Charles-Edouard Bréhier

Ludovic Goudenège

ANALYSIS OF SOME SPLITTING SCHEMES FOR THE STOCHASTIC ALLEN-CAHN EQUATION

Keywords: 1991 Mathematics Subject Classification. 60H15, 65C30, 60H35 Stochastic Partial Differential Equations, splitting schemes, Allen-Cahn equation

We introduce and analyze an explicit time discretization scheme for the onedimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution.

We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L 2 (Ω)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4.

The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

Introduction

In this article, we define and study new numerical schemes for the time discretization of the following Stochastic Partial Differential Equation (SPDE), ∂u(t, ξ) ∂t = ∂ 2 u(t, ξ) ∂ξ 2 + u(t, ξ) -u(t, ξ) 3 + Ẇ (t, ξ) driven by Gaussian space-time white noise, with ξ ∈ (0, 1) a one-dimensional space variable -and homogeneous Dirichlet boundary conditions.

The Allen-Cahn equation has been introduced [START_REF] Allen | A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening[END_REF] as a model for a two-phase system driven by the Ginzburg-Landau energy

E(u) = |∇u| 2 + 1 ε 2 V (u)
, where u is the ratio of the two species densities, and V = (u 2 -1) 2 is a double well potential. The first term in the energy models the diffusion of the interface between the two pure phases, and the second one pushes the solution to two possible stable states ±1 (named the pure phases, i.e. minima of V ). The behavior of the interface in the regime ε → 0 is described in terms of mean curvature flow, see for instance [START_REF] Chen | Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term[END_REF][START_REF] Chen | Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations[END_REF][START_REF] Evans | Phase transitions and generalized motion by mean curvature[END_REF][START_REF] Evans | Motion of level sets by mean curvature[END_REF][START_REF] Evans | Motion of level sets by mean curvature[END_REF].

The stochastic version of the Allen-Cahn equation models the effect of thermal perturbations by a additional noise term, see for instance [START_REF] Faris | Large fluctuations for a nonlinear heat equation with noise[END_REF][START_REF] Funaki | The scaling limit for a stochastic PDE and the separation of phases[END_REF][START_REF] Katsoulakis | Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem[END_REF]. The behavior as ε → 0 has been studied for instance in [START_REF] Funaki | The scaling limit for a stochastic PDE and the separation of phases[END_REF] in dimension 1 (with space-time white noise), and [START_REF] Funaki | Singular limit for stochastic reaction-diffusion equation and generation of random interfaces[END_REF][START_REF] Weber | On the short time asymptotic of the stochastic Allen-Cahn equation[END_REF] in higher dimension (with more regular noise).

The stochastic Allen-Cahn equation is also a popular model for the study and simulation of rare events in infinite dimensional stochastic system, see for instance [START_REF] Bréhier | CEMRACS 2013-modelling and simulation of complex systems: stochastic and deterministic approaches[END_REF][START_REF] Kohn | Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation[END_REF][START_REF] Rolland | Computing transition rates for the 1-d stochastic Ginzburg-Landau-Allen-Cahn equation for finite-amplitude noise with a rare event algorithm[END_REF][START_REF] Vanden-Eijnden | Rare event simulation of small noise diffusions[END_REF].

In this work, our aim is to study numerical schemes for the stochastic Allen-Cahn equation. In the theoretical analysis, we only focus on the temporal discretization. To perform numerical simulations, a spatial discretization is required: we use a standard finite difference method.

Numerical schemes for SPDEs have been extensively studied in the last two decades, see for instance the monographs [START_REF] Jentzen | Taylor approximations for stochastic partial differential equations[END_REF][START_REF] Kruse | Strong and weak approximation of semilinear stochastic evolution equations[END_REF][START_REF] Lord | An introduction to computational stochastic PDEs[END_REF]. Compared with the numerical discretization of Stochastic Differential Equations (SDEs), both temporal and spatial discretization are required. In addition, the temporal regularity of the solutions of SPDEs depends heavily on the spatial regularity of the noise perturbation, and this affects orders of convergence. For instance, consider equations with globally Lipschitz continuous coefficients. For SDEs, the solutions are Hölder continuous with exponents α < 1/2, and the Euler-Maruyama scheme has in general a strong order of convergence 1/2 and a weak ordre 1: see for instance the monographs [START_REF] Kloeden | Numerical solution of stochastic differential equations[END_REF][START_REF] Milstein | Numerical integration of stochastic differential equations[END_REF]. For SPDEs, driven by space-time white noise, the solutions are only Hölder continuous with exponents α < 1/4, and (explicit or implicit) Euler type schemes only have strong order 1/4 and weak order 1/2. We recall that strong convergence usually refers to convergence in mean-square sense: see for instance [START_REF] Davie | Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations[END_REF][START_REF] Gyöngy | Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise[END_REF][START_REF] Gyöngy | Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise[END_REF][START_REF] Gyöngy | On discretization schemes for stochastic evolution equations[END_REF][START_REF] Gyöngy | Rate of convergence of implicit approximations for stochastic evolution equations[END_REF][START_REF] Gyöngy | Implicit scheme for quasi-linear parabolic partial differential equations perturbed by space-time white noise[END_REF][START_REF] Gyöngy | Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise[END_REF][START_REF] Jentzen | Higher order pathwise numerical approximations of SPDEs with additive noise[END_REF][START_REF] Jentzen | Efficient simulation of nonlinear parabolic SPDEs with additive noise[END_REF][START_REF] Jentzen | Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise[END_REF][START_REF] Kloeden | Linear-implicit strong schemes for Itô-Galerkin approximations of stochastic PDEs[END_REF][START_REF] Lord | Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise[END_REF][START_REF] Printems | On the discretization in time of parabolic stochastic partial differential equations[END_REF][START_REF] Wang | Strong convergence rates of the linear implicit Euler method for the finite element discretization of SPDEs with additive noise[END_REF]. Weak convergence refers to convergence in distribution: recent contributions are [START_REF] Bréhier | Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient[END_REF][START_REF] Conus | Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients[END_REF][START_REF] Debussche | Weak approximation of stochastic partial differential equations: the nonlinear case[END_REF][START_REF] Jentzen | Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients[END_REF].

For discretization of equations such as the stochastic Allen-Cahn equation, the main difficulty is the polynomial coefficient, which is not globally Lipschitz continuous. Standard schemes with explicit discretization of such coefficients cannot be applied. Defining efficient numerical schemes for stochastic equations with non globally Lipschitz continuous coefficients is delicate: see for instance the recent works [START_REF] Hutzenthaler | Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients[END_REF], and the monograph [START_REF] Hutzenthaler | Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients[END_REF], and references therein. The general methodology has been recently applied to various examples of SPDEs: see for instance [START_REF] Becker | Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg-Landau equations[END_REF][START_REF] Hutzenthaler | Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-type approximations for stochastic Kuramoto-Sivashinsky equations[END_REF][START_REF] Jentzen | Strong convergence rates for an explicit numerical approximation method for stochastic evolution equations with non-globally Lipschitz continuous nonlinearities[END_REF][START_REF] Jentzen | Exponential moments for numerical approximations of stochastic partial differential equations[END_REF][START_REF] Jentzen | Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations[END_REF]. We also mention [START_REF] Gyöngy | Convergence of tamed Euler schemes for a class of stochastic evolution equations[END_REF] for the analysis of a tamed Euler scheme for a class of SPDEs.

The case of the Allen-Cahn equation has been treated in the recent work [START_REF] Becker | Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations[END_REF], with a scheme based on an exponential integrator and a tamed discretization of the nonlinear coefficient. References [START_REF] Kovács | On the backward Euler approximation of the stochastic Allen-Cahn equation[END_REF][START_REF] Kovács | On the discretisation in time of the stochastic Allen-Cahn equation[END_REF] present analysis of implicit schemes. Finally, a Wong-Zakai approximation has been considered in [START_REF] Liu | Wong-Zakai approximations of stochastic Allen-Cahn equation[END_REF].

In this work, we introduce new schemes, based on a splitting (also referred to as splittingup) strategy, see for instance [START_REF] Bessaih | Splitting up method for the 2D stochastic Navier-Stokes equations[END_REF][START_REF] Bessaih | On strong L2 convergence of numerical schemes for the stochastic 2d Navier-Stokes equations[END_REF][START_REF] Gyöngy | On the rate of convergence of splitting-up approximations for SPDEs[END_REF][START_REF] Gyöngy | On the splitting-up method and stochastic partial differential equations[END_REF] in the SPDE case, and [START_REF] Jeong | Comparison study of numerical methods for solving the Allen-Cahn equation[END_REF][START_REF] Lee | A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms[END_REF] for the deterministic Allen-Cahn equation. Indeed, the solution of the ordinary differential equation ż = z -z 3 has a known explicit expression. The splitting strategy then consists in solving separately the contributions of the linear coefficient with the noise, and of the nonlinear coefficient. Several schemes may be chosen to treat the first contribution: exponential and linear implicit Euler integrators may be used.

We mostly focus on the following scheme (other schemes will be defined when performing numerical simulations). Let us write the stochastic Allen-Cahn equation as an evolution equation in the sense of Da Prato-Zabczyk, [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]:

dX(t) = AX(t)dt + Ψ 0 (X(t))dt + dW (t),
with Ψ 0 (x) = x -x 3 . Then the numerical scheme, with time-step size ∆t > 0, is defined by the recursion:

Y n = Φ ∆t (X n ), X n+1 = S ∆t X n + S ∆t W ((n + 1)∆t) -W (n∆t) ,
where S ∆t = (I -∆tA) -1 corresponds to the choice of a linear implicit Euler integrator, and

Φ ∆t (z) = z z 2 + (1 -z 2 )e -2∆t
.

Observe that the discrete-time process X n n∈N may be interpreted as the solution of a standard linear implicit Euler scheme for a modified SPDE, with nonlinear coefficient Ψ ∆t (z) = ∆t -1 (Φ ∆t (z)-z), in the spirit of [START_REF] Higham | Strong convergence of Euler-type methods for nonlinear stochastic differential equations[END_REF]. The coefficient Ψ ∆t satisfies the same type of one-sided Lipschitz condition as Ψ 0 , uniformly with respect to ∆t.

Our first contribution is the analysis of the splitting scheme introduced above. We first prove moment bounds, uniform with respect to ∆t. Our main result, Theorem 4.1, is a strong convergence result, with order of convergence almost 1/4, localized on an event of arbitrarily large probability, in the spirit of [START_REF] Bessaih | Splitting up method for the 2D stochastic Navier-Stokes equations[END_REF]. We also state and prove several straightforward consequences of Theorem 4.1, related with other types of convergence:

• convergence in mean-square sense, with no order of convergence,

• convergence in probability of order almost 1/4, in the spirit of [START_REF] Printems | On the discretization in time of parabolic stochastic partial differential equations[END_REF],

• weak convergence, with order almost 1/4, rejecting exploding trajectories in the spirit of [START_REF] Milstein | Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients[END_REF]. We also provide numerical simulations to illustrate the rates of convergence of the scheme introduced above, and compare with a few variants.

These numerical experiments lead us to conjecture that our results may be improved as follows. First, we conjecture that the strong order is equal to 1/4, in the standard sense, i.e. that it is possible to get rid of the localization in Theorem 4.1. But we expect the analysis to be considerably more complex and similar to [START_REF] Becker | Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations[END_REF]. Second, we conjecture that the weak order is equal to 1/2, when considering sufficiently smooth test functions. Again the analysis requires more complex arguments. We plan to investigate these questions in future works. This article is organized as follows. The setting is introduced in Section 2. The splitting schemes are introduced in section 3. Results concerning the auxiliary flow map Φ ∆t are given in Section 3.3. A priori bounds on the moments of the numerical solutions are given in Section 3.4. Our main results are stated in Section 4.1, and their proofs are given in Sections 4.2 and 4.3. Numerical experiments to investigate strong and weak orders of convergence are reported in Section 5.

Setting

We work in the standard framework of stochastic evolution equations with values in infinite dimensional separable Hilbert and Banach spaces. We refer for instance to [START_REF] Cerrai | Second order PDE's in finite and infinite dimension[END_REF][START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] for details. Let H = L 2 (0, 1), and E = C([0, 1]). We use the following notation: for

x 1 , x 2 ∈ H, x ∈ E, x 1 , x 2 = 1 0 x 1 (ξ)x 2 (ξ)dξ , x 1 H = 1 0 x 1 (ξ) 2 dξ 1 2 , |x| E = max ξ∈[0,1] |x(ξ)|.
To simplify notation, we often write x = x H and |x| = |x| E .

2.1. Assumptions.

2.1.1. Linear operator. Let A denote the unbounded linear operator on H, with

D(A) = H 2 (0, 1) ∩ H 1 0 (0, 1) Ax = ∂ 2 ξ x, ∀ x ∈ D(A).
It is well-known that A generates a strongly continuous semi-group, both on H and on E. We use the notation e tA t≥0 . More precisely, it is an analytic semi-group. Finally, let e n = √ 2 sin(nπ•) and λ n = n 2 π 2 , for n ∈ N. Note that Ae n = -λ n e n , and that e n n∈N is a complete orthonormal system of H.

dX(t) = AX(t)dt + dW (t), Z(0) = 0, admits a unique (global) mild solution in H X(t) = t 0 e (t-s)A dW (s) = W A (t), called the stochastic convolution.
Moreover, this process is continuous with values in E. Moment estimates are satisfied: for all T ∈ (0, ∞) and all p ∈ N, there exists

C p (T ) ∈ (0, ∞) such that (1) E sup 0≤t≤T W A (t) 2p H + sup 0≤t≤T |W A (t)| 2p E ≤ C p (T ).

2.2.

Allen-Cahn equation. The potential energy function V : R → R is defined by

V (z) = z 4 4 - z 2 2 .
Then the function Ψ 0 = -V ′ satisfies a one-sided Lipschitz condition: for all z 1 , z 2 ∈ R,

Ψ 0 (z 2 ) -Ψ 0 (z 1 ) z 2 -z 1 ≤ |z 2 -z 1 | 2 .
However, Ψ 0 is not globally Lipschitz continuous.

In this article, we consider the stochastic Allen-Cahn equation, with additive space-time white noise, i.e. the Stochastic Partial Differential Equation ( 2)

dX(t) = AX(t)dt + Ψ 0 (X(t))dt + dW (t), X(0) = x 0 ,
with an initial condition x 0 ∈ E. We quote the following well-posedness result, see for instance [START_REF] Cerrai | Second order PDE's in finite and infinite dimension[END_REF]Chapter 6].

Proposition 2.1. Let T ∈ (0, ∞). There exists a unique global mild solution X(t) 0≤t≤T of (2), with values in E. Moreover, for every p ∈ N, there exists C p (T ) ∈ (0, ∞) such that

E sup 0≤t≤T |X(t)| 2p E ≤ C p (T ) 1 + |x 0 | 2p E .

Numerical schemes

Since the coefficient Ψ 0 is not globally Lipschitz continuous, it is well-known that explicit discretization schemes are not appropriate -unless combined with a taming strategy, as in [START_REF] Becker | Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations[END_REF][START_REF] Gyöngy | Convergence of tamed Euler schemes for a class of stochastic evolution equations[END_REF] for instance. Fully implicit schemes are expensive, and split-step schemes such as defined in [START_REF] Kovács | On the backward Euler approximation of the stochastic Allen-Cahn equation[END_REF][START_REF] Kovács | On the discretisation in time of the stochastic Allen-Cahn equation[END_REF], with an implicit discretization for the contributions depending on Ψ 0 , may be defined.

In the case of Allen-Cahn equations, our strategy, detailed below, consists in replacing these implicit steps with the exact solution of the flow associated with Ψ 0 , in the spirit of [START_REF] Jeong | Comparison study of numerical methods for solving the Allen-Cahn equation[END_REF][START_REF] Lee | A second order operator splitting method for Allen-Cahn type equations with nonlinear source terms[END_REF].

Splitting schemes. Introduce the auxiliary ordinary differential equation

ż = Ψ 0 (z), z(0) = z 0 ∈ R.
The flow of this equation is known: the unique solution z(t) t≥0 is given by ( 3)

z(t) = Φ t (z 0 ) = z z 2 0 + (1 -z 2 0 )e -2t
, t ≥ 0.

The splitting schemes we consider may be written in the following abstract form: let ∆t > 0 denote the time-step size, then (4)

Y n = Φ ∆t (X n ), X n+1 = Γ Y n , ∆t, (W (t)) n∆t≤t≤(n+1)∆t .
To complete the definition of the numerical schemes, it remains to provide the definition of the mapping Γ, corresponding to the approximation of the stochastic convolution. In the analysis below, three examples are considered:

     Γ exact y, ∆t, (W (t)) n∆t≤t≤(n+1)∆t = e ∆tA y + (n+1)∆t n∆t e ((n+1)∆t-t)A dW (t), Γ expo y, ∆t, (W (t)) n∆t≤t≤(n+1)∆t = S expo ∆t y + S expo ∆t ∆W n , Γ imp y, ∆t, (W (t)) n∆t≤t≤(n+1)∆t = S imp ∆t y + S imp ∆t ∆W n where ∆W n = W (n + 1)∆t -W n∆t
are Wiener increments, and with linear operators

S expo ∆t = e ∆tA , S imp ∆t = I -∆tA -1 .
Numerical experiments, see Section 5; will also be performed for other schemes, based on different splitting strategies.

When using the splitting scheme with Γ = Γ exact , both sub-steps are solved exactly. On the contrary, when using the other examples, there is an error due to the discretization of the stochastic convolution.

We use the notation X exact n , X expo n and X imp n , when choosing Γ = Γ exact , Γ expo and Γ = Γ imp respectively. To simplify, we do not mention the dependence with respect to ∆t.

Auxiliary SPDE.

Define auxiliary functions Ψ t , for t > 0, as follows: for all z ∈ R,

(5) Ψ t (z) = Φ t (z) -z t .
An important tool in the analysis is the auxiliary equation ( 6)

dX (∆t) (t) = AX (∆t) (t)dt + Ψ ∆t X (∆t) (t) dt + dW (t) , X (∆t) (0) = x 0 , with nonlinear coefficient Ψ 0 in (2) replaced with Ψ ∆t .
Observe that the numerical schemes defined by ( 4), based on the splitting method, can be interpreted as standard numerical schemes for the auxiliary equation ( 6):

     X exact n+1 = e ∆tA X exact n + ∆te ∆tA Ψ ∆t (X exact n ) + (n+1)∆t n∆t e ((n+1)∆t-t)A dW (t), X expo n+1 = S expo ∆t X expo n + ∆tS expo ∆t Ψ ∆t (X expo n ) + S expo ∆t ∆W n , X imp n+1 = S imp ∆t X imp n + ∆tS imp ∆t Ψ ∆t (X imp n ) + S imp ∆t ∆W n .
The schemes X exact and X expo correspond to versions of the exponential Euler scheme, applied to the auxiliary equation ( 6). The scheme X imp is the standard linear implicit Euler scheme, applied to the auxiliary equation [START_REF] Bréhier | Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient[END_REF].

The three schemes are well-defined, for any value of ∆t > 0. Indeed, the mapping Ψ ∆t is globally Lipschitz continuous.

Remark 3.1. One may also introduce the following scheme:

X acc n+1 = e ∆tA X acc n + (-A) -1 I -e ∆tA Ψ ∆t (X acc n ) + (n+1)∆t n∆t e ((n+1)∆t-t)A dW (t).
This scheme corresponds to the application of the accelerated exponential Euler scheme to the auxiliary equation. However, this scheme is not based on a splitting method.

3.3.

Results concerning the auxiliary coefficients Φ ∆t and Ψ ∆t . In this section, we state several results concerning the real valued mappings Φ ∆t and Ψ ∆t defined by ( 3) and ( 5), with t = ∆t. Proofs are postponed to the Appendix A. Note that the estimates below are uniform for ∆t ∈ [0, ∆t 0 ], for any arbitrary ∆t 0 > 0and without loss of generality assume ∆t 0 < 1. Moreover, the estimates are consistent when ∆t = 0, with Φ 0 (z) = z and Ψ 0 (z) = z -z 3 .

The first result yields global Lipschitz continuity of Φ ∆t .

Lemma 3.2. For every ∆t 0 ∈ (0, 1), for all ∆t ∈ [0, ∆t 0 ], the mapping Φ ∆t is globally Lipschitz continuous, and the Lipschitz constant is bounded from above uniformly for ∆t ∈ [0, ∆t 0 ]. More precisely, for all z 1 , z 2 ∈ R,

Φ ∆t (z 2 ) -Φ ∆t (z 1 ) ≤ e ∆t |z 2 -z 1 |.
The second result yields a one-sided Lipschitz condition for Ψ ∆t .

Lemma 3.3. For every ∆t 0 ∈ (0, 1), for all ∆t ∈ [0, ∆t 0 ], the mapping Ψ ∆t satisfies a one-sided Lipschitz condition, uniformly for ∆t ∈ [0, ∆t 0 ]. More precisely, for all z 1 , z 2 ∈ R,

Ψ ∆t (z 2 ) -Ψ ∆t (z 1 ) z 2 -z 1 ≤ e ∆t (z 2 -z 1 ) 2 .
In addition to the one-sided Lipschitz condition from Lemma 3.3 above, Ψ ∆t is locally Lipschitz continuous. Lemma 3.4. For every ∆t 0 ∈ (0, 1), there exists C(∆t 0 ) ∈ (0, ∞), such that for all ∆t ∈ [0, ∆t 0 ] and all z 1 , z 2 ∈ R,

Ψ ∆t (z 2 ) -Ψ ∆t (z 1 ) ≤ C(∆t 0 )|z 2 -z 1 | 1 + |z 1 | 3 + |z 2 | 3 .
In addition, for all z ∈ R,

|Ψ ∆t (z)| ≤ C(∆t 0 )(1 + |z| 4 ).
Finally, the following result makes precise the speed of convergence of Ψ ∆t to Ψ 0 , when ∆t → 0. Lemma 3.5. For every ∆t 0 ∈ (0, 1), there exists C(∆t 0 ) ∈ (0, ∞) such that, for all ∆t ∈ [0, ∆t 0 ] and z ∈ R,

|Ψ ∆t (z) -Ψ 0 (z)| ≤ C(∆t 0 )∆t(1 + |z| 5 ).
3.4. Moment bounds.

3.4.1.

Moment bounds for solutions of the auxiliary SPDE [START_REF] Bréhier | Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient[END_REF]. Using the one-sided Lipschitz condition for Ψ ∆t , from Lemma 3.3, the same arguments used to get Proposition 2.1 yield the following moment bounds for the solution X (∆t) of the auxiliary equation, uniformly in ∆t ∈ (0, ∆t 0 ], for arbitrary ∆t 0 ∈ (0, 1).

Proposition 3.6. Let T ∈ (0, ∞). There exists a unique global mild solution X (∆t) (t) 0≤t≤T of ( 6), with values in E. Moreover, for every p ∈ N, and every ∆t 0 ∈ (0, 1), there exists

C p (T, ∆t 0 ) ∈ (0, ∞) such that sup ∆t∈(0,∆t 0 ] E sup 0≤t≤T |X (∆t 0 ) (t)| 2p E ≤ C p (T, ∆t 0 ) 1 + |x 0 | 2p E .
Observe that, when ∆t > 0, the mapping Ψ ∆t is globally Lipschitz continuous, the existence of moments is thus a standard result. The one-sided Lipschitz condition ensures that the estimate is uniform for ∆t ∈ (0, ∆t 0 ].

3.4.2.

Moment bounds for solutions of the numerical schemes (4). Let ∆t 0 ∈ (0, 1), and ∆t ∈ (0, ∆t 0 ] denote a time-step size. Let X n n∈N 0 be defined by the numerical scheme (4), with the mapping Γ ∈ Γ exact , Γ expo , Γ imp . For T ∈ (0, ∞), let N T,∆t = ⌊ T ∆t ⌋. Proposition 3.7. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). For any p ∈ N, there exists C p (t, ∆t 0 ) ∈ (0, ∞) such that for all ∆t ∈ (0, ∆t 0 ] and all x 0 ∈ E,

E sup 0≤n≤N T,∆t |X n | 2p E ≤ C p (T, ∆t 0 )(1 + |x 0 | 2p E ).
The proof uses the following result, in the situation with Ψ 0 replaced with 0 in (2).

Lemma 3.8. Let ω n n=0,...,N T,∆t be defined by ω 0 = 0, and

ω n+1 = Γ ω n , ∆t, (W (t)) n∆t≤t≤(n+1)∆t
Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). For any p ∈ N, there exists C p (T, ∆t 0 ) ∈ (0, ∞) such that for all ∆t ∈ (0, ∆t 0 ],

E sup 0≤n≤N T,∆t |ω n | 2p E ≤ C p (T, ∆t 0 ).
Before sketching the proof of Lemma 3.8, we provide a detailed proof of Proposition 3.7.

Proof of Proposition 3.7. Let r n = X n -ω n , for n ∈ {0, . . . , N T,∆t }. Then r 0 = x 0 , and (with S ∆t = S exact ∆t = e ∆tA when Γ = Γ exact )

r n+1 = X n+1 -ω n+1 = S ∆t Φ ∆t (X n ) -ω n = S ∆t Φ ∆t (r n + ω n ) -Φ ∆t (ω n ) + S ∆t Φ ∆t (ω n ) -ω n .
In addition, the linear operator S ∆t ∈ S exact ∆t , S expo ∆t , S imp ∆t satisfies |S ∆t x| E ≤ |x| E , for all x ∈ E and all ∆t > 0, as a consequence of the maximum principle for the Laplace operator. On the one-hand, using Lipschitz continuity of Φ ∆t , see Lemma 3.2, then

S ∆t Φ ∆t (r n + ω n ) -Φ ∆t (ω n ) E ≤ e ∆t |r n | E .
On the other hand, thanks to Lemma 3.4, and the identity Φ ∆t (z) -z = ∆tΨ ∆t (z),

S ∆t Φ ∆t (ω n ) -ω n E ≤ C(∆t 0 )∆t(1 + |ω n | 4 E ).
The last two estimates prove that

|r n+1 | E ≤ e ∆t |r n | E + C(∆t 0 )∆t(1 + |ω n | 4
E ), and by discrete Gronwall's Lemma, for all n ∈ {0, . . . , N T,∆t },

|r n | E ≤ C(T, ∆t 0 ) 1 + |x 0 | E + sup 0≤m≤N T,∆t |ω m | 4 E .
Applying the estimate of Lemma 3.8 then concludes the proof of Lemma 3.7.

It remains to give a sketch of proof of Lemma 3.8.

Sketch of proof of Lemma 3.8. When Γ = Γ exact , then ω n = W A (n∆t), and thus the result is a straightforward consequence of (1). When Γ = Γ expo , define

ω(t) = e (t-n∆t)A ω n + t n∆t
e ∆tA dW (s) , n∆t ≤ t ≤ (n + 1)∆t.

When Γ = Γ imp , define

ω(t) = ω n + (t -n∆t)AS imp ∆t ω n + t n∆t S imp ∆t dW (s) , n∆t ≤ t ≤ (n + 1)∆t.
Note that ω is a continuous process, with values in E, and satisfies ω(n∆t) = ω n . For all t ≥ 0 and ξ ∈ [0, 1], let ω(t, ξ) = ω(t)(ξ).

We claim that, for all T ∈ (0, ∞), p ∈ N, ∆t 0 ∈ (0, 1), and α ∈ (0, 1 4 ), there exists C α,p (T, ∆t 0 ) such that for all ∆t ∈ (0, ∆t 0 ],

E ω(t, ξ 2 ) -ω(t, ξ 1 ) 2p ≤ C α,p (T, ∆t 0 )|ξ 2 -ξ 1 | 4αp , ∀ t ∈ [0, T ], ξ 1 , ξ 2 ∈ [0, 1], E ω(t 2 , ξ) -ω(t 1 , ξ) 2p ≤ C α,p (T, ∆t 0 )|t 2 -t 1 | 2αp , ∀ t 1 , t 2 ∈ [0, T ], ξ ∈ [0, 1].
The proof of this statement uses only standard arguments (see [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF]), it is left to the reader. Note that ω(0) = 0, and ω(t, 0) = ω(1) = 0 for all t ≥ 0. Then the application of the Kolmogorov regularity criterion concludes the proof.

Convergence analysis of the splitting schemes

Our main result, Theorem 4.1, and several consequences, are stated in Section 4.1. Section 4.2 is devoted to a detailed proof of Theorem 4.1, and the other results are then proved in Section 4.3. 4.1. Statements. Our main result is the following. Theorem 4.1. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1), and α ∈ (0, 1 4 ). There exists C α (T, ∆t 0 ) ∈ (0, ∞), such that, for every ∆t ∈ (0, ∆t 0 ], M ∈ N and x 0 ∈ E, with (-A) α x 0 H < ∞, then

E 1 Ω (∆t) M (T ) sup 0≤n≤N T,∆t X n -X(n∆t) 2 H ≤ K α (M, T, ∆t 0 )∆t 2α (1 + |x 0 | 8 E + (-A) α x 0 2 H ) with K α (M, T, ∆t 0 ) ≤ C α (T, ∆t 0 )(1 + M 6 ) exp(C α (T, ∆t 0 )M 6 ),
and

Ω (∆t) M (T ) = sup 0≤k∆t≤T |X k | E + sup 0≤t≤T |X (∆t) (t)| E ≤ M .
Moreover, there exists C(T, ∆t 0 ) such that for every x 0 ∈ E, ∆t ∈ (0, ∆t 0 ] and every M ∈ N P Ω

(∆t)

M (T ) c ≤ C(T, ∆t 0 )(1 + |x 0 | E ) M .
Remark 4.2. The condition (-A) α x 0 H < ∞ may be relaxed using standard arguments.

If one assumes (-A) β x 0 H < ∞ with β ∈ [0, α], a factor of the type t α-β n needs to be introduced. To simplify notation, we only consider the case β = α and leave the details of the general case to the interested readers.

Let us state three straightforward consequences of Theorem 4.1, presenting other standard ways to describe the error of the numerical scheme. Proofs are postponed to Section 4.3.

Corollary 4.3. The numerical scheme is mean-square convergent. Precisely, for every T ∈ (0, ∞), and any initial condition x 0 ∈ E, with (-A) α x 0 H < ∞, then

E sup 0≤n≤N T,∆t X n -X(n∆t) 2 H → ∆t→0 0.
Corollary 4.4. The numerical scheme converges in probability with order α, for all α < 1 4 . More precisely, for every α ∈ (0, 1 4 ), K ∈ (0, ∞), and any initial condition

x 0 ∈ E, with (-A) α x 0 H < ∞, then P sup 0≤n≤N T,∆t X n -X(n∆t) H ≥ K∆t α → ∆t→0 0.
Corollary 4.5. Let T ∈ (0, ∞), ∆t 0 ∈ (0, 1), and α ∈ (0, 1 4 ). For every ǫ ∈ (0, 1), there exists M = M α (ǫ, T, ∆t 0 ) ∈ (0, ∞) and C = C α (ǫ, T, ∆t 0 ) ∈ (0, ∞) such that for any bounded Lipschitz continuous function ϕ : H → R, with ϕ ∞ + Lip(ϕ) ≤ 1, and for every ∆t ∈ (0, ∆t 0 ] and

x 0 ∈ E, with (-A) α x 0 H < ∞, then E ϕ(X(T )) -E ϕ(X N T,∆t )1 Ω (∆t) M (T ) c ≤ ǫ + C∆t α .
This weak convergence result is not expected to be optimal. First, it is based on the concept of rejecting exploding trajectories: with a more technical analysis, it may be possible to remove the ǫ error term. Second, as will be confirmed by the numerical experiments below, the order of convergence α may be replaced with 2α, using the standard weak convergence analysis, for functions ϕ of class C 2 , bounded and with bounded derivatives. These improvements will be investigated in future works. 4.2. Proof of Theorem 4.1.

4.2.1.

Two auxiliary lemmas. We first state two auxiliary results. Lemma 4.6. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). For every α < 1 4 , there exists C α (T, ∆t 0 ) ∈ (0, ∞) such that for all ∆t ∈ (0, ∆t 0 ]

E sup 0≤n≤N T,∆t ω n -W A (n∆t) 2 H ≤ C α (T, ∆t 0 )∆t 2α .
Proof of Lemma 4.6. If Γ = Γ exact , then ω n = W A (n∆t), and there is nothing to prove. When Γ = Γ expo or Γ imp , then

ω n = n-1 k=0 (k+1)∆t k∆t S n-k ∆t dW (t).
and it is known that, for all p ∈ N, there exists C p,α (T, ∆t 0 ) such that for all ∆t ∈ (0, ∆t 0 ],

sup 0≤n≤N T,∆t E ω n -W A (n∆t) p H ≤ C p,α (T, ∆t 0 )∆t αp .
See for instance [START_REF] Printems | On the discretization in time of parabolic stochastic partial differential equations[END_REF] for details. It remains to control the expectation of the supremum. Let α ∈ (0, 1 4 ), and, set α = α + 1 p , with p ∈ N, p ≥ 2, chosen sufficiently large to have α < 1 4 . Then

E sup 0≤n≤N T,∆t ω n -W A (n∆t) p H ≤ E 0≤n≤N T,∆t ω n -W A (n∆t) p H ≤ N T,∆t C p, α(T , ∆t 0 )∆t αp ≤ T C p, α(T , ∆t 0 )∆t αp-1 , with αp -1 = pα. Then, since p ≥ 2 E sup 0≤n≤N T,∆t ω n -W A (n∆t) 2 H 1 2 ≤ E sup 0≤n≤N T,∆t ω n -W A (n∆t) p H 1 p ≤ C α (T, ∆t 0 )∆t α .
This concludes the proof.

Lemma 4.7. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). For every p ∈ N and α < 1 4 , there exists C p,α (T ) ∈ (0, ∞), such that for any initial condition x 0 with (-A) α x 0 H < ∞, and all ∆t ∈ (0, ∆t 0 ], then for all t, s ∈ [0, T ],

E X (∆t) (t) -X (∆t) (s) 2p H ≤ C p,α (T )(1 + (-A) α x 0 2p H )∆t 2pα .
Proof of Lemma 4.7. We give a sketch of proof, and give details only for p = 1.

Then, let 0 ≤ s < t ≤ T . Then

E X (∆t) (t) -X (∆t) (s) 2 H ≤ C(T ) e tA x 0 -e sA x 0 2 H + C(T )E W A (t) -W A (s) 2 H + C(T ) t s E Ψ ∆t (X (∆t) (r) 2 dr + C(T ) s 0 (t -s) 2α (s -r) 2α E Ψ ∆t (X (∆t) (r)) 2 H dr.
First, e tA x 0 -e sA x 0

H ≤ |t -s| α (-A) α x 0 H . The inequality E W A (t) -W A (s) 2 H ≤ C α (T ) ≤ |t -s| 2α
is a standard result. Finally, it remains to apply Lemma 3.4, and the moment bound from Proposition 3.6 to conclude.

Error between solutions of exact and auxiliary equations.

The following result states convergence of the X (∆t) to X when ∆t goes to 0. The order of convergence is 1, and there is no need for localization. Proposition 4.8. Let T ∈ (0, ∞) and ∆t 0 ∈ (0, 1). There exists C(T, ∆t 0 ) ∈ (0, ∞) such that for all x 0 ∈ E and ∆t ∈ (0, ∆t 0 ],

E sup 0≤t≤T X (∆t) (t) -X (0) (t) 2 H ≤ C(T, ∆t 0 ) 1 + |x 0 | 10 E ∆t 2 . Proof of Proposition 4.8. Let R (∆t) (t) = X (∆t) (t) -X (0) (t). Then dR (∆t) (t) = AR (∆t) (t)dt + Ψ ∆t (X (∆t) (t)) -Ψ 0 (X (0) (t)) dt,
with R (∆t) (0) = 0. As a consequence,

1 2 d R (∆t) (t) 2 dt = AR (∆t) (t), R (∆t) (t) + Ψ ∆t (X (∆t) (t)) -Ψ 0 (X (0) (t)), R (∆t) (t) ≤ Ψ ∆t (X (∆t) (t)) -Ψ ∆t (X (0) (t)), R (∆t) (t) + Ψ ∆t (X (0) (t)) -Ψ 0 (X (0) (t)) R (∆t) (t)
Using the one-sided Lipschitz condition from Lemma 3.3, and Lemma 3.5, then

1 2 d R (∆t) (t) 2 dt ≤ e ∆t 0 + 1 2 R (∆t) (t) 2 + 1 2 Ψ ∆t (X (0) (t)) -Ψ 0 (X (0) (t)) 2 ≤ C R (∆t) (t) 2 + C∆t 2 1 + |X (0) (t)| 10
E . Using Gronwall's lemma, and Proposition 2.1 then concludes the proof. 4.2.3. Proof of Theorem 4.1. We are now in position to prove the main result of this article, Theorem 4.1. Thanks to Proposition 4.8, it is sufficient to look at the error X n -X (∆t) (n∆t).

Define

r n = X n -ω n . Note that X n+1 -S ∆t X n -∆tS ∆t Ψ ∆t (X n ) = ω n+1 -S ∆t ω n , thus r n+1 = S ∆t r n + ∆tS ∆t Ψ ∆t (X n ).
This identity yields (since r 0 = X 0 = x 0 )

X n -ω n = r n = S n ∆t x 0 + ∆t n-1 k=0 S n-k ∆t Ψ ∆t (X k ).
We are now in position to make precise the decomposition of the error:

X n -X (∆t) (n∆t) = S n ∆t -e n∆t x 0 + ω n -W A (n∆t) + n-1 k=0 (k+1)∆t k∆t S n-k ∆t Ψ ∆t (X k ) -e (n∆t-t)A Ψ ∆t (X (∆t) (t)) dt.
First, there exists C α ∈ (0, ∞), such that for all n ∈ N,

S n ∆t -e n∆t x 0 H ≤ C α ∆t α (-A) α x 0 H .
In addition, the error term ω n -W A (n∆t) is controlled thanks to Lemma 4.6.

It remains to deal with

n-1 k=0 (k+1)∆t k∆t S n-k ∆t Ψ ∆t (X k ) -e (n∆t-t)A Ψ ∆t (X (∆t) (t)) dt = ∆t n-1 k=0 S n-k ∆t Ψ ∆t (X k ) -Ψ ∆t (X (∆t) (k∆t)) + n-1 k=0 (k+1)∆t k∆t S n-k ∆t Ψ ∆t (X (∆t) (k∆t) -Ψ ∆t (X (∆t) (t)) dt + n-1 k=0 (k+1)∆t k∆t S n-k ∆t -e (n∆t-t)A Ψ ∆t (X (∆t) (t))dt
Since Ψ ∆t is not globally Lipschitz continuous uniformly in ∆t ∈ (0, ∆t 0 ), a localization argument is introduced.

For M ∈ N, and n ∈ {0, . . . , N T,∆t }, let

Ω (∆t) n,M = sup 0≤k≤n |X k | E + sup 0≤t≤n∆t |X (∆t) (t)| E ≤ M .
Note that for all k ∈ {0, . . . , n -1}, 0 ≤ 1 Ω (∆t)

n,M ≤ 1 Ω (∆t) k,M ≤ 1. Let ǫ n = 1 Ω (∆t) n,M X n -X (∆t) (n∆t) 2 H . Then E sup 0≤m≤n ǫ m ≤ C sup 0≤m≤n S m ∆t -e m∆t x 0 2 H + CE sup 0≤m≤n ω m -W A (m∆t) 2 H + CT ∆t n-1 k=0 E 1 Ω (∆t) k,M Ψ ∆t (X k ) -Ψ ∆t (X (∆t) (k∆t)) 2 + CT n-1 k=0 (k+1)∆t k∆t E Ψ ∆t (X (∆t) (k∆t) -Ψ ∆t (X (∆t) (t)) 2 dt + CT n-1 k=0 (k+1)∆t k∆t E S n-k ∆t -e (n∆t-t)A Ψ ∆t (X (∆t) (t)) 2 H dt ≤ C α (T )(1 + (-A) α x 0 2
H )∆t 2α + C(1 + M 6 )T ∆t n-1 k=0 E[ sup 0≤m≤k ǫ m ] + C α (T )(1 + M 6 )∆t 2α + C α (T )(1 + |x 0 | 8 E )∆t 2α .
We have first used the estimate above and Lemma 4.6. Moreover, if

x 1 , x 2 ∈ E satisfy |x 1 | E ≤ M, |x 2 | E ≤ M, then Ψ ∆t (x 2 ) -Ψ ∆t (x 1 ) H ≤ C(∆t 0 )(1 + M 3 ) x 2 -x 1 H . Thus E 1 Ω (∆t) k,M Ψ ∆t (X k ) -Ψ ∆t (X (∆t) (k∆t)) 2 ≤ C(1 + M 6 )E 1 Ω (∆t) k,M X k -X (∆t) (k∆t) 2 ≤ C(1 + M 6 )E[ sup 0≤m≤k ǫ m ].
Finally, we have used Lemma 4.7, and the standard estimates to control S n-k ∆t -e (n∆t-t)A L(H) . Applying the discrete Gronwall's Lemma,

E sup 0≤m≤n ǫ m ≤ C α (T, M)∆t 2α (1 + |x 0 | 8 E + (-A) α x 0 2 H ), with K α (T, M) ≤ C α (T )(1 + M 6 ) exp(C α (T )M 6 T ).
To conclude, note that Ω (∆t)

N T,∆t ,M = Ω (∆t)
M (T ), and that

E 1 Ω (∆t) N T,∆t ,M sup 0≤n≤N T,∆t X n -X (∆t) (n∆t) 2 H ] ≤ E[ sup 0≤n≤N T,∆t ǫ n ].
In addition,

1 -P Ω (∆t) N T,∆t ,M ≤ E sup 0≤k≤N T,∆t |X k | E + sup 0≤t≤T |X (∆t) (t)| E M ≤ C(T, ∆t 0 )(1 + |x 0 | E ) M ,
thanks to Propositions 3.6 and 3.7. These estimates, combined with Proposition 4.8, conclude the proof of Theorem 4.1.

Proof of the Corollaries.

Proof of Corollary 4.3. Let M be an arbitrary integer. Then, for all ∆t ∈ (0, ∆t 0 ],

E sup 0≤n≤N T,∆t X n -X(n∆t) 2 H ] = E 1 Ω (∆t) M (T ) sup 0≤n≤N T,∆t X n -X(n∆t) 2 H ] + E 1 Ω (∆t) M (T ) c sup 0≤n≤N T,∆t X n -X (∆t) (n∆t) 2 H ] ≤ C α (M, T, ∆t 0 , x)∆t 2α + P Ω (∆t) M (T ) c 1 2 E sup 0≤n≤N T,∆t X n 4 H + X(n∆t) 4 H ] 1 2 ≤ C α (M, T, ∆t 0 , x)∆t 2α + C α (T, ∆t 0 , x) √ M ,
thanks to Theorem 4.1, Cauchy-Schwarz inequality, and Propositions 2.1 and 3.7. Thus, letting first ∆t → 0, then M → ∞, yields

lim sup ∆t→0 E sup 0≤n≤N T,∆t X n -X(n∆t) 2 H ] ≤ C α (T, ∆t 0 , x) √ M → M →∞ 0,
which concludes the proof of Corollary 4.3.

Proof of Corollary 4.4. Let K ∈ (0, ∞) be an arbitrary positive real number, and M ∈ N be an arbitrary integer. Let α ∈ (α, 1 4 ). Then

P sup 0≤n≤N T,∆t X n -X(n∆t) H ≥ K∆t α ≤ P Ω (∆t) M (T ) c + P sup 0≤n≤N T,∆t X n -X(n∆t) H ≥ K∆t α ∩ Ω (∆t) M (T ) ≤ C α (T, ∆t 0 , x) M + 1 K 2 ∆t 2α E 1 (Ω (∆t) M (T ) sup 0≤n≤N T,∆t X n -X(n∆t) 2 H ≤ C α (T, ∆t 0 , x) M + 1 K 2 K α(M, T, ∆t 0 )∆t 2(α-α) ,
thanks to Theorem 4.1. Thus, letting first ∆t → 0, then M → ∞, yields

lim sup ∆t→0 P sup 0≤n≤N T,∆t X n -X(n∆t) H ≥ K∆t α ≤ C α (T, ∆t 0 , x) M → M →∞ 0,
which concludes the proof of Corollary 4.4.

Proof of Corollary 4.5. Let ǫ ∈ (0, 1), and M ∈ N be such that, for all ∆t ∈ (0, ∆t 0 ],

P Ω (∆t)

M (T ) c ≤ C(T, ∆t 0 )(1 + |x 0 | E ) M ≤ ǫ. Then E ϕ(X(T )) -E ϕ(X N T,∆t )1 Ω (∆t) M (T ) ≤ E ϕ(X(T )) -ϕ(X N T,∆t ) 1 Ω (∆t) M (T ) + E ϕ(X(T ))1 Ω (∆t) M (T ) c ≤ E X(T ) -X N T,∆t H 1 Ω (∆t) M (T ) + ǫ ≤ C α (T, ∆t 0 , x)∆t α + ǫ,
thanks to the assumption that ϕ is bounded and Lipschitz continuous, with ϕ ∞ +Lip(ϕ) ≤ 1, and to Theorem 4.1. This concludes the proof of Corollary 4.5.

Numerical experiments

This section is devoted to numerical simulations, in order to investigate the properties of the numerical scheme (4), with the choice Γ = Γ imp of the linear implicit Euler scheme:

(7) Y n = Φ ∆t (X n ), X n+1 = S ∆t Y n + S ∆t W ((n + 1)∆t) -W (n∆t) ,
with S ∆t = (I -∆tA) -1 . All simulations are performed with this choice of integrator. Indeed, we expect that there is no gain in the orders of convergence when using the version Γ = Γ exp , with S ∆t = e ∆tA . In addition, computing such exponential operators may be expensive in more complex situations, for instance where eigenvalues and eigenfunctions of A are not explicitly known, or in higher dimensional domains. It is thus natural to restrict our simulations to the linear implicit Euler scheme. Spatial discretization is performed using a standard finite differences scheme, with a fixed mesh size. The dependence of the error with respect to this spatial discretization parameter is not studied in this article: we only focus on the temporal discretization error.

Variants of the scheme [START_REF] Bréhier | CEMRACS 2013-modelling and simulation of complex systems: stochastic and deterministic approaches[END_REF] are introduced below, in Section 5.1. They are based on other splitting strategies. The numerical simulations allow us to compare the orders of convergence of these methods.

First, in Section 5.2, strong orders of convergence of the schemes are compared. We observe that in practice the result of Theorem 4.1 holds true without requiring the introduction of the set Ω (∆t) M (T ), and that all the methods are expected to have the same order of convergence, equal to 1/4. We conjecture that the strong order of convergence of the scheme ( 7) is equal to 1/4.

Second, in Section 5.3, weak orders of convergence of the schemes are compared. Note that the rejection of exploding trajectories, as suggested by Corollary 4.5, is not performed: we may take ǫ = 0. Moreover, the test function is of class C 2 , bounded and with bounded derivatives. In addition, one of the alternative splitting schemes defined below has a lower weak error. Based on these numerical simulations, we conjecture that the weak order of convergence is then equal to 1/2 for the scheme [START_REF] Bréhier | CEMRACS 2013-modelling and simulation of complex systems: stochastic and deterministic approaches[END_REF]. This question will be studied in future works.

We also plan to study generalizations in higher dimension.

5.1. Variants of the numerical scheme (7). We define three numerical schemes, for each value of the time-step size ∆t > 0. We recall that S ∆t = (I -∆tA) -1 , and use the notation ∆W n = W ((n + 1)∆t) -W (n∆t) for Wiener increments.

Method 1, given by the scheme (8), corresponds with the scheme studied above, (4), with the linear implicit Euler integrator. The definition of Method 3, given by the scheme [START_REF] Chen | Asymptotic behavior of solutions of an Allen-Cahn equation with a nonlocal term[END_REF], is motivated by [START_REF] Kovács | On the discretisation in time of the stochastic Allen-Cahn equation[END_REF]. The numerical experiments below show that the error is reduced when using this scheme, but the order of convergence seems to be the same. Finally, the definition of Method 3 is motivated by [START_REF] Bréhier | CEMRACS 2013-modelling and simulation of complex systems: stochastic and deterministic approaches[END_REF]. We have checked that the three variants give consistent results. In addition, the observations are stable with respect to the choice of the mesh size.

Method 1. (8) Y 1 n = Φ ∆t (X 1 n ), X 1 n+1 = S ∆t X 1 n + S ∆t ∆W n , Method 2.
(

)      Y 2,1 n = S ∆t 2 X 2 n Y 2,2 n = Φ ∆t (Y 2,1 n ) X 2 n+1 = S ∆t 2 Y 2,2 n + ∆W n . Method 3. 9
     Y 3,1 n = S ∆t 2 X 3 n + 1 2 ∆W n Y 3,2 n = Φ ∆t (Y 3,1 n ) X 3 n+1 = S ∆t 2 Y 3,2 n + 1 2 ∆W n . Remark 5.1. (10)
Using different splitting strategies yields other numerical schemes. For instance, we may have considered the scheme defined by

     Y 1 n = Φ∆t 2 (X n ) Y 2 n = S ∆t Y 1 n + ∆W n ) X n+1 = Φ∆t 2 (Y 2 n )
. Numerical experiments for this scheme are not reported, since they do not differ from Method 1.

Strong convergence.

In order to study the strong order of convergence, one needs to compare trajectories computed using the same Wiener path, which constraints the construction of the associated Wiener increments. It is customary to compare the numerical solution computed with time-step size ∆t, with a reference solution computed using a much smaller time-step size. Instead, we estimate the mean-square error

E X (∆t) N -X ( ∆t 2 ) 2N 2
where X (∆t) N is the numerical scheme, with time-step size ∆t, and N∆t = T . The solutions are computed using the same Wiener path for one value of ∆t, and using independent Wiener paths when changing ∆t. One needs to check that this error is bounded from above by C α (T )∆t 2α : by a telescoping sum argument, since by Corollary 4.4 the scheme is meansquare convergent, this property is equivalent to a standard error estimate In addition, note that the order of convergence of the error E X

E X (∆t) N -X(T ) 2 ≤ C ′ α (T )∆t 2α .
(∆t) N -X ( ∆t 2 ) 2N
2 is exactly what matters in the analysis of Multilevel Monte Carlo algorithms, which are used in the context of weak convergence -see Section 5.3 below.

The simulations are performed with T = 1, a mesh size ∆x = 2.5 10 -4 , and computing Monte-Carlo averages over 10 5 independent realizations. The numerical results, in logarithmic scale, are reported in Figure 1. We observe that the mean-square error converges with order 2α = 1/2, for the three methods. 5.3. Weak convergence. Weak orders of convergence deal to the behavior of the error

E ϕ X (∆t) N -E ϕ(X(T ))
where ϕ : H → R is a test function, with appropriate regularity properties. Precisely, in the numerical experiments below, the test function is given by ϕ(x) = exp -5 x 2 H which is of class C 2 , bounded and with bounded derivatives. Our aim is to check that the weak order of convergence is equal to 2α = 1/2, where α = 1/4 is the strong order.

Experiments to identify weak rates of convergence are plagued by statistical error, and thus we need to use a variance reduction strategy. Instead of directly comparing E ϕ(X (∆t) N ) with using the same Wiener paths (as explained in Section 5.2), but different time-step sizes, respectively ∆t and ∆t 2 . Between two successive levels, the time-step size ∆t is decreased, and computations at different levels use independent Wiener paths. Contrary to the standard Multilevel Monte Carlo strategy, the number of realizations per level is not optimized (it is the same at each level): still the computational cost is significantly reduced (thanks to the strong convergence property checked in Section 5.2), and the observation of the weak orders of convergence is improved a lot.

The comparison of E ϕ X (∆t) N with a reference value E ϕ X ( ∆t 2 K ) N estimated with a smaller time step ∆t 2 K is performed using a straightforward telescoping sum procedure. The simulations are performed with T = 1, ∆x = 2.5 10 -4 , and with 10 5 independent Monte Carlo realizations at each level. The numerical results, in logarithmic scale, are reported in Figure 2. We observe that the weak error converges with order 2α = 1/2 for three methods, and that Method 3 seems more efficient. Then, for fixed z ∈ R, and all t ≥ 0, compute

f ′ z (t) = f z (t) -2 + 3(1 -z 2 )e -2t z 2 + (1 -z 2 )e -2t = f z (t) - 3z 2 f z (t) z 2 + (1 -z 2 )e -2t ≤ f z (t).
Thanks to Gronwall's Lemma, and f z (0) = 1, one gets for all t ≥ 0, and fixed z ∈ R, Indeed, for fixed z ∈ R, and t ≥ 0, using that f z (t) ∈ [0, e t ],

f ′ z (t) ≤ f z
f ′ z (t) ≤ f z (t) +

Figure 1 .

 1 Figure 1. Mean square error order for T = 1, ∆x = 2.5 10 -4 and 10 5 independent realizations.

Figure 2 .

 2 Figure 2. Weak error order for T = 1, ∆x = 2.5 10 -4 and 10 5 independent realizations.

d dz z ∆t 1 ee

 1 To get this estimate, first computedΨ ∆t (z) dz = -2∆t + (1 -e -2∆t )z 2 --2∆t + (1 -e -2∆t )z 2 3/2 -1 = 1 ∆t f z (∆t) -f z (0) ,where, for all t ≥ 0 and z ∈ R,f z (t) = e -2t z 2 + (1 -z 2 )e -2t 3/2 .

  (t) ≤ e t .Then, for all z ∈ R,dΨ ∆t (z) dz = f z (∆t) -f z (0) ∆t ≤ e ∆twhich concludes the proof of the claim. Concluding the proof of Lemma 3.3 is then straightforward.Proof of Lemma 3.4. Following the computations from the proof of Lemma 3.3 above, for all z ∈ R,dΨ ∆t (z) dz = 1 ∆t f z (∆t) -f z (0) ≤ 3e 3∆t (1 + |z| 2 ).

3|z| 2 f 1 √ z 2 +

 212 z (t) e -2t + (1 -e -2t )|z| 2 ≤ e t + 3|z| 2 e 3t .This concludes the proof of the first estimate. The second estimate is straightforward, since Ψ ∆t (0) = 0.Proof of Lemma 3.5. Let z ∈ R be fixed. Note thatΨ ∆t (z) -Ψ 0 (z) = z ∆t g z (∆t) -g z (0) -∆tg ′ z (0)with g z (t) = (1-z 2 )e -2t , and g ′ z (0) = 1 -z 2 .

  This series does not converge in H. However, if H is an Hilbert space, and L ∈ L 2 (H, H) is a linear, Hilbert-Schmidt, operator, then LW (t) is a Wiener process on H, centered and with covariance operator LL ⋆ . 2.1.3. Stochastic convolution. The linear equation, with additive noise,

	2.1.2. Wiener process. Let Ω, F , P denote a probability space, and consider a family β n n∈N of independent standard real-valued Wiener processes. Then set
	W (t) =	β n (t)e n .
	n∈N

Appendix A. Proof of auxiliary results

Let ∆t 0 ∈ (0, 1), and ∆t ∈ (0, 1). Note that the properties are straightforward when ∆t = 0: then Φ 0 (z) = z and Ψ 0 (z) = z -z 3 .

Recall (see (3) and ( 5)) that, for all ∆t ≥ 0 and z ∈ R,

Proof of Lemma 3.2. The mapping Φ ∆t is of class C 1 , and for all z ∈ R,

The conclusion is straightforward.

Proof of Lemma 3.3. We claim that, for all z ∈ R,

The derivatives of g z satisfy, for all t ≥ 0,

Note that z 2 + (1 -z 2 )e -2t = e -2t + (1 -e -2t )z 2 ≥ e -2t ≥ e -2∆t 0 , when 0 ≤ t ≤ ∆t ≤ ∆t 0 . Then it is straightforward to check that for all t ∈ [0, ∆t]

), and applying Taylor's formula then concludes the proof.

Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR5208, Institut Camille Jordan, F-69622 Villeurbanne, France E-mail address: brehier@math.univ-lyon1.fr

Université Paris-Saclay, CNRS -FR3487, Fédération de Mathématiques de Centrale-Supélec, CentraleSupélec, 3 rue Joliot Curie, F-91190 Gif-sur-Yvette, France E-mail address: ludovic.goudenege@math.cnrs.fr


Christian Gérard
email: christian.gerard@math.u-psud.fr

On the Hartle-Hawking-Israel states for spacetimes with static bifurcate Killing horizons

Keywords: Mathematics Subject Classification. 81T20, 35S05, 35S15 Hartle-Hawking state, Killing horizons, Hadamard states, pseudodifferential calculus, Calderón projector

We revisit the construction by Sanders [S1] of the Hartle-Hawking-Israel state for a free quantum Klein-Gordon field on a spacetime with a static, bifurcate Killing horizon and a wedge reflection. Using the notion of the Calderón projector for elliptic boundary value problems and pseudodifferential calculus on manifolds, we give a short proof of its Hadamard property.

Introduction

Let (M, g) be a globally hyperbolic spacetime, with a bifurcate Killing horizon, see [KW], [S1] or Subsect. 2.1 for precise definition. The bifurcate Killing horizon H is generated by the bifurcation surface B = {x ∈ M : V (x) = 0}, where V is the Killing vector field. It allows to split (M, g) into four globally hyperbolic regions, the right/left wedges M + , M -and the future/past cones F , P, each invariant under the flow of V . An important object related with the Killing horizon H is its surface gravity κ, which is a scalar, constant over all of H .

Let us consider on (M, g) a free quantum Klein-Gordon field associated to the Klein-Gordon equation -2 g φ(x) + m(x)φ(x) = 0, where m ∈ C ∞ (M, R), m(x) > 0 is invariant under V , and its associated free field algebra.

If V is time-like in (M + , g), ie if (M + , g, V ) is a stationary spacetime, there exists (see [S2]) for any β > 0 a thermal state ω + β at temperature β -1 with respect to the group of Killing isometries of (M + , g) generated by V .

It was conjectured by Hartle and Hawking [HH] and Israel [I] that if β = 2π κ is the inverse Hawing temperature, denoted by β H in the sequel, then ω + β can be extended to the whole of M as a pure state, invariant under V , the Hartle-Hawking-Israel state, denoted in the sequel by ω HHI .

The rigorous construction of the HHI state was first addressed by Kay in [K4], who constructed the HHI state in the Schwarzschild double wedge of the Kruskal spacetime. In such a double wedge, the HHI state is a double KMS state, see [K2, K3]. Later Kay and Wald [KW] considered the more general case of spacetimes with a bifurcate Killing horizon, and study general properties of stationary states on this class of spacetimes. They emphasized in particular the importance of the Hadamard condition. They proved that a specific sub-algebra of the free field algebra has at most one state invariant under V and Hadamard. They also showed that if M admits a wedge reflection (see Subsect. 2.2) the restriction of such a state to M + will necessarily be a β H -KMS state. These results were later improved in [K1].

The existence of such a state, ie of the HHI state, was however not proved in [HH]. The first proof of the existence of ω HHI was given by Sanders in the remarkable paper [S1], if the bifurcate Killing horizon is static, ie if V is static in M + , assuming also the existence of a wedge reflection. Sanders showed that there exists a unique Hadamard state ω HHI on M extending the double β H -KMS state ω β on M + ∪M -. The double β H -KMS state ω β is a pure state on M + ∪ M -which is the natural extension of ω + β defined using the wedge reflection, see [K2, K3]. It is an exact geometrical analog of the Fock vacuum vector in the Araki-Woods representation of a thermal state.

1.1. Content of the paper. In this paper we revisit the construction in [S1] of the Hartle-Hawking-Israel state in a spacetime with a static bifurcate Killing horizon. Using the notion of the Calderón projector (see Sect. 5), which is a standard tool in elliptic boundary value problems, we significantly shorten the proof of the Hadamard property of ω HHI .

In [S1] the fact that ω HHI is Hadamard was proved by a careful comparison of the Hadamard parametrix construction for the D'Alembertian -2 g + m associated to the Lorentzian metric g and for the Laplacian -∆ ĝ + m associated to the Riemannian metric ĝ obtained from g by Wick rotation in the Killing time coordinate.

In our paper we avoid working with the spacetime covariances of states and instead systematically work with the Cauchy surface covariances (see Subsect.

= u ∂Ω ∂ ν u ∂Ω for u ∈ C ∞ (Ω) the Calderón projector D is a map from C ∞ c (∂Ω) ⊗ C 2 to C ∞ (∂Ω) ⊗ C 2 defined by: Df • • = γ • G(f 1 (dV ol ĝ ) -1 dS -f 0 (dV ol -1 ĝ )∂ * ν dS), f = f 0 f 1 ∈ C ∞ c (∂Ω) ⊗ C 2 ,
where

G = P -1 . It is easy to see that f ∈ C ∞ (Σ) ⊗ C 2 equals γu for some u ∈ C ∞ (Ω) solution of P u = 0 in Ω if and only if Df = f .
In our case we take N = S β × Σ + , where S β is the circle of length β and

Σ + = Σ ∩ M + is the right part of the Cauchy surface Σ. The Riemannian metric is ĝ = v 2 (y)dτ 2 +h ij (y)dy i dy j , obtained by the Wick rotation t = • • iτ of the Lorentzian metric g = -v 2 (y)dt 2 + h ij (y)dy i dy j on M + ∼ R × Σ + where M + is identified to R × Σ + using the Killing time coordinate t.
The existence of an extension of ω βH to M is then an almost immediate consequence of the fact that (N, ĝ) admits a smooth extension (N ext , ĝext ) if and only if β = β H , a well-known result which plays also a role in [S1].

In fact this geometrical fact implies that D, viewed as an operator defined on C ∞ c (Σ\B)⊗C 2 uniquely extends to a Calderon projector D ext , defined on C ∞ c (Σ)⊗ C 2 . From D ext one can then easily obtain a pure quasi-free state ω HHI on the whole of M .

The Hadamard property of ω HHI follows then from the well-known fact that D ext , being a Calderón projector, is a 2 × 2 matrix of pseudodifferential operators on Σ, and of the Hadamard property of ω β in M + ∪ M -.

Beside shortening the proof of the Hadamard property of ω HHI , we think that our paper illustrates the usefulness of pseudodifferential calculus for the construction and study of Hadamard states, see also [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF], [START_REF] Gérard | Hadamard states for the linearized Yang-Mills equation on curved spacetime[END_REF], [START_REF] Gérard | Hadamard property of the in and out states for Klein-Gordon fields on asymptotically static spacetimes[END_REF], [GOW] for other applications. We believe that Calderón projectors could also be used to construct the Hartle-Hawking-Israel state in the still open case of spacetimes with a Killing horizon that is only stationary.

1.2. Plan of the paper. Let us now briefly give the plan of the paper. In Sect. 2 we recall the notion of a static bifurcate Killing horizon, following [S1] and introduce the associated Klein-Gordon equation.

Sect. 3 is devoted to background material on CCR * -algebras, bosonic quasi-free states and their spacetime and Cauchy surface covariances in the case of quantum Klein-Gordon fields. We use the framework of charged fields, which is in our opinion more elegant, even when considering only neutral field equations. We also recall the notion of pseudodifferential operators on a manifold, which will be useful later on and formulate a consequence of [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF] which states that the Cauchy surface covariances of any Hadamard state for Klein-Gordon fields is given by a matrix of pseudodifferential operators.

In Sect. 4 we define various 'Euclidean' Laplacians, K = -∆ ĝ + m acting on N = S β × Σ + and a related operator K, obtained from Wick rotation of the Lorentzian metric on M in the Killing time coordinate, which are considered in [S1]. It is sufficient for us to define these Laplacians by quadratic form techniques, which simplifies some arguments.

In Sect. 5 we recall the definition of the Calderón projector, which is a standard notion in elliptic boundary value problems. In Sect. 6, using the explicit expression for K-1 , we show that the projection associated to the double β-KMS state ω β equals to the Calderón projector D associated to K and the open set

Ω =]0, β/2[×Σ + .
In Sect. 7, we recall the well-known fact that a smooth extension (N ext , ĝext ) of (N, ĝ) exists iff β = β H . The extended Calderón projector D ext generates a pure state on M , which is the Hartle-Hawking-Israel state ω HHI . In Prop. 7.4, we show that such an extension is unique among quasi-free states whose spacetime covariances map C ∞ c (M ) into C ∞ (M ) continuously. Finally we give in the proof of Thm. 7.5 a new and elementary proof of the Hadamard property of ω HHI , using the pseudodifferential calculus on Σ.

2. spacetimes with a static bifurcate Killing horizon 2.1. Static bifurcate Killing horizons. We consider as in [S1] a globally hyperbolic spacetime (M, g) with a static bifurcate Killing horizon. We recall, see [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Def. 2.2], that this is a triple (M, g, V ), such that (1) the Lorentzian manifold (M, g) is globally hyperbolic, (2) V is a complete Killing vector field for (M, g), (3) B • • = {x ∈ M : V (x) = 0} is an compact, orientable submanifold of codimension 2, (4) there exists a Cauchy hypersurface Σ containing B, (5) V is g-orthogonal to Σ, see Figure 1 below where the vector field V is represented by arrows.

Σ M + M - F P H + H - H - H + B Figure 1.
For simplicity we will also assume that the bifurcation surface B is connected. Denoting by n the future pointing normal vector field to Σ one introduces the lapse function:

(2.1) v(x) • • = -n(x)•g(x)V (x), x ∈ Σ,
and Σ decomposes as

Σ = Σ -∪ B ∪ Σ + ,
where Σ ± = {x ∈ Σ : ±v(x) > 0}. The spacetime M splits as

M = M + ∪ M -∪ F ∪ P,
where the future cone F

(1) R • R = Id, (2)
R is an isometry of (M + ∪ M -, g) onto itself, which reverses the time orientation, [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Prop. 2.7], and we denote by r the restriction of R to Σ. Denoting by h the induced Riemannian metric on Σ one has:

(3) R = Id on B, (4) R * V = V on M + ∪ M -. It follows that R preserves Σ, see [
(2.2) r * h = h, r * v = -v.

Killing time coordinate.

Denoting by Φ V s : M → M the flow of the Killing vector field V , we obtain a diffeomorphism

χ : R × (Σ\B) (t, y) → Φ V t (y) ∈ M + ∪ M -,
which defines the coordinate t on M + ∪ M -called the Killing time coordinate. The metric g on M + ∪ M -pulled back by χ takes the form (see [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Subsect. 2.1]):

(2.3)

g = -v 2 (y)dt 2 + h ij (y)dy i dy j ,
where the Riemannian metric h ij (y)dy i dy j is the restriction of g to Σ.

2.4. Klein-Gordon operator. We fix a real potential m ∈ C ∞ (M ). As in [S1] we assume that m is stationary w.r.t. the Killing vector field V and invariant under the wedge reflection, ie:

(2.4)

V a ∇ a m(x) = 0, m • R(x) = m(x), x ∈ M + ∪ M -∪ U.
For simplicity we also assume that (2.5) m(x) ≥ m 2 0 > 0, x ∈ M, ie we consider only massive fields. Note that in [S1] the weaker condition m(x) > 0 was assumed. We consider the Klein-Gordon operator (2.6) P = -2 g + m.

Free Klein-Gordon fields

In this section we briefly recall some well-known background material on free quantum Klein-Gordon fields on globally hyperbolic spacetimes. We follow the presentation in [GW1, Sect. 2] based on charged fields.

Charged CCR algebra.

3.1.1. Charged bosonic fields. Let Y a complex vector space, Y * its anti-dual. Sesquilinear forms on Y are identified with elements of L(Y, Y * ) and the action of a sesquilinear form β is correspondingly denoted by y 1 •βy 2 for y 1 , y 2 ∈ Y. We fix q ∈ L h (Y, Y * ) a non degenerate hermitian form on Y, ie such that Ker q = {0}.

The CCR * -algebra CCR(Y, q) is the complex * -algebra generated by symbols 1, ψ(y), ψ * (y), y ∈ Y and the relations:

ψ(y 1 + λy 2 ) = ψ(y 1 ) + λψ(y 2 ), y 1 , y 2 ∈ Y, λ ∈ C, ψ * (y 1 + λy 2 ) = ψ * (y 1 ) + λψ * (y 2 ), y 1 , y 2 ∈ Y, λ ∈ C, [ψ(y 1 ), ψ(y 2 ] = [ψ * (y 1 ), ψ * (y 2 )] = 0, [ψ(y 1 ), ψ * (y 2 )] = y 1 •qy 2 1, y 1 , y 2 ∈ Y, ψ(y) * = ψ * (y), y ∈ Y. A state ω on CCR(Y, q) is (gauge invariant) quasi-free if ω( p i=1 ψ(y i ) q i=1 ψ * (y j )) = 0 if p = q, σ∈Sp p i=1 ω(ψ(y i )ψ * (y σ(i) )) if p = q.
There is no loss of generality to restrict oneself to charged fields and gauge invariant states, see eg the discussion in [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF]Sect. 2]. It is convenient to associate to ω its (complex) covariances λ ± ∈ L h (Y, Y * ) defined by:

ω(ψ(y 1 )ψ * (y 2 )) = • • y 1 •λ + y 2 , ω(ψ * (y 2 )ψ(y 1 )) = • • y 1 •λ -y 2 , y 1 , y 2 ∈ Y.
The following results are well-known, see eg [START_REF] Derezinski | Mathematics of Quantization and Quantum Fields[END_REF]Sect. 17

.1] or [GW1, Sect. 2]: -two hermitian forms λ ± ∈ L h (Y, Y * ) are the covariances of a quasi-free state ω iff (3.1) λ ± ≥ 0, λ + -λ -= q.
-Let Y ω be the completion of Y for the Hilbertian scalar product λ + + λ -. If there exist linear operators c ± ∈ L(Y ω ) such that

c + + c -= 1, (c ± ) 2 = c ± ,
(ie c ± is a pair of complementary projections) and λ ± = ±q • c ± , then ω is a pure state.

3.1.2. Neutral bosonic fields. We complete this subsection by explaining the relationship with the formalism of neutral fields, see eg [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF]Subsect. 2.5]. Let X a real vector space, X # its dual, and σ ∈ L a (X , X # ) a symplectic form on X . The * -algebra CCR(X , σ) is the complex * -algebra generated by symbols 1, φ(x), x ∈ X and relations:

φ(x 1 + λx 2 ) = φ(x 1 ) + λφ(x 2 ), x 1 , x 2 ∈ X , λ ∈ R, [φ(x 1 ), φ(x 2 ] = ix 1 •σx 2 1, x 1 , x 2 ∈ X , φ(x) * = φ(x), x ∈ X .
To relate the neutral to the charged formalism one sets Y = CX and for

β ∈ L(X , X # ) denote by β C ∈ L(Y, Y * ) its sesquilinear extension. Y R ∼ X ⊕ X is the real form of Y, ie Y R = Y as a real vector space. Then (Y R , Reσ C ) ∼ (X , σ)⊕(X , σ) is a real symplectic space and we denote by φ(y), y ∈ Y R the selfadjoint generators of CCR(Y R , Reσ C ). Under the identificationφ(y) ∼ φ(x) ⊗ 1 + 1 ⊗ φ(x ) for y = x + ix we can identify CCR(Y R , Reσ C ) with CCR(X , σ) ⊗ CCR(X , σ) as * -algebras.
Note also that under the identification

ψ(y) ∼ 1 √ 2 (φ(y) + iφ(iy)), ψ * (y) ∼ 1 √ 2 (φ(y) -iφ(iy)), y ∈ Y we can identify CCR(Y R , Reσ C ) with CCR(Y, q) for q = iσ C . A quasi-free state ω on CCR(X , σ) is determined by its real covariance η ∈ L s (X , X # ) defined by: ω(φ(x 1 )φ(x 2 )) = • • x 1 •ηx 2 + i 2 x 1 •σx 2 , x 1 , x 2 ∈ X . A symmetric form η ∈ L s (X , X # ) is the covariance of a quasi-free state iff η ≥ 0, |x 1 •σx 2 | ≤ 2(x 1 •ηx 1 ) 1 2 (x 2 •ηx 2 ) 1 2 , x 1 , x 2 ∈ X .
To such a state ω we associate the quasi-free state ω on CCR(Y R , Reσ C ) with real covariance Reη C . Then its complex covariances λ ± are given by (see [GW1, Subsect. 2.5]):

(3.2) λ ± = η C ± 1 2 iσ C .
Applying complex conjugation, we immediately see that in this case

(3.3) λ + ≥ 0 ⇔ λ -≥ 0,
so it suffices to check for example that λ + ≥ 0.

3.2.

Free Klein-Gordon fields.

Let P = -2 g + m(x), m ∈ C ∞ (M, R)
a Klein-Gordon operator on a globally hyperbolic spacetime (M, g) (we use the convention (1, d) for the Lorentzian signature). Let E ± be the advanced/retarded inverses of P and E • • = E + -E -. We apply the above framework to

Y = C ∞ c (M ) P C ∞ c (M ) , [u]•q[u] = i(u|Eu) M ,
where (u|v) M = ´M uvdV ol g . One restricts attention to quasi-free states on CCR(Y, q) whose covariances are given by distributions on M × M , ie such that there exists

Λ ± ∈ D (M × M ) with (3.4) ω(ψ([u 1 ])ψ * ([u 2 ])) = (u 1 |Λ + u 2 ) M , ω(ψ * ([u 2 ])ψ([u 1 ])) = (u 1 |Λ -u 2 ) M , u 1 , u 2 ∈ C ∞ c (M ).
In the sequel the distributions Λ ± ∈ D (M × M ) will be called the spacetime covariances of the state ω.

In (3.4) we identify distributions on M with distributional densities using the density dV ol g and use the notation

(u|ϕ) M , u ∈ C ∞ c (M ), ϕ ∈ D (M )
for the duality bracket. We have then

(3.5) P (x, ∂ x )Λ ± (x, x ) = P (x , ∂ x )Λ ± (x, x ) = 0, Λ + (x, x ) -Λ -(x, x ) = iE(x, x ).
Such a state is called a Hadamard state , (see [R] for the neutral case and [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF] for the complex case) if

(3.6) WF(Λ ± ) ⊂ N ± × N ± ,
where WF(Λ) denotes the 'primed' wavefront set of Λ, ie

S • • = {((x, ξ), (x , -ξ )) : ((x, ξ), (x , ξ )) ∈ S} for S ⊂ T * M × T * M
, and N ± are the two connected components (positive/negative energy shell) of the characteristic manifold:

(3.7) N • • = {(x, ξ) ∈ T * M \{0}) : ξ µ g µν (x)ξ ν = 0}.
3.3. Cauchy surface covariances. Denoting by Sol sc (P ) the space of smooth space-compact solutions of P φ = 0, it is well known that

[E] : C ∞ c (M ) P C ∞ c (M ) [u] → Eu ∈ Sol sc (P ) is bijective, with i(u 1 |Eu 2 ) = Eu 1 •qEu 2 , u i ∈ C ∞ c (M ), for (3.8) φ 1 •qφ 2 • • = i ˆΣ(∇ µ φ 1 φ 2 -φ 1 ∇ µ φ 2 )n µ dσ,
where Σ is any spacelike Cauchy hypersurface, n µ is the future directed unit normal vector field to Σ and dσ the induced surface density. Setting

ρ : C ∞ sc (M ) φ → φ Σ i -1 ∂ ν φ Σ = f ∈ C ∞ c (Σ) ⊕ C ∞ c (Σ)
Since the Cauchy problem

P φ = 0, ρu = f as a unique solution φ ∈ Sol sc (P ) for f ∈ C ∞ c (Σ) ⊕ C ∞ c (Σ) the map C ∞ c (M ) P C ∞ c (M ) [u] → ρEu ∈ C ∞ c (Σ) ⊕ C ∞ c (Σ)
is bijective, and i(u|Eu

) M = ρEu•qρEu, for (3.9) f •qf • • = ˆΣ f 1 f 0 + f 0 f 1 dσ Σ , f = f 0 f 1 .
It follows that to a quasi-free state with spacetime covariances Λ ± one can associate its Cauchy surface covariances λ ± defined by:

(3.10)

Λ ± = • • (ρE) * λ ± (ρE).
Using the canonical scalar product

(f |f ) Σ • • = ´Σ f 1 f 1 +f 0 f 0 dσ Σ we identify λ ± with operators, still denoted by λ ± , belonging to L(C ∞ c (Σ) ⊕ C ∞ c (Σ), D (Σ) ⊕ D (Σ)).
A more explicit expression of λ ± in terms of Λ ± is as follows, see eg [START_REF] Gérard | Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry[END_REF]Thm. 7.10]: let us introduce Gaussian normal coordinates to Σ

U (t, y) → χ(t, y) ∈ V, where U is an open neighborhood of {0} × Σ in R × Σ and V an open neighborhood of Σ in M , such that χ * g = -dt 2 + h ij (t,
y)dy i dy j . We denote by Λ ± (t, y, t , y ) ∈ D (U × U ) the restriction to U × U of the distributional kernel of Λ ± . By (3.5) and standard microlocal arguments, their restrictions to fixed times t, t , denoted by Λ ± (t, t ) ∈ D (Σ ⊗ Σ) are well defined.

We know also that

∂ k t ∂ k t Λ ± (0, 0) ∈ D (Σ × Σ) is well defined for k, k = 0, 1. Then setting λ ± = • • ±q • c ± we have: (3.11) c ± = ± i∂ t Λ ± (0, 0) Λ ± (0, 0) ∂ t ∂ t Λ ± (0, 0) i -1 ∂ t Λ ± (0, 0)
.

Large classes of Hadamard states were constructed in terms of their Cauchy surface covariances in [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF][START_REF] Gérard | Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry[END_REF] using pseudodifferential calculus on Σ, see below for a short summary.

3.4. Pseudodifferential operators. We briefly recall the notion of (classical) pseudodifferential operators on a manifold, referring to [START_REF] Shubin | Pseudodifferential Operators and Spectral Theory[END_REF]Sect. 4.3] for details.

For m ∈ R we denote by Ψ m (R d ) the space of classical pseudodifferential operators of order m on R d , associated with poly-homogeneous symbols of order m see eg [START_REF] Shubin | Pseudodifferential Operators and Spectral Theory[END_REF]Sect. 3.7].

Let N be a smooth,

d-dimensional manifold. Let U ⊂ N a precompact chart open set and ψ : U → Ũ a chart diffeomorphism, where Ũ ⊂ R d is precompact, open. We denote by ψ * : C ∞ c ( Ũ ) → C ∞ c (U ) the map ψ * u(x) • • = u • ψ(x). Definition 3.1. A linear continuous map A : C ∞ c (N ) → C ∞ (N ) belongs to Ψ m (N ) if the following condition holds: (C) Let U ⊂ N be precompact open, ψ : U → Ũ a chart diffeomorphism, χ 1 , χ 2 ∈ C ∞ c (U ) and χi = χ i • ψ -1 . Then there exists à ∈ Ψ m (R d ) such that (3.12) (ψ * ) -1 χ 1 Aχ 2 ψ * = χ1 à χ2 .
Elements of Ψ m (N ) are called (classical) pseudodifferential operators of order m on N . The subspace of Ψ m (N ) of pseudodifferential operators with properly supported kernels is denoted by

Ψ m c (N ). Note that if Ψ ∞ (c) (N ) • • = m∈R Ψ m (c) (N ), then Ψ ∞ c (N ) is an algebra, but Ψ ∞ (N )
is not, since without the proper support condition, pseudodifferential operators cannot in general be composed.

We denote by T * N \{0} the cotangent bundle of N with the zero section removed.

To A ∈ Ψ m (N ) one can associate its principal symbol σ pr (A) ∈ C ∞ (T * N \{0}), which is homogeneous of degree m in the fiber variable ξ in T * M , in {|ξ| ≥ 1}. A is called elliptic in Ψ m (N ) at (x 0 , ξ 0 ) ∈ T * N \{0} if σ pr (A)(x 0 , ξ 0 ) = 0. If A ∈ Ψ m (N ) there exists (many) A c ∈ Ψ m c (N ) such that A -A c has a smooth kernel.
Finally one says that

(x 0 , ξ 0 ) ∈ essupp(A) for A ∈ Ψ ∞ (N ) if there exists B ∈ Ψ ∞ c (N ) elliptic at (x 0 , ξ 0 ) such that A c • B is smoothing, where A c ∈ Ψ ∞ c (N ) is as above, ie A -A c is smoothing.
3.5. The Cauchy surface covariances of Hadamard states. We now state a result which follows directly from a construction of Hadamard states in [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF]Subsect. 8.2].

Theorem 3.2. Let ω be any Hadamard state for the free Klein-Gordon field on (M, g) and Σ a spacelike Cauchy hypersurface. Then its Cauchy surface covariances λ ± are 2 × 2 matrices with entries in Ψ ∞ (Σ).

Proof. It is well known (see eg [R]) that if ω 1 , ω 2 are Hadamard states, then Λ ± 1 -Λ ± 2 are smoothing operators on M . Using (3.10) this implies that λ ± 1 -λ ± 2 are matrices of smoothing operators on Σ. From the definition of Ψ ∞ (Σ) it hence suffices to construct one Hadamard state ω whose Cauchy surface covariances λ ± are matrices of pseudodifferential operators. The state constructed in [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF]Subsect. 8.2] has this property, as can be seen from [START_REF] Gérard | Construction of Hadamard states by pseudo-differential calculus[END_REF]Equ. (8.2)]. 2

Euclidean operators

The construction of the β-KMS state on M + with respect to the Killing vector field V relies on the Wick rotation, where (R × Σ + , g) is replaced by (S β × Σ + , ĝ):

(4.1) ĝ = v 2 (y)dτ 2 + h ij (y)dy i dy j ,
is the Riemannian metric obtained from (2.3) by setting t = iτ and S β = [0, β[ with endpoints identified is the circle of length β.

In this section we recall various 'Euclidean' operators related to ĝ appearing in [S1, S2]. It will be convenient to construct them by quadratic form techniques.

We set N • • = S β × Σ + , whose elements are denoted by (τ, y). We equip N with the Riemannian metric ĝ in (4.1) and the associated density dV ol ĝ = |v|(y)|h| 1 2 (y)dτ dy. The hypersurface Σ + is equipped with the induced density dV ol h = |h| 1 2 (y)dy.

4.1.

Euclidean operator on N . We consider the operator

K • • = -∆ ĝ + m(y),
for m as in Subsect. 2.4. Note that m depends only on y since m is invariant under the Killing flow. We have

K = -v -2 (y)∂ 2 τ -|v| -1 (y)|h| -1 2 (y)∂ y i |v|(y)|h| 1 2 (y)h ij (y)∂ y j + m(y).
K is well defined as a selfadjoint operator on L 2 (N, dV ol ĝ ) obtained from the quadratic form:

(4.2) Q(u, u) • • = ˆN |v| -2 |∂ τ u| 2 + ∂ i uh ij ∂ j u + m|u| 2 dV ol ĝ ,
which is closeable on C ∞ c (N ), since K is symmetric and bounded from below on this domain. Denoting its closure again by Q and the domain of its closure by Dom Q, K is the selfadjoint operator associated to Q, ie the Friedrichs extension of its restriction to

C ∞ (S β ) ⊗ C ∞ c (Σ + ). We know that u ∈ Dom K, Ku = f iff (4.3) u ∈ Dom Q and Q(w, u) = (w|f ) L 2 (N ) , ∀w ∈ C ∞ c (N ). From (2.5) we know that K ≥ m 2 0 hence is boundedly invertible and we set G • • = K -1 . 4.2. Change of volume form. Let us set Q(u, u) = Q(vu, vu), Dom Q = {u ∈ L 2 (N ) : vu ∈ Dom Q}. By (2.5) we have Q(u, u) ≥ m 2 0 vu 2 . If u n ∈ Dom Q, u ∈ L 2 (N ) with u n -u → 0 and Q(u n -u m , u n -u m ) → 0 then from the inequality above we obtain that vu ∈ L 2 (N ) and v(u n -u) → 0. Since Q is closed we obtain that u ∈ Dom Q and Q(u n -u, u n -u) → 0, ie Q is closed.
Let K be the injective selfadjoint operator associated to Q, (which is formally equal to vKv) and let Ĝ = K-1 . We claim that (4.4)

G = v Ĝv, on v -1 L 2 (N ).
This follows easily from the caracterization (4.3) of G and similarly of Ĝ.

Let now

U : L 2 (N ) → L 2 (S β ) ⊗ L 2 (Σ + ) the unitary map given by U u = v 1 2 u. We set K • • = U KU * .
We have K = -∂ 2 τ + 2 (y, ∂ y ), where:

2 (y, ∂ y ) = -|v| 1 2 (y)|h| -1 2 (y)∂ y i |v|(y)|h| 1 2 (y)h ij (y)∂ y j |v| 1 2 (y) + v 2 (y)m(y),
is obtained as above from the quadratic form (4.5)

ˆΣ+ ∂ i |v| 1 2 ũ|v|h ij ∂ i |v| 1 2 u + |v| 2 m|u| 2 |h| 1 2 dy.
If G • • = K-1 we have by (4.4):

(4.6) G = |v| 1/2 G|v| 3/2 , on v -3/2 L 2 (N ).
We now recall a well known expression for G. Let

F (τ ) = e -τ + e (τ -β) 2 (1 -e -β ) , τ ∈ [0, β[, extended to τ ∈ R by β-periodicity.
In particular we have:

(4.7) F (τ ) = e -|τ | + e (|τ |-β) 2 (1 -e -β ) , τ ∈ [-β, β]
The following expression for G is well-known (see eg [START_REF] Derezinski | Mathematics of Quantization and Quantum Fields[END_REF]Sect. 18.3.2]):

(4.8) Gũ(τ ) = ˆSβ F (τ -τ )ũ(τ )dτ , ũ ∈ L 2 (S β ) ⊗ L 2 (Σ\B).
Note that since 2 ≥ mv 2 by (4.5), we have also -2 ≤ m -1 v -2 by Kato-Heinz theorem hence C ∞ c Σ + ) ⊂ Dom F (τ ).

Calderón projectors

In this section we recall some standard facts on Calderón projectors. We refer the reader to [START_REF] Chazarain | Introduction to the Theory of Linear Partial Differential Operators[END_REF] for details. 5.1. The Calderón projector. Let (N, h) a complete Riemannian manifold and P = -∆ h + m, where m ∈ C ∞ (N ) is a real potential with m(x) ≥ m 2 0 > 0. As in Sect. 4 we construct P as a selfadjoint operator on L 2 (N, dV ol h ) using the quadratic form

(5.1) Q(u, u) = ˆN ∂ i uh ij ∂ j u + m(x)|u| 2 (x)dV ol h .
We obtain that 0 ∈ ρ(P ), hence G • • = P -1 is a bounded operator on L 2 (N, dV ol h ), defined by

(5.2) Q(Gv, w) = (v|w) L 2 (N ) , ∀w ∈ C ∞ c (N ). Let Ω ⊂ N an open set such that ∂Ω = S = n 1 S i
, where S i are the connected components of S and are assumed to be smooth hypersurfaces. We denote by C ∞ (Ω) the space of restrictions to Ω of functions in C ∞ (N ).

We associate to S i the distribution density dS i defined by:

dS i |u • • = ˆSi udσ (i) h , u ∈ C ∞ c (N ),
where dσ

(i)
h is the induced Riemannian density on S i and we set

dS = n i=1 dS i .
We denote by ∂ ν the unit exterior normal vector field to S and set

∂ * ν dS|u • • = dS|∂ ν u , u ∈ C ∞ c (N ). For u ∈ C ∞ (Ω) we set γu • • = u S ∂ ν u S =: γ 0 u γ 1 u . For v ∈ C ∞ c (S) we denote by ṽ ∈ C ∞ c (N ) an extension of v to N such that ũ S = u, ∂ ν ũ S = 0. Definition 5.1. Let f = f 0 f 1 ∈ C ∞ c (S) ⊕ C ∞ c (S)
. We set:

Df • • = γ • G( f1 (dV ol h ) -1 dS -f0 (dV ol -1 h )∂ * ν dS). -The operator D : C ∞ c (S) ⊕ C ∞ c (S) → C ∞ (S) ⊕ C ∞ (S)
is continuous and is called the Calderón projector associated to (P, S).

-The operator D is a 2 × 2 matrix of pseudodifferential operators on S.

Note that dS and ∂ * ν dS are distributional densities, hence (dV ol h ) -1 dS and (dV ol h ) -1 ∂ * ν dS are distributions on N , supported on S. Note also that the Calderón projector is obviously covariant under diffeomorphisms: if χ : (N, h) → (N , h ) is an isometric diffeomorphism with S = χ(S), P = χ * P , then D = ψ * D , where ψ : S → S is the restriction of χ to S.

Expression in Gaussian normal coordinates.

Let U i be a neighborhood of {0} × S i in R × S i and V i a neighborhood of S i in N such that Gaussian normal coordinates to S i induce a diffeomorphism:

χ i : U i x → (s, y) ∈ V i from U i to V i , and ds 2 + k s (y)dy 2 = χ * i h on U i . Then for f ∈ C ∞ c (S i ) ⊗ C 2 we have (5.3) χ * i f1 (dV ol h ) -1 dS -f0 (dV ol h ) -1 ∂ * ν dS = δ 0 (s) ⊗ (f 1 (y) -r 0 (y)f 0 (y)) -δ 0 (s) ⊗ f 0 (y), where r s (y) = |k s | -1 2 (y)∂ s |k s | 1 2 . If ϕ ∈ C ∞ c (R) with ϕ ≥ 0, ´ϕ(s)ds = 1, setting ϕ n (s) = nϕ(ns), we can compute Df for f ∈ C ∞ c (S i ) ⊗ C 2 as (5.4) Df = lim n→+∞ γ • G(ϕ n (s) ⊗ (f 1 (y) -r 0 (y)f 0 (y)) -ϕ n (s) ⊗ f 1 (y)),
where the limit takes place in C ∞ (S) ⊕ C ∞ (S). Note that it is not obvious that Df ∈ C ∞ (S) ⊕ C ∞ (S). To prove it one can first replace G by a properly supported pseudodifferential parametrix P (-1) ∈ Ψ -2 c (N ). Using then Gaussian normal coordinates near a point x 0 ∈ S, one is reduced locally to N = R d , S = {x 1 = 0}. The details can be found for example in [START_REF] Chazarain | Introduction to the Theory of Linear Partial Differential Operators[END_REF].

Another useful identity is the following: for u ∈ C ∞ (Ω) let Iu be the extension of u by 0 in N \Ω. Then (5.5) P Iu = f1 (dV ol h ) -1 dS -f0 (dV ol -1 h )∂ * ν dS + IP u, for f = γu.

The double β-KMS state

In this section we consider the double β-KMS state ω β in M + ∪ M -. It is obtained as the natural extension to M + ∪ M -of the state ω + β in M + , which is a β-KMS state in M + with respect to the Killing flow . Its construction, for the more general stationary case is given in [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Thm. 3.5].

Since Σ\B is a Cauchy surface for M + ∪ M -, we associate to ω β its (complex) Cauchy surface covariances on Σ\B λ ± , and (since ω β is a pure state), the pair of complementary projections c ± = ±q -1 • λ ± , see Subsect. 3.1. We will study in details the projection c + .

We identify

C ∞ c (Σ\B) with C ∞ c (Σ + ) ⊗ C 2 using the map (6.1) R : C ∞ c (Σ + ) ⊗ C 2 → C ∞ c (Σ + ) ⊕ C ∞ c (Σ -) g = g (0) ⊕ g (β/2) → f = g (0) ⊕ r * g (β/2) ,
where r : Σ → Σ is the restriction to Σ of the wedge reflection R, see Subsect. 2.2.

We will show that

C • • = R-1 • c + • R
is exactly the Calderón projector for the Euclidean operator K + acting on (N, ĝ), see Subsect. 4.1, and the open set

Ω • • = {(τ, y) ∈ N : 0 < τ < β/2}.
6.1. The double β-KMS state. We recall now the expression of ω β given by Sanders, see [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Sect. 3.3].

There are some differences in signs and factors of i with the expression given by Sanders in [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Sect. 3.3]. They come from two differences between our convention for quantized Klein-Gordon fields and the one of Sanders:

-our convention for Cauchy data of a solution of P u = 0 is given the map

ρu = u Σ i -1 ∂ ν u Σ =:
f , which is more natural for complex fields and leads to a more symmetric formulation of the Hadamard condition, while Sanders uses ρu = u Σ ∂ ν u Σ = g, so f = 1 0 0 -i g.

-we use as complex symplectic form f •σf = (f 1 |f 0 ) -(f 0 |f 1 ), while Sanders uses g •σg = (g 0 |g 1 ) -(g 1 |g 0 ). In terms of spacetime fields, we use i -1 E, Sanders uses iE.

Let us unitarily identify L 2 (Σ, |h|

1 2 dy) with L 2 (Σ + , |h| 1 2 dy) ⊕ L 2 (Σ -, |h| 1 2 dy), by u → u + ⊕ u -, u ± = u Σ ± .
Under this identification the action of the wedge reflection r * u = u • r will be denoted by T , with:

(6.2) T (u + ⊕ u -) • • = r * u -⊕ r * u + .
A direct comparison with the formulas in [S1, Sect. 3.3], using the identity (3.2) gives the following proposition.

Proposition 6.1. The double β-KMS state on M + ∪ M -is given by the Cauchy

surface covariance λ + = λ + 00 λ + 01 λ + 10 λ + 11
where:

(6.3)

λ + 00 = 1 2 |v| 1 2 -1 coth( β 2 ) + -1 T sh -1 ( β 2 ) |v| 1 2 , λ + 11 = 1 2 |v| -1 2 coth( β 2 ) -T sh -1 ( β 2 ) |v| -1 2 , λ + 01 = λ + 10 = 1 2 1.
As in Subsect. 3.1 we have λ -= λ + -q, where the charge q = iσ is given by the matrix q = 0 1 1 0 . We introduce the operators c ± • • = ±q -1 λ ± and obtain (6.4)

c + = 1 2 λ + 00 λ + 11 1 2 . Note that if b 0 = -1 coth( β 2 ) + -1 T sh -1 ( β 2 ), b 1 = coth( β 2 ) -T sh -1 ( β 2 ),
then using that [T, ] = 0 we obtain that

b 0 b 1 = b 1 b 0 = coth( β 2 ) 2 -sh -1 ( β 2 ) 2 = 1,
from which it follows easily that c ± are (formally) projections. This is expected since the double β-KMS state ω β is a pure state in M + ∪ M -.

6.2. Conjugation by R. The map R defined in (6.1) allows to unitarily identify

L 2 (Σ + ) ⊗ C 2 with L 2 (Σ + ) ⊕ L 2 (Σ -).
We have:

(6.5) R-1 R = + ⊕ + , R-1 T R = 0 1 1 0 .
Denoting by c + ij for i, j ∈ {0, 1} the entries of the matrix c + and setting

C ij • • = R-1 • c + ij • R
, we obtain after an easy using (6.3), (6.1): (6.6)

C 00 g 0 = 1 2 g (0) 0 ⊕ 1 2 g (β/2) 0 , C 11 g 1 = 1 2 g (0) 1 ⊕ 1 2 g (β/2) 1 , C 01 g 1 = 1 2 |v| 1 2 -1 + coth( β 2 + )|v| 1 2 g (0) 1 + 1 2 |v| 1 2 -1 + sh -1 ( β 2 + )|v| 1 2 g (β/2) 1 ⊕ 1 2 |v| 1 2 -1 + coth( β 2 + )|v| 1 2 g (β/2) 1 + 1 2 |v| 1 2 -1 + sh -1 ( β 2 + )|v| 1 2 g (0) 1 , C 10 g 0 = 1 2 |v| -1 2 + coth( β 2 + )|v| -1 2 g (0) 0 -1 2 |v| -1 2 + sh -1 ( β 2 + )|v| -1 2 g (β/2) 0 ⊕ 1 2 |v| -1 2 + coth( β 2 + )|v| -1 2 g (β/2) 0 -1 2 |v| -1 2 + sh -1 ( β 2 + )|v| -1 2 g (0)
0 . In (6.6) the upper indices (0), (β/2) refer to the two connected components {τ = 0} and {τ = β/2} of ∂Ω, while the lower indices 0, 1 refer to the two components of g.

6.

3. The Calderón projector. We now compute the Calderón projector for K + , associated to the Riemannian manifold (N, ĝ). We choose

Ω = {(τ, y) ∈ N : 0 < τ < β/2}. We have S = ∂Ω = S 0 ∪S β/2 and we write f ∈ C ∞ c (S)⊕C ∞ c (S) as f = f (0) ⊕f (β/2) for f (i) ∈ C ∞ c (S i ) ⊕ C ∞ c (S i
). We denote by γ (i) , i = 0, β/2 the trace operators on S i defined by γu = γ (0) u ⊕ γ (β/2) u for u ∈ C ∞ (Ω). We have: (6.7) τ, y) .

γ (0) u = lim τ →0 + u(τ, y) -|v(y)| -1 ∂ τ u(τ, y) , γ (β/2) u = lim τ →(β/2) - u(τ, y) |v(y)| -1 ∂ τ u(
We denote similarly by ∂

N ext → R with m ext ≥ m 2 0 > 0 such that ψ Σ\B = χ • r, ψ * m ext = m N . Σ + Σ - B S β Σ + ∼r(Σ -) Σ + S β × Σ + R 2 × B χ Figure 2. The embedding χ
This fundamental fact is well explained in [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Subsect. 2.2]. Let us briefly recall the construction of χ following [S1]: one introduces Gaussian normal coordinates to B in (Σ, h), where h is the Riemannian metric induced by g on Σ. We choose the unit normal vector field to B pointing towards Σ + . Using these coordinates we can, since B is compact, identify a small neighborhood U of B in Σ with ] -δ, δ[×B.

Denoting by ω local coordinates on B we have a map

φ :] -δ, δ[×B (s, ω) → y = exp h ω (s) ∈ U, U + = φ(]0, δ[×B), φ * h = ds 2 + k αβ (s, ω)dω α dω β ,
for U + = Σ + ∩ U . In the local coordinates (τ, s, ω) on S β × U + the embedding χ takes the form:

(7.2) χ : S β ×]0, δ[×B → B 2 (0, δ) × B (τ, s, ω) → (s cos(β(2π) -1 τ ), s sin(β(2π) -1 τ ), ω) = • • (X, Y, ω),
where [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Subsect. 2.2] shows that ĝ admits a smooth extension ĝext to N ext iff β = (2π)κ -1 .

B 2 (0, δ) = {(X, Y ) ∈ R 2 : 0 < X 2 +Y 2 < δ 2 }. A straighforward computation performed in [
7.2. The extension of ω β to M . We recall from Subsect. 4.1 that K is defined from the closure Q of the quadratic form Q on C ∞ c (N ). Similarly K ext = -∆ ĝext +m ext , acting on N ext is defined using the corresponding quadratic form Q ext .

The following lemma is equivalent to [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Prop. 5.2], for completeness we give a short proof using quadratic form arguments (note that we assume the stronger condition that inf m(x) > 0).

Lemma 7.2. Let U : C ∞ c (N ) → C ∞ c (N ext \B ext ) defined by U u = u • χ -1 .
Then U extends as a unitary operator U :

L 2 (N ) → L 2 (N ext ) with K ext = U KU * .
Proof. U clearly extends as a unitary operator. To check the second statement it suffices, taking into account the way K and K ext are defined, to prove that

C ∞ c (N ext \B ext ) is a form core for Q ext . The domain of Q ext is the Sobolev space H 1 (N ext ) associated to ĝext , so we need to show that C ∞ c (N ext \B ext ) is dense in H 1 (N ext ). Using the coordinates (X, Y, ω) near B ext ∼ {0} × B, this follows from the fact that C ∞ c (R 2 \{0}) is dense in H 1 (R 2
), see eg [A,Thm. 3.23]. 2 We recall that the projection c + associated to the double β-KMS state ω β was defined in (6.4). Let us identify in the sequel Σ with

Σ ext = ψ(Σ) ⊂ N ext . Theorem 7.3. Let D ext the Calderón projector for (K ext , Σ). Then for f, g ∈ C ∞ c (Σ\B) ⊗ C 2 we have: (g|c + g) L 2 (Σ) = (g ext |D ext f ext ) L 2 (Σ) ,
where

f ext = (ψ * ) -1 f , g ext = (ψ * ) -1 g.
Proof. This follows from Prop. 6.2, the fact that R is implemented by the embedding r of Σ\B into N , (see (7.1)) and Lemma 7.2. 2 7.3. Uniqueness of the extension. We discuss now the uniqueness of extensions of ω β to M . Other types of uniqueness results were obtained before in [KW] and [K1].

Proposition 7.4. There exists at most one quasi-free state ω for the Klein-Gordon field on M such that:

(1) the restriction of ω to M + ∪ M -equals ω β , (2) the spacetime covariances

Λ ± of ω map C ∞ c (M ) into C ∞ (M ). Proof.
Let ω a quasi-free state for the Klein-Gordon operator P in M , with spacetime covariances Λ ± . We assume that Λ

± : C ∞ c (M ) → C ∞ (M ).
Denoting by Λ ± (x, x ) their Schwartz kernels, we have P (

x, ∂ x )Λ ± (x, x ) = P (x , ∂ x )Λ ± (x, x ) = 0, which implies that (7.3) WF(Λ ± ) ⊂ N × N ,
where N is defined in (3.7). We claim that the entries c

± k,k , k, k = 0, 1 of c ± defined in (3.11) map C ∞ c (Σ) into C ∞ (Σ).
In fact by (7.3) we have Λ ± = Λ ± • A modulo smoothing, where A ∈ Ψ 0 (M ) is a pseudodifferential operator with essupp(A) included in an arbitrary small conical neighborhood of N . For u ∈ C ∞ c (Σ) we have, modulo factors of i:

c ± k,k u = ∂ k t Λ ± • A(-∂ k t δ 0 ⊗ u) t=0 , see (3.11). Since WF((-∂ k t δ 0 ) ⊗ u) ⊂ N * Σ, where N * Σ ⊂ T * M is the conormal bundle to Σ and Σ is spacelike, we have N * Σ ∩ N = ∅, hence A(-∂ k t δ 0 ⊗ u) ∈ C ∞ (M ), which proves our claim.
Let now ω i , i = 1, 2 be two quasi-free states as in the proposition. Since (u|(Λ

+ 1 - Λ + 2 )v) L 2 (M ) = 0 for u, v ∈ C ∞ c (M + ∪M -) we obtain that (f |(λ + 1 -λ + 2 )g) L 2 (Σ)⊗C 2 = 0 for f, g ∈ C ∞ c (Σ\B) ⊗ C 2 hence supp(λ + 1 -λ + 2 )g ⊂ B for g ∈ C ∞ c (Σ\B) ⊗ C 2 . Since we have seen that λ + i : C ∞ c (Σ) ⊗ C 2 → C ∞ (Σ) ⊗ C 2 this implies that (λ + 1 - λ + 2 )f = 0 for f ∈ C ∞ c (Σ\B) ⊗ C 2 . Since λ + i are selfadjoint for L 2 (Σ, dV ol h ) ⊗ C 2 this implies that supp(λ + 1 -λ + 2 )f ⊂ B for f ∈ C ∞ c (Σ) ⊗ C 2 , hence (λ + 1 -λ + 2 )f = 0 using again that λ + i : C ∞ c (Σ) ⊗ C 2 → C ∞ (Σ) ⊗ C 2 . 2 7
• • = q • D ext , λ - HHI • • = λ + HHI -q
, where D ext is the Calderón projector for (K ext , Σ) and the charge quadratic form q is defined in (3.8). Then:

(1) λ ± HHI are the Cauchy surface covariances for the Cauchy surface Σ of a quasifree state ω HHI for the free Klein-Gordon field on M . where ũ is identified with the map S β τ → ũ(τ ) ∈ L 2 (S β ; L 2 (Σ + )). This proves (7.4). By Lemma 7.2 and using that G ext = K -1 ext is bounded on L 2 (N ext ), we deduce from (7.4) that G ext is also reflection positive, ie (7.6) (R ext u|G ext u ext ) L 2 (Next) ≥ 0, u ∈ L 2 (N ext ), suppu ⊂ N + ext = χ([0, β/2] × Σ + ), for R ext = U RU * . By the remark before [START_REF] Sanders | On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon[END_REF]Thm. 5.3], if (s, y) are Gaussian normal coordinates to Σ in N ext we have R ext u(s, y) = u(-s, y), ie R ext is given by the reflection in Gaussian normal coordinates. This map is a isometry of (N ext , ĝext ), which implies that if ĝext = ds 2 + h ext (s, y)dy 2 near Σ, we have h ext (s, y) = h ext (-s, y) hence if r s (y) = |h ext (s, y)| -1 2 ∂ s |h ext (s, y)| 1 2 we have r 0 (y) ≡ 0.

If f ∈ C ∞ c (Σ) ⊗ C 2 it follows from (5.3) that D ext f = γG ext T f for T f = δ 0 (s) ⊗ f 1 -δ 0 (s) ⊗ f 0 .

We have R ext T f = δ 0 (s) ⊗ f 1 + δ 0 (s) ⊗ f 0 . Applying the reflection positivity (7.6) to u = T f we obtain that:

(R ext T f |G ext T f ) L 2 (Next) = (f |qD ext f ) L 2 (Σ) ≥ 0,
which proves the positivity of λ + HHI . To make the argument rigorous is suffices to approximate δ 0 by a sequence ϕ n as in (5.4). This completes the proof of (1).

Let us now prove (2). The fact that ω HHI is the unique extension of ω β to M with the stated properties has been proved in Prop. 7. Moreover we know that the restriction of ω HHI to M + ∪ M -is a Hadamard state. The same is obviously true of the restriction of ω ref to M + ∪ M -. Going to Cauchy surface covariances, this implies that if

χ ∈ C ∞ c (Σ ± ) then χ • (c ± HHI -c ± ref )
• χ is a smoothing operator on Σ. We claim that this implies that c ± HHI -c ± ref is smoothing, which will imply that ω HHI is a Hadamard state.

If fact let a be one of the entries of c ± HHI -c ± ref , which is a scalar pseudodifferential operator belonging to Ψ m (Σ) for some m ∈ R. We know that χ • a • χ is smoothing for any χ ∈ C ∞ c (Σ\B). Then its principal symbol σ pr (a) vanishes on T * (Σ\B) hence on T * Σ by continuity, so a ∈ Ψ m-1 (Σ). Iterating this argument we obtain that a is smoothing, which proves our claim and completes the proof of the theorem. 2

  3.3) associated with a Cauchy surface Σ containing the bifurcation surface B. It turns out that the Cauchy surface covariances λ ± of the double β-KMS state ω β are related to a Calderón projector D. Let us informally recall what is the Calderón projector associated to a elliptic boundary value problem, see Sect. 5 for more details: let (N, ĝ) be a complete Riemannian manifold and P = -∆ ĝ + m(x) for m ∈ C ∞ (N ), m(x) > 0 a Laplace-Beltrami operator. Let also Ω ⊂ M a smooth open set. To Ω is naturally associated the canonical surface density dS, defined by dS|u = ´∂Ω udσ, for u ∈ C ∞ c (M ), where dσ is the induced surface element on ∂Ω. If ∂ ν is the external normal derivative to ∂Ω and γu

ν

  the exterior normal derivatives on S i . Proposition 7.1. [S1, Subsect. 2.2] Assume that β = (2π)κ -1 . Then there exists a smooth complete Riemannian manifold (N ext , ĝext ) and (1) a smooth isometric embedding ψ : Σ → N ext , (2) a smooth isometric embedding χ : (N, ĝ) → (N ext \B ext , ĝext ) for B ext = ψ(B), (3) a smooth function m ext :

  (2) the Hartle-Hawking-Israel state ω HHI is a pure Hadamard state and is the unique extension to M of the double β-KMS state ω β with the property that its spacetime covariancesΛ ± HHI map continuously C ∞ c (M ) into C ∞ (M ). Proof.Let us first prove (1). By (3.3) it suffices to check the positivity of λ + HHI . This was shown in [S1, Thm. 5.3] using reflection positivity. For the reader's convenience, let us briefly repeat the argument:for u ∈ L 2 (N ) we set Ru(τ, y) = u(-τ, y), for τ ∈ [-β/2, β/2] ∼ S β . The operator G = K -1 is reflection positive, i.e. (7.4) (Ru|Gu) L 2 (N ) ≥ 0, ∀u ∈ L 2 (N ), suppu ⊂ [0, β/2] × Σ + .In fact setting ũ = |v| 3/2 u, (7.4) is equivalent to(7.5) (Rũ| Gũ) L 2 (S β )⊗L 2 (Σ + ) ≥ 0, ∀ũ ∈ L 2 (S β ) ⊗ L 2 (Σ + ), suppũ ⊂ [0, β/2] × Σ + .Using (4.7) we obtain(Rũ| Gũ) L 2 (S β )⊗L 2 (Σ + ) e -β ) u 0 ) L 2 (Σ + ) + (u β | 1 2 (1-e -β ) u β ) L 2 (Σ +) , for u 0 = ˆSβ e -τ ũ(τ )dτ, u β = ˆSβ e (τ -β/2) ũ(τ )dτ

  4. It remains to prove that ω HHI is a pure Hadamard state in M . The fact that ω HHI is pure follows from the fact that D ext is a projection. To prove the Hadamard property let us fix a reference Hadamard state ω ref for the Klein-Gordon field in M . By Thm. 3.2 its Cauchy surface covariances on Σ λ ± ref are matrices of pseudodifferential operators on Σ. The same is true of c ± ref = ±q -1 •λ + ref and of c ± HHI , since Calderón projectors are given by matrices of pseudodifferential operators on Σ.

  .4. The Hartle-Hawking-Israel state.

	Theorem 7.5 ([S1]). Let us set
	λ + HHI

Acknowledgments. It is a pleasure to thank Michał Wrochna, Ko Sanders and Francis Nier for useful discussions. We are also grateful to Ko Sanders for his kind comments on a previous version of the manuscript.

We compute the Calderón projector D defined in Subsect. 5.1 using the coordinates (τ, y). Since dS i = |h| 1 2 (y)dy and dV ol ĝ = |v| 1 2 (y)|h| 1 2 (y)dy, we obtain: (6.8) Df = D (0) f ⊕ D (β/2) f, for (6.9)

for i, j ∈ {0, β/2} and k, l ∈ {0, 1} the various entries of D, we obtain: (6.10)

We also set

Proof. We recall that C ij are the entries of R-1 • c + • R. We compute D (i)(j) kl using (6.10) and the explicit formulas (4.7), (4.8) for the kernel G(τ, τ ) of G. Computation of D 00 :

Computation of D 11 :

Hence

D 10 g 0 = C 10 g 0 . This completes the proof of the proposition. 2 7. The Hartle-Hawking-Israel state and its properties 7.1. The smooth extension of (N, ĝ) and the Hawking temperature. The existence of the Hartle-Hawking-Israel state and the definition of the Hawking temperature T H = κ(2π) -1 (where κ is the surface gravity) rely on the existence of an extension (N ext , ĝext ) of (N, ĝ) such that the two components S 0 , S β/2 ∼ Σ + of ∂Ω are smoothly glued together into Σ ⊂ N ext .

The extended Riemannian metric ĝext is smooth iff β = (2π)κ -1 (for other values of β (N ext , ĝext ) has a conic singularity on B).

Let us embed Σ\B into N by: r :

x → (0, x) for x ∈ Σ + , x → (β/2, r(x)) for x ∈ Σ -, Note that for R defined in (6.1) we have (7.1) R = (r) * .

We recall that the function m : Σ → R + was introduced in Subsect. 2.4.