `σϕ m 0 0 ptqqq σ´1
						ˆf 1 pû p¨`st `σϕ m 0 ptqqq ũm 0 pt, 0	˙, ¨q	(4.2a)
	S m 2 pt, ¨q :" σ	´1 `f 1 pû p¨`st `σϕ m 0 ptqqq ´f 1 pû p¨`stqq ˘ˆũ m 0 pt, 0	˙, ¨q	(4.2b)
	S m 3 pt, ¨q :" 9 ϕ m 0 ptq	´d dξ p¨`stq X	´d dξ p¨`st `σϕ m X 0 ptqq ¯.	(4.2c)
						(2.18) " T st P # spt´t 1 q	d dξ X	(2.18),(2.20) "	d dξ p¨`stq, X
	so that
	ˆt 0	9 ϕ m 0 pt 1 qP st,st 1 d dξ p¨`st 1 qdt 1 " X	ˆt 0	9 ϕ m 0 pt 1 q dt 1 d dξ p¨`stq " ϕ m X 0 ptq d dξ p¨`stq X
	and we arrive at (3.6b).
	which ensures smallness of Xpt, ¨q and ϕ m 0 ptq.	q,σ	(	" T	( ,
	We consider the remainder
	σS m pt,	¨q (3.8) " (2.11) "	Xpt, ¨q ´X p¨`st `σϕ m 0 ptqq ´σX m 0 pt, Xpt, ¨q `X p¨`stq ´X p¨`st `σϕ m 0 ptqq ´σX m ¨q 0 pt, ¨q,
	so that with
	dXpt,	¨q (2.13),(2.15) "	L st Xpt, ¨q dt	`ˆσ 0	˙dW pt, ¨q,
						`ˆf pupt, ¨q `ûp¨`stqq ´f pûp¨`stqq ´f 1 pûp¨`stqq upt, 0	¨q ˙dt
	d X p¨`stq	(1.2),(2.14) "	L st X p¨`stq dt	0 `ˆf pû p¨`stqq ´f 1 pû p¨`stqq û p¨`stq ˙dt,
	as well as
	d X p¨`st `σϕ m 0 ptqq

1 p1, 0q t dW pt 1 , ¨q, P-almost surely. With help of Proposition 2.8 (a) and (b) it follows P st,st 1 d X dξ p¨`st 1 q Proof of Theorem 3.3 (b). In what follows, we restrict ourselves to paths on min τ q,σ , τ m (1.2),(2.14)

"

L st X p¨`st `σϕ m 0 ptqq dt `ˆf pû

  ´1 ` û L 8 pRq ` ũm 0 pt, ¨q H 1 pRq ¯ ũm 0 pt, ¨q H 1 pRq |ũ m 0 pt, xq| , where the Sobolev embedding in form of ũm 0 pt, ¨q L 8 pRq ď ũm 0 pt, ¨q H 1 pRq has been applied. ´1 ` û L 8 pRq ` ũm 0 pt, ¨q H 1 pRq

	1h),(4.2a) ď ď σ ´1η 2 This already yields σ ´1η 2 p1 `|û px `st `σϕ m 0 ptqq| `|ũ m 0 pt, xq|q |ũ m 0 pt, xq| 2
	S m 1 pt, ¨q H	(2.6a) ď σ	´1?	εZ η 2 ¯	ũm 0 pt, ¨q H 1 pRq	ũm 0 pt, ¨q L 2 pRq . (4.4)
	On the other hand, a direct computation gives (suppressing the arguments in S m 1 " S m 1 pt, xq, u m 0 " u m 0 pt, xq, û " û px `st `σϕ m 0 ptqq, and ũm 0 " ũm 0 pt, xq)
	σB x S m 1	(4.2a) "	´f 1 pû `ũ m 0 q ´Bx	ũm 0	`dû dξ ¯´f 1 pûq dû dξ	´f 2 pûq dû dξ	ũm 0	´f 1 pûq B x	ũm 0 ¯p1, 0q t
		"

  pt, ¨q H 1 pRq |B x ũm 0 pt, xq| ´1 `2 û L 8 pRq ` ũm 0 pt, ¨q H 1 pRq ¯, where ũm 0 pt, ¨q L 8 pRq ď ũm 0 pt, ¨q H 1 pRq has been utilized once again. Hence, B x S m 1 pt, ¨q H pt, ¨q H 1 pRq B x ũm 0 pt, ¨q L 2 pRq ´1 `2 û L 8 pRq ` ũm 0 pt, ¨q H 1 pRq

	p1 `2 |û px `st `σϕ m 0 ptqq| `|ũ m 0 pt, xq|q η 6 ũm 0 (2.6a) ď ď σ ´1? εZ η 5 ũm 0 pt, ¨q H 1 pRq dû dξ L 8 pRq ũm 0 pt, ¨q L 2 pRq `σ´1 ? εZ η 6 ũm 0
	0	˘dû dξ	(2.1k) ď	η 5 pũ m 0 pt, xqq 2 dû dξ px `st `σϕ m 0 ptqq
	`f 1 pû `ũ m 0 q ´f 1 pûq ˘Bx	ũm 0	ď (2.1l) ď	η 5 η 6 |ũ m ũm 0 pt, ¨q H 1 pRq 0 pt, xq| |B x ũm 0 pt, xq| dû dξ L 8 pRq	|ũ m 0 pt, xq| ,

2 pûq ũm

  1 pû px `st `σϕ m 0 ptqqq ´f 1 pûpx `stqq

	(2.6a) ď and the combination with (4.6) yields ? εZ η 7 dû dξ 2 L 8 pRq `?εZ f 2 pûq L 8 pRq σ |ϕ m 0 ptq| ũm 0 pt, ¨q L 2 pRq d 2 û dξ 2 σ |ϕ m 0 ptq| ũm 0 pt, ¨q L 2 pRq L 8 pRq `?εZ η 6 dû ´1 `2 û L 8 pRq ¯σ |ϕ m 0 ptq| B x ũm 0 pt, ¨q L 2 pRq dξ L 8 pRq
					S m 2 pt, ¨q V	(2.10b) ď C 2 |ϕ m 0 ptq| ũm 0 pt, ¨q H 1 pRq ,
								(2.1l) ď η 6 |û px `st `σϕ m 0 ptqq ´ûpx `stq| p1 `|û px `st `σϕ m 0 ptqq| `|ûpx `stq|q ď η 6 dû dξ L 8 pRq ´1 `2 û L 8 pRq ¯σ |ϕ m 0 ptq| ,
	so that
	S m 2 pt, ¨q H	(2.6a),(4.2b) ď	?	εZ η 6	dû dξ L 8 pRq	´1 `2 û L 8 pRq ¯|ϕ m 0 ptq| ũm 0 pt, ¨q L 2 pRq .	(4.6)
	On the other hand, for the derivative we get
	B x " "`f 1 pûpx `st `σϕ m 0 ptqqq ´f 1 pûpx `stqq ˘ũ m 0 pt, xq ‰ " f 2 pû px `st `σϕ m 0 ptqqq dû dξ px `st `σϕ m 0 ptqq ´f 2 pûpx `stqq dû dξ px `stq ı `"f 1 pû px `st `σϕ m 0 ptqqq ´f 1 pûpx `stqq ‰ B x ũm 0 pt, xq " " f 2 pû px `st `σϕ m 0 ptqqq ´f 2 pûpx `stqq ‰ dû dξ px `st `σϕ m 0 ptqq ũm 0 pt, xq `f 2 pûpx `stqq " dû ı dξ px `st `σϕ m 0 ptqq ´dû dξ px `stq ũm 0 pt, xq `"f 1 pûpx `st `σϕ m 0 ptqqq ´f 1 pûpx `stqq ‰ B x ũm 0 pt, xq,	ũm 0 pt, xq
	so that
	B x (2.1l),(2.1m) "`f 1 pûpx `st `σϕ m 0 ptqqq ´f 1 pûpx `stqq ˘ũ m 0 pt, xq ‰ ď η 7 |ûpx `st `σϕ m 0 ptqq ´ûpx `stq| dû dξ px `st `σϕ m 0 ptqq |ũ m 0 pt, xq|
	ď	` f 2 pûpx `stqq dû dξ px `st `σϕ m 0 ptqq ´dû dξ px `stq |ũ m 0 pt, xq| `η6 |ûpx `st `σϕ m 0 ptqq ´ûpx `stq| p1 `|ûpx `st `σϕ m 0 ptqq| `|ûpx `stq|q |B x ũm 0 pt, xq| η 7 dû dξ 2 L 8 pRq σ |ϕ m 0 ptq| |ũ m 0 pt, xq| ` f 2 pûq L 8 pRq d 2 û dξ 2 L 8 pRq σ |ϕ m 0 ptq| |ũ m 0 pt, xq|
		`η6	dû dξ L 8 pRq
								H

´1 `2 û L 8 pRq ¯σ |ϕ m 0 ptq| |B x ũm 0 pt, xq| This gives B x "`f 1 pûp¨`st `σϕ m 0 ptqqq ´f 1 pûp¨`stqq ˘ũ m 0 pt, ¨q‰

  1 S m 3 pt 1 , ¨q dt 1

		V	, observe that
	pB t ´sB x q ´X px `st `σϕ m 0 ptqq ´Xpx `stq	´d dξ px `stqσϕ m X 0 ptq "
	σ 9 ϕ m 0 ptq	´d dξ px `st `σϕ m X 0 ptqq	´d dξ px `stq X ¯(4.2c) " ´σS m 3 pt, ¨q.
	Using integration by parts, this gives

  ´Π#,0 p1, 0q t pW pt, ¨´stq ´W pt 1 , ¨´st 1 qq , d

	so that we may conclude
										X	¯H
					M :" sup t,t 1 Pr0,T s	|t ´t1 | α	dξ	ă 8,
	P-almost surely, i.e.,
		m	ˆt 0	e ´mpt´t 1 q ´Π#,0 p1, 0q t `W pt, ¨´stq ´W pt 1 , ¨´st 1 q ˘, d dξ ¯H dt 1 X
							ˆmT
			ď m ´αM	0	e ´τ τ α dτ ď m ´αM Γp1 `αq Ñ 0 as m Ñ 8, P-almost surely.
	By continuity, furthermore
	sup tPrδ,T s (3.1) e ´mt ´Π0 st p1, 0q t W pt, ¨q, d dξ p¨`stq ¯H X ď e ´mδ Π #,0 LpHq ? εZ sup tPrδ,T s W pt, ¨q L 2 pRq	d dξ H X	Ñ 0 as m Ñ 8, P-almost surely,
	and
	sup tPrδ,T s	s	ˆt 0	e ´mpt´t 1 LpHq	?	εZ sup tPr0,T s	pB x W qpt, ¨q L 2 pRq	d dξ H X	Ñ 0 as m Ñ 8, P-almost surely.
	As a result, we infer that
							sup tPrδ,T s	|pϕ 8 0	´ϕm 0 q ptq| Ñ 0 as m Ñ 8, P-almost surely.	(4.13)
	From (4.10) and (4.12) we further deduce that sup tPr0,T s |pϕ 8 0 g :" Π #,0 LpHq X p0q 0 H d X dξ H ` Π #,0 LpHq ? εZ sup ´ϕm 0 q ptq| ď g, where d W pt, ¨q L 2 pRq dξ H X tPr0,T s `|s| T Π #,0 LpHq ? εZ sup tPr0,T s pB x W qpt, ¨q L 2 pRq d X dξ H .
	With help of the Burkholder-Davis-Gundy inequality [23, Lemma 7.7], we infer that
							E sup tPr0,T s	W pt, ¨q H 1 pRq ď C	?	T	a Q
										continuous with
	2 , so that in particular exponent α ă 1 sup t,t 1 Pr0,T s ´Π#,0 p1, 0q t pW pt, ¨´stq ´W pt 1 , ¨´stqq , d dξ X |t ´t1 | α (2.6a) ď Π #,0 LpHq d X dξ H ? εZ sup t,t 1 Pr0,T s W pt, ¨q ´W pt 1 , ¨q L 2 pRq ¯H |t ´t1 | α	ă 8, P-almost surely.
	Furthermore,
						dξ ´Π#,0 p1, 0q t `W pt, ¨´stq ´W pt, ¨´st 1 q ˘, d X	¯H
							"		s	ˆt t 1
										d dξ H X	t ´t1 ,

1

, P-almost surely. Now, note that r0, T s Q t Þ Ñ W pt, ¨q P H 1 pRq is, P-almost surely, Hölder ´Π#,0 p1,

0q t pB x W qpt, ¨´st 2 q, d X dξ ¯H dt 2 (2.6a) ď |s| Π #,0 LpHq ? εZ pB x W qpt, ¨q L 2 pRq q ´Π#,0 p1, 0q t pB x W qpt 1 , ¨´st 1 q, d X dξ ¯H dt 1 ď m ´1 |s| Π #,0 L 2 pL 2 pRq;H 1 pRqq ă 8,

i.e., E rgs ă 8. Hence, with (4.13) and dominated convergence, we conclude that (3.11a) holds true.

1 ,

 1 P-almost surely, where we have integrated by parts in the last step. Hence, for δ ą 0 it holds sup tPr0,δs |ϕ m 0 ptq| ď gpδq, P-almost surely, where It is obvious that g is, P-almost surely, non-decreasing and by the Burkholder-Davis-Gundy inequality[START_REF] Da Prato | Stochastic Equations in Infinite Dimensions[END_REF] Lemma 7.7], as in the proof of Theorem 3.4 (b), we see that

	gptq :" ´Π#,0 X `|s| t Π #,0 p0q 0 , d dξ X LpHq ¯H ? ` Π #,0 εZ sup t 1 Pr0,ts	LpHq pB x W qpt 1 , ¨q L 2 pRq sup W pt 1 , ¨q H t 1 Pr0,ts d X dξ H d X dξ H
	is m-independent. E	« tPr0,δs sup	|ϕ m 0 ptq|	ff	ď E rgpδqs Ñ ´Π#,0 X	p0q 0 , d dξ X	¯H	as δ OE 0.
	Since we conclude that				gpδq ě gp0q " ´Π#,0 X	p0q 0 , d dξ X	¯H	,
			«						ff
		P	sup tPr0,δs	|ϕ m 0 ptq| ě ´Π#,0 X	p0q 0 , d dξ X	¯H	`c 2	Ñ 0 as δ OE 0
	and hence, for any p P p0, 1s there exists δ P r0, T s small enough such that sup tPr0,δs |ϕ m 0 ptq| ď ´Π#,0 X p0q 0 , d ¯H X dξ `c 2 ă σ
	sup tPrδ,T s P rE p s " 1 ´P rΩzE p s ě 1 ´P rΩzE p |ϕ m 0 ptq| ď sup tPrδ,T s 1 s ´P rΩzE p |ϕ 8 0 ptq| `c ď σ p 2 , then 1 X E p 2 s " ´1 `P rE p 1 s `P rE p 2 s ě 1 ´p. In total, on min τ q,σ , τ 8 q,σ,c ( " T ( X E p we can apply the multiscale decomposition (3.8) of Theorem 3.3 (b) in order to obtain
	σ S 8 pt, ¨q V (3.13)

´q on an event E p 1 P F with P rE p 1 s ě 1 ´p 2 . Using the convergence (3.11a), we infer that for any p P p0, 1s we find m P p0, 8q sufficiently large with ´q on an event E p 2 P F with P rE p 2 s ě 1 ´p 2 . Define E p :" E

  We will use Theorem A.2 to prove Proposition 2.4 by verifying Assumptions A.1.Proof of Proposition 2.4. The Hilbert spaces U " L 2 pRq and H " H 1 pRq ? ε kL 2 pRq is reflexive, and as the test functions are dense in H and V , also V is dense in H. Next, we concentrate on verifying (LR1)-(LR6) with the choices

	ously separable, V	(2.5)				(2.5) " L 2 pRq ?	ε kL 2 pRq are obvi-
	α :" 2,							(A.1a)
	β :" 4,							(A.1b)
	θ :" ν,							(A.1c)
	C 1 :" max	"	2η 1 , 1, η 1 `ν, 4ν 2 ,	64η 2 4 ε 2 Z 2	*	,	(A.1d)
	C 2 :" 1,							(A.1e)
	g :" 0,							(A.1f)
								1, for any initial datum
					Xp0q " X p0q P L p pΩ, F 0 , P; Hq ,
	equation (2.4) has a unique variational solution pXpt, ¨qq tPr0,T s (cf. Definition 2.3), which addi-tionally satisfies
					«			ff
				E	sup tPr0,T s	Xpt, ¨q p H	`ˆT 0	Xpt, ¨q α V dt	ă 8.

  pw 1 `τ w 2 `ûp¨`stqq ´f pûp¨`stqq , wyH 1 pRq ´εZ H ´1pRq xq 1 `τ q 2 , wy H 1 pRq `εZ L 2 pRq xw 1 `τ w 2 ´γ pq 1 `τ q 2 q , qy L 2 pRq " τ εZ ´´ν pB x w 2 , B x wq L 2 pRq ´pq 2 ,wq L 2 pRq `pw 2 ´γq 2 , qq L 2 pRq εZ pf pw 1 `τ w 2 `ûp¨`stqq ´f pûp¨`stqq , wq L 2 pRq ´εZν pB x w 1 , B x wq L 2 pRq ´εZ pq 1 , wq L 2 pRq `εZ pw 1 ´γq 1 , qq L 2 pRq . Ñ R, τ Þ Ñ hpτ q :" εZ pf pw 1 `τ w 2 `ûp¨`stqq ´f pûp¨`stqq , wq L 2 pRq is continuous. This is true because for τ 1 , τ 2 P R we have |hpτ 1 q ´hpτ 2 q| " εZ pf pw 1 `τ1 w 2 `ûp¨`stqq ´f pw 1 `τ2 w 2 `ûp¨`stqq , wq L 2 pRq

	Hence, hemicontinuity follows if
	h : R (2.1j) ď	εZη 4 ´1 `6 û 2 L 8 pRq `6 w 1	2 L 8 pRq

  8b) " ´εZν pB x w, B x wq L 2 pRq `εZ pf pw `ûp¨`stqq ´f pûp¨`stqq , wq L 2 pRq ´εZ pq, wq L 2 pRq `εZ pw, qq L 2 pRq ´εZγ pq, qq L 2 pRq

	(2.1j)

  Notably, Bpt, Xpt, ¨qq (cf. (2.8c)) does for P r0, T s neither depend on t nor on X and takes values in L 2 pU ; Hq. Hence, Bpt, Xpt, ¨qq is deterministic and in particular Sobolev embedding in form of upt, ¨q L 8 pRq ď upt, ¨q H 1 pRq has been used and finiteness of the terms in the last line follows from the regularity of the variational solution stated in Definition 2.3 and Proposition 2.4. Hence, also (A.2) is satisfied.

										"ˆT			
									P	0	Bpt, Xpt, ¨qq L 2 pU ;Hq dt ă 8	" 1
	trivially holds true. Furthermore, by the regularity of the variational solution stated in Defini-
	tion 2.3, we have
							"ˆT 0	Xpt, ¨q V dt		ď	?	T	ˆE "ˆT 0	Xpt, ¨q 2 V dt	˙1 2	ă 8,
	which is why obviously also
										"ˆT	
										P	0		Xpt, ¨q H dt ă 8	" 1
	is valid. In view of the above estimate, the mixing of u and v in the two components of Apt, Xptqq
	(cf. (2.8b)) is immaterial, so that it remains show that
							"ˆT						
					P	0		f pupt, ¨q `ûp¨`stqq ´f pûp¨`stqq L 2 pRq dt ă 8	" 1	(A.2)
	holds true. Indeed, we can estimate
		"ˆT											 1 2
	E	0	f pupt, ¨q `ûp¨`stqq ´f pûp¨`stqq L 2 pRq dt
		(2.1j) ď η 4 E	« tPr0,T s sup	upt, ¨q L 2 pRq	ˆT 0	´1 `2 û 2 L 8 pRq `3 upt, ¨q 2 H 1 pRq ¯dt	ff 1 2
					«					ff
		ď	η 4 2	E	sup tPr0,T s	upt, ¨q L 2 pRq	2 `η4 T	´1 `2 û 2 L 8 pRq	¯`3η 4 2	E	0 "ˆT	upt, ¨q 2 H 1 pRq dt	
	where the

  ¨q, P-almost surely, where K t pxq " e

	´x2 4νt

  Hq, P-almost surely, and therefore u P L 2 `r0, T s; H 1 pRq ˘X C 0 `r0, T s; L 2 pRq ˘, P-almost surely, we conclude that there exists a constant C ă 8, P-almost surely, with C Ñ 0 as T OE 0, P-almost surely, such thattpB x uqpt, ¨q L 2 pRq ď C `2η 3 ˆt 0 B x upt 1 , ¨q2L 2 pRq t 1 B x upt 1 ,¨q L 2 pRq dt, P-almost surely. Grönwall's inequality implies tpB x uqpt, ¨q L 2 pRq ď C exp ˆ2η 3 ˆt 0 B x upt 1 , ¨q

2

L 2 pRq dt 1 ˙ă 8, P-almost surely, whence tpB x uq P L 8 `r0, T s; L 2 pRq ˘, P-almost surely. Furthermore, since C Ñ 0 as T OE 0, P-almost surely, r0, T s Q t Þ Ñ tB x upt, ¨q P L 2 pRq is, P-almost surely, continuous in t " 0.

  ´t1 ˚pB x uqpt 1 , ¨q ´Bx upt 1 , ¨q L 2 pRq Ñ 0 as t 2 Ñ t 1 , P-almost surely, follows from tu P L 8 pr0, T s; L 2 pRqq, P-almost surely, and the fact that pK t ˚qtě0 Ă L `L2 pRq ˘is strongly continuous in t " 0. The other lines of (A.6) can be estimated as before. Altogether, we obtain t 2 B x upt 2 , ¨q ´t1 B x upt 1 , ¨q L 2 pRq Ñ 0 as t 2 Ñ t 1 , P-almost surely, which implies tu P C 0 `r0, T s; H 1 pRq ˘, P-almost surely. Finally, by (A.4) and (A.5) it follows that v P C 0 `r0, T s; H 1 pRq ˘, P-almost surely.

	¨q, P-almost surely.	(A.6)
	For the first line of (A.6) observe that
	K t 2

If u p0q P H 1 pRq, an analogous reasoning without time weight yields that u P C 0 `r0, T s; H 1 pRq ˘, P-almost surely, too.

  1 " Y 2 distributionally and hence in particular ν d 2 w 1 dx 2 " w 2 ``1 ´κ ´f 1 p0q ˘w1 `q1 P H 1 pRq, giving w 1 P H 3 pRq and thus Y 1 P D pL ˘8q. The arguments have shown that L ˘8 `pκ 1q id V : D pL ˘8q Ñ V is a bijection. The Lumer-Philips theorem [74, Chapter 1, Theorem 4.3] yields that L ˘8 `κ id V generates a 0 -semigroup of contractions in V. Hence, L ˘8 generates a C 0 -semigroup `etL ˘8 ˘tě0 in V with bound e tL ˘8

  1 p0q W 1,8 pRq ´min ´f 1 p0q, εγ ( . A.2.2. Moving frame. In this section, we prove spectral properties of the frozen-wave operator L # (cf. (2.17)), as stated in Proposition 2.8. We view L # as a perturbation of the limiting operator L # ˘8 (cf. (2.17)) Proof. By density, we may assume Y " pw, qq t P pC 8 c pR; Cqq 2 and obtain Cq is dissipative, so that, as in the proof of Proposition 2.6, we deduce with the Lax-Milgram and the Lumer-Philips theorem [74, Chapter 1, Theorem 4.3] that L # ˘8 generates a 0 -semigroup of contractions pP ˘8 st q tě0 in H C meeting (A.10).

	Proof of Proposition 2.8 (b). We have
					´Bt	´L#	¯T´st P st,st 1 T st 1	(2.19)
										L # " L # ˘8 `R# ,	(A.8a)
	where
	L # ˘8 :"	ˆνB 2 ξ	`f 1 p0q ´sB ξ ε	´1 ´εγ ´sB ξ	˙and R # :"	ˆf 1 pûq ´f 1 p0q 0 0 0	˙.	(A.8b)
	Lemma A.3. For Y P D `L# ˘" H 2 pR; Cq ?	ε kH 1 pR; Cq we have
	´L# ˘8Y, Y	¯HC		ď ´κ Y 2 H C ´εZν dw dξ	2 L 2 pRq	with κ	(2.22) " min ´f 1 p0q, εγ	(	.	(A.9)
	In particular, L # ˘8 generates a C 0 -semigroup `P st ˘tě0 of contractions in H C satisfying ˘8
										P ˘8 st	LpH C q ď e ´κt .	(A.10)
	˘8Y, Y	¯HC	" εZ ´ν d 2 w dξ 2 " εZ ˆ´ν dw `f 1 p0qw ´s dw dξ ´q, w ¯L2 pR;Cq 2 dξ L 2 pR;Cq	`Z ´εpw ´γqq ´s dq dξ , q ¯L2 pR;Cq
	This implies
	Re ˆ´L # ˘8Y, Y	¯HC	˙(2.21b) ď	´min ´f 1 p0q, εγ	(	Y 2 H C	´Zs 2	ˆR d dξ ´ε |w| 2 `|q| 2 ¯dξ
										"	2 L 2 pRq H C ´εZν dw ´εZν dw dξ ´κ Y 2 dξ	2 L 2 pRq	,
	which is (A.9). From (A.9), we recognize that
	L # ˘8 `κ id H Proof of Proposition 2.8 (a). Since R # is a bounded operator in H C with
										R #	LpH C q	ď f 1 pûq ´f 1 p0q L 8 pRq ,
	we conclude with Lemma A.3 and [74, Chapter 5, Theorem 2.3] that L # generates a 0 -
	semigroup ´P # st ¯tě0	in H C meeting the bound
						P # st LpH C q

`f 1 p0q w 2 L 2 pR;Cq ˙´εZγ q 2 L 2 pR;Cq `2iεZ Im ´pw, qq L 2 pR;Cq ¯´εZs ˆR ´dw dξ ¯w dξ ´Zs ˆR ´dq dξ ¯q dξ. C : D ´L# ¯" H 2 pR; Cq ? ε kH 1 pR; Cq Ñ H C (2.21a) " L 2 pR; Cq ? ε kL 2 pR; ď e ´ρt with ρ :" κ ´ f 1 pûq ´f 1 p0q L 8 pRq . (A.11) Since L # has real coefficients (cf. (2.17)), the statement for H instead of C is immediate.

  4. We have σ ess pL # ˘8q Ď tλ P C : Re λ ď ´κu , where κ ˘8¯( cf. Definition 2.7 (d) and (e), and [52, Lemma 4.1.9]). Then the result is immediate from (2.1d), Definition 2.7 (e), and Lemma A.

	(2.22) " min ´f 1 p0q, εγ ˘8q lies to the left of the imaginary axis. In particular, σ ess pL #	(	.
		´L#	´L#
	Proof. We first use that σ ess	˘8¯Ď R H C

  1 1`ξ 2 dξ, d dy " `1 `ξ2 ˘d dξ , `ξ2 `f 1 pûpξqq ´f 1 p0q ˘wn pξq ``1 `ξ2 ˘3 2 f 2 pûpξqq dû dξ pξqw n pξq ``1 `ξ2 ˘3 2 `f 1 pûpξqq ´f 1 p0q ˘dwn

		dφn dy	" ξ	a 1 dξ pξq,
	and therefore
	φ n	2 H 1 pp´π 2 , π 2 q;Cq "	ˆπ 2 ´π 2	ˆ|φ n | 2 ` dφn dy	2	˙dy ď C	ˆR ˆ|w n | 2 ` dwn dξ	2	˙dξ " C w n	2 H 1 pR;Cq

  Then, it suffices to show that sup nPN w n H 1 pR;Cq ă 8. Observe that by interpolation, we have for Y " pw, qq t P pC 8 c pR; Cqq 2 with C 1 ą 0 and C 2 ă 8 independent of n. Since Y n L 2 pR;Cq , we may conclude with (2.1d) that indeed sup nPN w n H 1 pR;Cq ă 8 holds true. Proof of Proposition 2.8 (ci). If σ ess pL # q " σ ess pL # ˘8q (A.12) holds true, it follows in particular that σ ess pL # q Ď tλ P C : Re λ ď ´κu , where κ (2.22) " min ´f 1 p0q, εγ ( by Lemma A.4. The equality (A.12), on the other hand, follows by Weyl's essential spectrum theorem [52, Theorem 2.2.6] because the operators L # , L # ˘8 : D ´L# ¯" H 2 pR; Cq ? ε kH 1 pR; Cq Ñ H C

	ˆ ˙ă 8. L # Y n H C ` L # ˘8Y n H C ˘8Y 2 H C " ε ν d 2 w dξ 2 `f 1 p0qw ´s dw dξ ´q 2 L 2 pR;Cq ` εw ´εγq ´s dq dξ ě C 1 dw dξ 2 L 2 pR;Cq ´C2 ´ w 2 L 2 pR;Cq L 2 pR;Cq ` q 2 ¯,	2 L 2 pR;Cq
	2 H C	(2.21b) " εZ w n	2 L 2 pR;Cq `Z q n	2
		(2.21a)

  First, assume Y p0q " `wp0q , q p0q ˘t P pC 8 c pR; Cqq 2 . Then, by construction B t Y ˘8 ´L# Y p0q for all j, k P N 0 and hence ´L# ˘8¯k Y ˘8 P C 8 pr0, 8q; H C q for any k P N 0 . The coercivity/dissipativity estimate (A.9) of Lemma A.3 then yields w ˘8 P C 8 `r0, 8q; H k pRq ˘for any k P N 0 .Having established qualitative regularity for regular initial data, we next prove a-priori estimates. Indeed, by testing (A.13) it followsH C `κ Y pt, ¨q 2 H C `εZν B ξ wpt, ¨q 2 L 2 pR;Cq ď 0.To obtain a point-wise bound in time on B ξ w ˘8pt, ¨q 2 L 2 pRq , observe that from (A.8b) and (A.13) it followsB t B ξ w ˘8 ´νB 3 ξ w ˘8 ´f 1 p0qB ξ w ˘8 `sB 2 ξ w ˘8 `Bξ q ˘8 " 0.Testing with tB ξ w ˘8 gives with Re `B2 ξ w ˘8pt, ¨q, B ξ w ˘8pt, ¨q˘L 2 pR;Cq " 1 2 Re ˆR B ξ |B ξ w ˘8pt, ξq| 2 dξ " 0 and Re pB ξ q ˘8pt, ¨q, B ξ w ˘8pt, ¨qq L 2 pRq ď The combination with (2.21b) and (A.14) yields that there exists a constant C ă 8 such that w ˘8pt, ¨q L 2 pR;Cq ď C Y p0q 2 The statement of the lemma now follows by density of pC 8 c pR; Cqq 2 in H C . Proof of Proposition 2.8 (d). We loosely follow the approach in [2, Section 3.2, Proposition 3.5]. First observe that by Lemma A.5 and Lemma A.6 the operator R # P ˘8 st Π # P LpH C q is compact for any t ą 0. This compactness implies thanks to [2, Proposition 3.4] that for every t ě 0 the operator P # st Π # ´P ˘8 st Π # is compact as well. By estimate (A.10) of Lemma A.3 and the Neumann series, we recognize that the operator ˘8 st Π # has no spectrum outside the disc ! µ P C : |µ| ď e ´κt Π # only contains point spectrum σ p ´P # st Π # ¯(cf. Definition 2.7 and [52, Theorem 2.2.6]). Using [74, Chapter 2, Theorem 2.4], we infer that σ p ´P # st Π # ¯Ď t0u Y ! e λt : λ P σ p ´L# Π # ¯) Ď ! µ P C : |µ| ď e maxt´κ,λ ˚pεqut ) , where we have used Proposition 2.8 (cii) in the last inclusion and that σ p `L# Π # ˘" σ p `L# ˘zt0u since 0 is not an eigenvalue of L # Π # . Altogether, we have σ ´P # st Π # ¯Ď " µ P C : |µ| ď e maxt´κ,λ ˚pεqut Π # Now, let maxt´κ, λ ˚pεqu ă ´ϑ ă 0. Since the spectral radius lim nÑ8 ´P # st Π # ´P # st Π # ¯) ď e maxt´κ,λ ˚pεqut Π # LpH C q ď e ´ϑt for t and n large enough. Thus there exists C ϑ ă 8 such that (2.24) holds true. Proof of Proposition 2.8 (e). Suppose that Y P H is real-valued. Then

	which implies							˘8Y ˘8 " 0,	(A.13)
	B j t so that with (A.9) of Lemma A.3 we deduce that ´L# ˘8¯k Y ˘8 " ´L# ˘8¯j `k Y ˘8 " P ˘8 st ˘8¯j `k 1 ´L# 2 d dt Y ˘8 2 H C ´´L # 0 Y ˘8, Y ˘8¯H C 1 2 d dt Y pt, ¨q 2 because Π # LpH C q ě 1. P # st Π # meets lim nÑ8 ´P # st Π # ¯n 1 n LpH C q " lim nÑ8 P # stn Π # " 0, 1 n LpH C q LpH C q and	*	¯n 1 n LpH C q	of
	Integrating in time yields 1 2 Y ˘8pt, ¨q 2 H C `ˆt 0 ´κ lim nÑ8 ´P # st Π # ¯n 1 n LpH C q Y ˘8pt 1 , ¨q " lim 1 we have P # stn Π # n	2 H C	`εZν B ξ w ˘8pt 1 , ¨q	2 L 2 pR;Cq ¯dt 1 ď	1 2	Y p0q 2 H C LpH C q . (A.14) ,
	Π #,0 Y	(2.23) "	1 2πi	‰ |λ|"r	X λ dλ "	2π r	ˆ2π
										ν 2	B 2 ξ w ˘8pt, ¨q	2 L 2 pR;Cq `1 2ν	q ˘8pt, ¨q 2 L 2 pR;Cq
	that
	t 2	B ξ w ˘8pt, ¨q 2 L 2 pR;Cq	`ˆt 0	t 1	´ν 2	B 2 ξ w ˘8pt 1 , ¨q	2 L 2 pRq	´f 1 p0q B ξ w ˘8pt 1 , ¨q	2 L 2 pRq ¯dt 1
		ď	t 2ν ˆt 0	q ˘8pt 1 , ¨q
						ess-sup	? 1 `?t t	B H C	.
						LpH C q

1`?t P L 8 `r0, 8q; L 2 pR; Cq ˘with w ˘8 L 8 pr0,8q;L 2 pR;Cqq ` q ˘8 L 8 pr0,8q;L 2 pR;Cqq ` ? tB ξ w ˘8 1 `?t L 8 pr0,8q;L 2 pR;Cqq ď C Y p0q H C , where C ă 8 is independent of Y p0q . Proof. 2 L 2 pR;Cq dt 1 `1 2 ˆt 0 B ξ w ˘8pt 1 , ¨q 2 L 2 pR;Cq dt 1 . ξ ) . Now, since P # st Π # is a compact perturbation of P ˘8 st Π # , the spectrum of P # st Π # in " µ P C : |µ| ą e ´κt Π # LpH C q * nÑ8 max ! |µ| : µ P σ 0 X re iτ e iτ dτ, where X λ :" ´λ id H ´L# ¯´1 Y.

  X re iτ e iτ dτ " Π #,0 Y. By (2.23) also Π # Y " Π # Y , i.e., Π #,0 , Π # : H Ñ H are well-defined and the estimates of the operator norms are trivial, too.

	"	r 2π	ˆ2π 0 ˆ2π 0	X re iτ e ´iτ dτ "	r 2π	ˆ2π 0	X re ´iτ e ´iτ dτ "	r 2π ˆ0 ´2π	X re iτ e iτ dτ

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(Katharina Eichinger) CEREMADE, Université Paris Dauphine, PSL, Pl. de Lattre Tassigny, 75775 Paris Cedex 16, France and INRIA-Paris, MOKAPLAN, 2 Rue Simone IFF, 75012 Paris, France Email address: eichinger@ceremade.dauphine.fr (Manuel V. Gnann) Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Sciences, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, Netherlands Email address: M.V.Gnann@tudelft.nl (Christian Kuehn) Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85747 Garching near Munich, Germany Email address: ckuehn@ma.tum.de

anonymous reviewers have helped to improve the content and presentation of this revised version. KE acknowledges partial support from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement # 754362. KE and MVG have been partially supported by the Deutsche Forschungsgemeinschaft (German Research Foundation -DFG) under project # 334362478. CK is supported by the Volkswagen-Stiftung through a Lichtenberg Professorship. KE appreciates the kind hospitality of Delft University of Technology. MVG appreciates the kind hospitality of Heidelberg University and the Technical University of Munich.

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