Michel Caffarel
email: caffarel@irsamc.ups-tlse.fr

Pierre Del Moral
email: pierre.del-moral@inria.fr

Luc De Montella
email: luc.de-montella@inria.fr

On the Mathematical foundations of Diffusion Monte Carlo

The Diffusion Monte Carlo method with constant number of walkers, also called Stochastic Reconfiguration as well as Sequential Monte Carlo, is a widely used Monte Carlo methodology for computing the ground-state energy and wave function of quantum systems. In this study, we present the first mathematically rigorous analysis of this class of stochastic methods on non necessarily compact state spaces, including linear diffusions evolving in quadratic absorbing potentials, yielding what seems to be the first result of this type for this class of models. We present a novel and general mathematical framework with easily checked Lyapunov stability conditions that ensure the uniform-in-time convergence of Diffusion Monte Carlo estimates towards the top of the spectrum of Schrödinger operators. For transient free evolutions, we also present a divergence blow up of the estimates w.r.t. the time horizon even when the asymptotic fluctuation variances are uniformly bounded. We also illustrate the impact of these results in the context of generalized coupled quantum harmonic oscillators with non necessarily reversible nor stable diffusive particle and a quadratic energy absorbing well associated with a semi-definite positive matrix force.

Introduction

The many-body Schrödinger equation describes interacting quantum particles. Depending on the domain of application, these particles may represent electrons in solid-state physics or quantum chemistry, nucleons in nuclear physics, atoms in quantum liquid physics, or coupled modes of oscillators in molecular spectroscopy, among the main applications. Except for trivial quantum systems, it is impossible to solve this equation analytically. The diffusion Monte Carlo method (abbreviated DMC) provides a powerful stochastic approach to numerically approximate the ground state energy and wave function of Schrödinger operators.

The DMC methodology has a long and rich history, dating back to its first mention in 1949 by Ulam and Metropolis in [START_REF] Metropolis | The Monte Carlo method[END_REF]. The idea was first implemented by Donsker and Kac [START_REF] Donsker | A sampling method for determining the lowest eigenvalue and the principal eigenfunction of schrödinger's equation[END_REF], and by Kalos [START_REF] Malvin | Monte Carlo calculations of the ground state of three-and fourbody nuclei[END_REF] in the early 1960s. Over the years, the physics community has proposed numerous variants of Diffusion Monte Carlo, known by various names such as Green's function Monte Carlo, [START_REF] Malvin | Monte Carlo calculations of the ground state of three-and fourbody nuclei[END_REF][START_REF] Malvin | Monte Carlo Methods in Statistical Physics[END_REF] Fixed-Node Diffusion Monte Carlo, [START_REF] Reynolds | Fixed-node quantum Monte Carlo for molecules[END_REF], Pure Diffusion Monte Carlo, [START_REF] Caffarel | Treatment of the Schrödinger equation through a Monte Carlo method based upon the generalized Feynman-Kac formula[END_REF][START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF] Stochastic Reconfiguration Monte Carlo, [START_REF] Jack | Observations on the statistical iteration of matrices[END_REF][START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration[END_REF][START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem[END_REF][START_REF] Assaraf | Diffusion Monte Carlo methods with a fixed number of walkers[END_REF] and Reptation Monte Carlo, [START_REF] Stefano | Reptation Quantum Monte Carlo: A method for unbiased ground-state averages and imaginary-time correlations[END_REF] to cite the main ones. Despite their apparent diversity, all these approaches are fundamentally based, in one way or another, on the stochastic simulation of a specific implementation of the Feynman-Kac formula with importance sampling.

For a more detailed discussion on the origins and the applications of these Monte Carlo techniques in physics we refer the reader to the recent review article [START_REF] Mareschal | The early years of quantum Monte Carlo (1): the ground state[END_REF] as well as to [START_REF] Ma | Ground state baryons in the fluxtube three-body confinement model using diffusion Monte Carlo[END_REF][START_REF] Matthew Wmc Foulkes | Quantum Monte carlo simulations of solids[END_REF] and references therein.

The version of interest employed here is the DMC method with a fixed number of walkers, commonly known in physics as Stochastic Reconfiguration Monte Carlo; see the pioneering article by Hetherington [START_REF] Jack | Observations on the statistical iteration of matrices[END_REF], followed by Sorella and co-authors [START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration[END_REF][START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem[END_REF] and by the first author and his co-workers in [START_REF] Assaraf | Diffusion Monte Carlo methods with a fixed number of walkers[END_REF].

In mathematics, the methodology may also be referred to by different names, such as genetic algorithm with selection and mutation, population Monte Carlo or sequential Monte Carlo [START_REF] Del | Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups[END_REF][START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF][START_REF] Rousset | On the control of an interacting particle estimation of schrödinger ground states[END_REF][START_REF] Cancés | Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation[END_REF][START_REF] Makrini | Diffusion monte carlo method: Numerical analysis in a simple case[END_REF]. For a more thorough discussion on these application model areas we refer to the books [START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF][START_REF] Del | Mean field simulation for Monte Carlo integration[END_REF] and references therein.

These sequential Monte Carlo methods do not rely on biased variational techniques. They can be seen as a sophisticated genetic-type Monte Carlo methodology to simulate interacting quantum many-body systems. Various asymptotic results have been derived, including central limit theorems and large deviation principles, see for instance [START_REF] Del | Large deviations for interacting particle systems: applications to non-linear filtering[END_REF][START_REF] Donald | Large deviations for interacting processes in the strong topology[END_REF] and [START_REF] Del | Central limit theorem for nonlinear filtering and interacting particle systems[END_REF][START_REF] Chopin | Central limit theorem for sequential Monte Carlo methods and its application to bayesian inference[END_REF], as well as the books [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF][START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF][START_REF] Del | Mean field simulation for Monte Carlo integration[END_REF] for an overview.

Our work concerns less studied non-asymptotic and time-uniform problems. Recalling that the estimation of ground state energies relies on the limiting behavior of the walkers' evolution in the DMC method, it is therefore crucial to obtain uniform-in-time convergence estimates. Despite its importance, there is a notable gap in the literature and very few results have been proven in this respect. To the best of our knowledge, such uniform controls are mainly valid for compact state space models, see for instance [START_REF] Del | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF][START_REF] Del | Mean field simulation for Monte Carlo integration[END_REF] as well as [START_REF] Whiteley | Sequential Monte Carlo samplers: error bounds and insensitivity to initial conditions[END_REF]. Surprisingly, the theoretical efficiency of the DMC method has never been verified rigorously even in basic linear-Gaussian scenarios such as the simple and well known harmonic oscillator. In this paper, we address this gap by establishing the first uniform-in-time convergence estimates that apply to general state space models including the coupled harmonic oscillators presented in [START_REF] Del | Coupled quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles[END_REF].

Our approach is partly based on recent developments on the stability of positive semigroups presented in [START_REF] Arnaudon | A lyapunov approach to stability of positive semigroups: An overview with illustrations[END_REF][START_REF] Del Moral | On the stability of positive semigroups[END_REF], see also the analysis of generalized coupled harmonic oscillators presented in [START_REF] Del | Coupled quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles[END_REF]. In the present article, we provide a natural Lyapunov condition that ensures the exponential stability of possibly time varying positive semigroups on non necessarily compact state spaces (cf. [START_REF] Reynolds | Fixed-node quantum Monte Carlo for molecules[END_REF] and the local conditions [START_REF] Caffarel | Treatment of the Schrödinger equation through a Monte Carlo method based upon the generalized Feynman-Kac formula[END_REF]). In the context of time homogeneous positive semigroups, these conditions ensure the existence of an unique leading eigen-triple (see for instance [START_REF] Assaraf | Diffusion Monte Carlo methods with a fixed number of walkers[END_REF]). We underline that these results do not rely on any reversibility-type condition, nor on some spectral theorem. They can be seen as an extended version of Perron-Frobenius and Krein-Rutman theorems for possibly time varying positive operators.

We present a nonlinear Markov chain interpretation of the DMC methodologies. In this interpretation, the genetic type evolution of the walkers can be seen as a mean field particle simulation of a nonlinear Markov chain (see Section 2.4). In this context, we present an auxiliary Lyapunov condition that depends on the potential function and the free evolution of the walkers that ensures the time uniform performance of the DMC methodology (cf. condition [START_REF] Reynolds | Fixed-node quantum Monte Carlo for molecules[END_REF], as well as Theorem 1 and Corollary 2). We illustrate this condition in the context of generalized coupled harmonic oscillators for a linear diffusive-type particle and a quadratic energy absorbing well associated with a semidefinite positive matrix force. In this context, we also show that the DMC methodology may diverge when the free evolution of the walkers is unstable for any fixed number of walkers, even if the asymptotic variance of the Central Limit Theorem is uniformly bounded with respect to the time parameter (see Proposition 1, as well as Section 5.3 and Proposition 2).

In the context of absorbing wells centered at the origin, this study leads us to conjecture that stable free evolution transitions is a necessary and sufficient condition for the DMC method to be uniformly convergent w.r.t. the time horizon.

Additionally, we propose and, to some extent, establish the validity of an importance sampling transformation to overcome this difficulty. This type of technique is related but not identical to the use of guiding wave functions in physics to direct the Monte Carlo moves to improve the efficiency of the DMC method [START_REF] Malvin | Monte Carlo calculations of the ground state of three-and fourbody nuclei[END_REF][START_REF] Malvin | Helium at zero temperature with hard-sphere and other forces[END_REF]. In contrast with conventional guiding waves techniques our approach is based on conditional free evolutions transitions and survival weight potential functions (cf. Section 2.4 and Section 5.2).

The rest of the article is organized as follows: In Section 2, we provide a detailed description of the general framework in which our study is set, as well as the theoretical foundations on which our proof will be based.

Section 3 is devoted to the presentation of our main results. Section 4 is mainly concerned the detailed proofs of time-uniform estimates.

Section 5 is devoted to the application of our convergence result to generalized coupled harmonic oscillators [START_REF] Del | Coupled quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles[END_REF]35,[START_REF] Griffiths | Introduction to quantum mechanics[END_REF]. These models arise in various fields such as molecular spectroscopy [START_REF] Herzberg | Molecular Spectra and Molecular Structure: Infrared and Raman of Polyatomic Molecules[END_REF], quantum optics [START_REF] Dmitry | Quantum entanglement and reflection coefficient for coupled harmonic oscillators[END_REF], quantum cryptography [START_REF] Ekert | Quantum cryptography based on bell's theorem[END_REF] and photosynthesis [START_REF] Romero | Quantum coherence in photosynthesis for efficient solar energy conversion[END_REF]. In signal processing, the harmonic oscillator and the DMC methods coincides with the Kalman and the particle filter [START_REF] Del | Non linear filtering: Interacting particle solution[END_REF][START_REF] Kálmán | New results in linear filtering and prediction theory[END_REF].

Description of the models 2.1 Free evolution semigroups

Consider a Markov chain X n indexed by n ∈ N and taking values in a locally compact Polish space (E, E), where E is the Borel σ-field on E. Let C(E) be the algebra of continuous measurable functions on E. We also define C b (E) ⊂ C(E) as the sub-algebra of bounded measurable continuous functions endowed with the supremum norm ∥.∥. With a slight abuse of notation, we denote by 0 and 1 the null and unit scalars as well as the null and unit functions on E and we denote by I : x ∈ E → I(x) = x the identity function on E.

For n ∈ N * , we consider the Markov transitions P n associated with X n , and assume that they are Feller; in the sense that for any f ∈ C b (E) we have P n (f ) ∈ C b (E), with the function P n (f ) defined for any x ∈ E by the integral operator

P n (f )(x) := E P n (x, dy)f (y) = E(f (X n ) | X n-1 = x).
Let C ∞ (E) ⊂ C(E) be the sub-algebra of uniformly positive continuous functions V that grow at infinity; that is, for any r ≥ V ⋆ := inf E V > 0, the r-sub-level set V(r) := {V ≤ r} ⊂ E is a non-empty compact subset. We further assume that there exists a P -Lyapunov function V ∈ C ∞ (E); in the sense that V (E) ⊂ [1, ∞) and there exists ϵ ∈ [0, 1) and c ∈ R such that for any n ∈ N * we have

P n (V ) ≤ ϵV + c. (1)
Let C V (E) ⊂ C(E) be the sub-space of functions f ∈ C(E) such that f /V is bounded, equipped with the norm ∥f ∥ V := ∥f /V ∥. The Markov semigroup associated with the Markov chain X n is defined for any f ∈ C V (E) by

P k,n (f )(x) := E(f (X n ) | X k = x).
Condition (1) ensures that P k,n is V -Feller in the sense that for f ∈ C V (E) we have P k,n (f ) ∈ C V (E). To ensure the semigroup P k,n is exponentially stable [START_REF] Arnaudon | A lyapunov approach to stability of positive semigroups: An overview with illustrations[END_REF], we assume the integral operator P n (x, dy) = p n (x, y)ν(dy) has a density p n w.r.t. some Radon measure ν satisfying for some r 1 > 0 and for any r ≥ r 1 the local minorization condition

0 < inf n∈N * inf V(r) 2 p n ≤ sup n∈N * sup V(r) 2 p n < ∞ and 0 < ν(V(r)) < ∞. (2)
The V -norm semigroup contraction techniques developed in Section 8.2 in [START_REF] Del | Stochastic Processes: From Applications to Theory[END_REF] (see also Lemma 2.3 in [32] and Theorem 2.2 in [START_REF] Arnaudon | A lyapunov approach to stability of positive semigroups: An overview with illustrations[END_REF]), ensure that for any µ ∈ P V (E), there exists some parameters a, b > 0 such that for any k ≤ n, and any µ 1 , µ 2 ∈ P V (E) we have

||µ 1 P k,n -µ 2 P k,n || V ≤ ae -b(n-k) ||µ 1 -µ 2 || V . (3)
Note that the r.h.s condition in ( 2) is met as soon as V has compact sub-level sets with non empty interior and ν is a Radon measure with full support; that is ν is finite on compact sets and strictly positive on non-empty open sets. For time-homogeneous models, the l.h.s. minorization condition is satisfied as soon as (x, y) ∈ (E • ) 2 → p n (x, y) is a continuous positive function on the interior E • of the set E.

Feynman-Kac semigroups

We associate with a sequence of strictly positive functions (G n ) n∈N ∈ C V (E) N the discrete generation Feynman-Kac semigroups

Q k,n (f )(x) = E   f (X n ) n-1 p=k G p (X p ) | X k = x   and Q k,n (f ) := G -1 k Q k,n (G n f ), with G -1 k := 1/G k .
To simplify notation, for k = (n -1) sometimes we write Q n and Q n instead of Q n-1,n and Q n-1,n . In this notation, we have

Q n (f )(x) = G n-1 (x) P n (f )(x) and Q n (f )(x) = P n (G n f )(x).
We also use the convention Q n,n = P n,n = Id, the identity operator.

Let M b (E) be the set of bounded signed measures on E. Also, let P(E) ⊂ M b (E) be the convex subset of probability measures on E and denote by P V (E) the convex set of probability measures µ ∈ P(E) such that µ(V ) < +∞. The left action of Q n on P V (E) is given for any (η, f ) ∈ (P V (E), C V (E)) by the formula

(ηQ n )(f ) := η(Q n (f )) = η(dx)Q n (f )(x) = η(dx)Q n (x, dy)f (y). (4)
By Fubini's theorem, the integration order doesn't matter. Thus to simplify notation, we sometimes write ηQ n (f ) instead of (ηQ n )(f ) or η(Q n (f )). We denote by C 0 (E) := {1/V : V ∈ C ∞ (E)} ⊂ C b (E) the sub-algebra of bounded continuous positive functions h that vanish at infinity; that is, for any 0 < ϵ ≤ ∥h∥ < ∞ the ϵ-super-level set {h ≥ ϵ} ⊂ E is a non empty compact subset.

We further assume the Lyapunov function V introduced in (1) is a Q-Lyapunov function in the sense that (1) holds and there exists Θ ∈ C 0 (E) and a compact subset K ⊂ E such that for any n ≥ 1 we have

Q n (V )/V ≤ Θ and (G n-1 (x) -G n-1 (y))(1 E\K (x)V (x) -1 E\K (y)V (y)) ≤ 0. (5)
Note that the l.h.s. condition in (5) holds as soon as there exists G ∈ C 0 (E) such that for any G n ≤ G, for any n ≥ 0. This condition ensures that for any positive function f ∈ C V (E) and n ≥ 1 we have

Q n (f )/V ≤ Q n (V )/V ≤ Θ.
By (2) the integral operator Q n (x, dy) = q n (x, y) ν(dy) also has a density given by q n (x, y) = G n-1 (x)p n (x, y) and for any r ≥ r 1 we have the local condition

0 < inf V(r) 2 q n ≤ sup V(r) 2 q n < ∞. (6)
Consider the normalized measure valued process η n ∈ P V (E) starting at η 0 ∈ P V (E) defined for any n ≥ 1 by

η n+1 = ϕ n+1 (η n ) := ψ Gn (η n )P n+1 and η n := ψ Gn (η n ), (7)
with the updated Boltzmann-Gibbs transformations ψ Gn associated with the potential function G n defined by

ψ Gn (η n )(dx) := 1 η(G n ) G n (x) η(dx).
We readily check that the evolution semigroup ϕ k,n = ϕ k+1,n • ϕ k associated with the flow of measure η n is given for any k ≤ n by the formula [START_REF] Metropolis | The Monte Carlo method[END_REF] .

ϕ k,n (η k ) = η k Q k,n η k Q k,n
Note that for any µ ∈ P V (E) we have the updating formula

ϕ k,n (η k ) = ψ H µ k,n (η k ) Qk,n , (8)
with the Markov operator

Qk,n (f ) := Q k,n (f )/Q k,n (1) and H
µ k,n (x) := Q k,n (1)(x) ϕ 0,k (µ)Q k,n (1)
.

By Theorem 4.2 in [START_REF] Del Moral | On the stability of positive semigroups[END_REF] (see also Theorem 1 in [START_REF] Whiteley | Stability properties of some particle filters[END_REF] in the context of nonlinear filtering), for any µ ∈ P V (E), there exists some parameters a, b > 0 such that for any k ≤ n, and any µ 1 , µ 2 ∈ P V (E) we have

µ 1 Qk,n -µ 2 Qk,n V ≤ ae -b(n-k) ||µ 1 -µ 2 || V /H µ k,n . (10)
Note that for constant potential functions G n (x) = G n (y) we have Qk,n = P k,n and the above contraction estimates resume to (3). For time homogeneous models Q n = Q, Theorem 4.4 in [START_REF] Del Moral | On the stability of positive semigroups[END_REF] ensures the existence of a leading eigen triple (h,

E 0 , η ∞ ) ∈ C V (E) × R * + × P V (E) , such that Q(h) = E 0 h , η ∞ Q = E 0 η ∞ and η ∞ (h) = 1. ( 11
)

Schrödinger semigroups

The objects defined in the previous subsection are core to a variety of physics problem. Indeed, consider an Hamiltonian differential operator H given by the formula

H := -L + U,
where U is a potential energy function from E to R + , and L is a kinetic energy operator acting on a subset D(L) of C(E). The time dependent Schrödinger equation and the imaginary time version associated with the hamiltonian H are given, respectively, by the equations i ∂ t ψ t (x) = H(ψ t )(x) and -∂ t φ t (x) = H(φ t )(x), with prescribed initial conditions (ψ 0 , φ 0 ). In the above display, i ∈ C stands for the imaginary unit. The right-hand side equation is obtained via a formal time change by setting φ t (x) = ψ -it (x), and can be equivalently written in the following form

∂ t φ t (x) = L(φ t )(x) -U (x)φ t (x) with initial condition φ 0 . ( 12
)
For a twice differentiable function φ 0 , the solution of ( 12) is given by the Feynman-Kac path integral formula

φ t (x) = Q t (φ 0 )(x) := Q t (x, dy) φ 0 (y) (13) = E φ 0 (X t ) exp - t 0 U (X s ) ds | X 0 = x .
In the above display, X t stands for a time homogeneous stochastic process X t on E, with generator L. To facilitate the interpretation of the theoretical and numerical physics in the measure theoretical framework used in this article, we note that the Feynman-Kac propagator defined by the integral operator ( 13) is sometimes written in terms of the exponential of the Hamiltonian operator with the exponential-type symbol

Q t := e -tH or in the bra-kets formalism Q t (φ 0 ) = |e -tH |φ 0 ⟩.
The integral operator Q t is sometimes called the Feynman-Kac propagator. For any s, t ≥ 0 the integral operators Q t satisfy the semigroup property

Q s+t (x, dz) = (Q s Q t )(x, dy) := Q s (x, dz) Q t (z, dy) =⇒ φ s+t = Q s (φ t ).
In terms of left action bra-kets, defining µ φ (dx) := φ(x)dx, Fubini's theorem yields

⟨φ|e -sH |φ t ⟩ = dx φ(x) Q s (x, dy) φ t (dy) = (µ φ Q s )(ψ t ) = µ φ ((Q s Q t )(φ 0 )) = µ φ Q s+t (φ 0 ) = ⟨φ|e -(s+t)H |φ 0 ⟩.
The exponential notation is compatible with finite space models and the matrix notation of the continuous one-parameter semigroup for time homogeneous models. The bra-ket notation (a.k.a. Dirac notation) is also used to represents linear projection forms acting on Hilbert spaces associated with some reversible or some stationary measure, such as the Lebesgue measure for the harmonic oscillator.

The present article deals with different types of non necessarily stationary stochastic processes, including the free evolution process X t discussed in [START_REF] Mareschal | The early years of quantum Monte Carlo (1): the ground state[END_REF]. Apart in the reversible situation in which spectral theorems are stated on the Hilbert space associated with a reversible measure, the use of the exponential symbol or the use of the brakets formalism is clearly not adapted to represent different expectations with respect to different types of stochastic and non-necessarily reversible processes.

To analyze these general stochastic models, we have chosen to only use elementary and standard measure theory notation such as [START_REF] Malvin | Monte Carlo Methods in Statistical Physics[END_REF]. The integral actions of a given integral operator Q t on the right for functions and on the left for measures are clearly compatible with finite space models and matrix notation. The left action µ → µQ t maps measures into measures, while the right action f → Q t (f ) maps functions into functions.

(µQ t )(dy) := E µ(dx)Q t (x, dy) and Q t (f )(x) := E Q t (x, dy)f (y).
In this context the normalized measures are defined for any s ≤ t by the flow of measures

µ t (f ) := Φ s,t (µ s )(f ) := µ s Q t-s (f )/µ s Q t-s (1), ( 14
)
where µ 0 stands for the distribution of the initial random state X 0 . Note that

∂ t µ 0 Q t (1) = -E U (X t ) exp - t 0 U (X s ) ds = -µ 0 Q t (U ) =⇒ ∂ t log µ 0 Q t (1) = -µ t (U ) =⇒ µ 0 Q t (1) = exp - t 0 µ s (U )ds . (15)
The operator Q t defined in ( 13) is sometimes called a Feynman-Kac propagator. However, despite its mathematical elegance, it can rarely be solved analytically. Under some regularity conditions (cf. for instance [START_REF] Del Moral | On the stability of positive semigroups[END_REF]) the flow of measures µ t converge as t → ∞ to some limiting fixed point measure µ ∞ = Φ s,t (µ ∞ ) (a.k.a. quasi-invariant measure). In this case, choosing µ 0 = µ ∞ in [START_REF] Matthew Wmc Foulkes | Quantum Monte carlo simulations of solids[END_REF] we have

µ ∞ Q t (1) = e -λ 0 t with λ 0 := µ ∞ (U ).
Whenever it exists, the ground state h 0 is the leading eigen-function associated with λ 0 ; that is, we have

Q t (h 0 ) = e -λ 0 t h 0 .
On a given time mesh t n = nδ, with time step t n -t n-1 = δ > 0 we clearly have

µ tn (f ) = µ t n-1 Q δ (f ) µ t n-1 Q δ (1) and Q δ (x, dy) = G δ (x) P δ (x, dy),
with the function

G δ (x) := Q δ (1)(x) and the Markov transition P δ (x, dy) := Q δ (x, dy) Q δ (1)(x) . Note that µ ∞ (G δ ) = µ ∞ Q δ (1) = e -λ 0 δ as well as Q δ (h 0 ) = e -λ 0 δ h 0 and µ ∞ Q δ = e -λ 0 δ µ ∞ .
In other words, choosing (G, P ) = (G δ , P δ ) in the time-homogeneous Feynman-Kac model discussed in [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF] and [START_REF] Assaraf | Diffusion Monte Carlo methods with a fixed number of walkers[END_REF], we obtain the leading triple

(η ∞ , E 0 , h) = (µ ∞ , e -λ 0 δ , h 0 ).
This yields the formula

- 1 δ log µ ∞ (G δ ) = λ 0 = µ ∞ (U ).
Unfortunately, with the notable exception of coupled harmonic models (cf. [START_REF] Del | Coupled quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles[END_REF] as well as Proposition 7.1 in [START_REF] Arnaudon | A lyapunov approach to stability of positive semigroups: An overview with illustrations[END_REF]), the potential function G δ can rarely be evaluated and the Markov transition P δ cannot be sampled. The Feynman-Kac measure η n introduced in ( 7) can also be interpreted as the solution of a discrete-time approximation of the formula [START_REF] Mareschal | The early years of quantum Monte Carlo (1): the ground state[END_REF]. Indeed, consider a discrete time approximation X tn of the process X tn and let

Q(x, dy) = G(x)P (x, dy), with G(x) = exp (-U (x)δ) and P (x, dy) = P(X δ tn ∈ dy | X δ t n-1 = x). (16)
In this situation, choosing n = ⌊t/δ⌋ we have

η 0 Q 0,n (f ) = E   f (X δ tn ) 0≤k<n G(X δ t k )   = E   f (X δ tn ) exp    - 0≤k<n U (X δ t k )(t k+1 -t k )      ≃ δ↓0 η 0 Q t (f ).
The leading triple (η ∞ , E 0 , h) associated with the discrete time Feynman-Kac approximation model now depends on the time step δ and we have

(η ∞ , h) ≃ δ↓0 (µ ∞ , h 0 ) and - 1 δ log η ∞ (G) = - 1 δ log η ∞ (e -U δ ) ≃ δ↓0 λ 0 = µ ∞ (U ).
It is clearly out of the scope of this article to analyze the bias introduced by the discrete time approximation discussed above.

Diffusion Quantum Monte Carlo

The Diffusion Quantum Monte Carlo methodology relies on the fact that the flow of measures η n introduced in (7) can be interpreted as the probability distributions η n = Law(X n ) of the random states X n of a nonlinear Markov chain X n . The choice of the Markov chain is far from unique. For instance, we have

η n+1 = ϕ n+1 (η n ) = η n K n+1,ηn ,
with the local Markov transition K n+1,ηn (x, dz) := (S n,ηn P n+1 ) (x, dz)

:= S n,ηn (x, dy) P n+1 (y, dz) = P X n+1 ∈ dz | X n = x .
In the above display, S n,ηn stands for the Markov transition

S n,ηn (x, dy) := ϵ n (η n )G n (x)δ x (dy) + (1 -ϵ n (η n )G n (x)) ψ Gn (η n )(dy).
for some tuning parameter

ϵ n (η n ) ∈ [0, 1] chosen such that ϵ n (η n )G n (x) ∈ [0, 1].
For instance, for ]0, 1]-valued potential functions, we can choose ϵ n (η n ) = 0 as well as ϵ n (η n ) = 1. For more general models, we can also choose the inverse of the η n -essential supremum of G n . Note that the transition X n ⇝ X n+1 depends on the probability distributions η n of the random states X n . In reference with similar nonlinear Markov chain models arising in fluid mechanics, the Markov chain X n is called a McKean interpretation of the flow of measures [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF].

The mean field particle interpretation associated with a given McKean model is defined by a discrete-time system of N walkers ξ n = ξ i n 1≤i≤N . The system starts with N independent copies of a random variable X 0 = X 0 with distribution η 0 . Given the system ξ n at some time n ≥ 0, we sample N conditionally independent walkers ξ i n+1 with their respective distribution

K n+1,η N n (ξ i n , dx) with η N n := 1 N 1≤i≤N δ ξ i n .
In other words, the DMC method consists of approximating the measure η n by using the occupation measure η N n associated with a system of N walkers. The initial positions of the walkers are randomly chosen from the distribution η 0 . The evolution of each walker follows then the following selection/mutation steps:

• Selection: We evaluate the current position ξ i n of a walker and its potential value

G n (ξ i n ). With probability 1 -ϵ n (η N n )G n (ξ i n ) , ξ i
n is killed and instantly replaced by another walker say ξ i n = ξ j n with a probability proportional to G n (ξ j n ) and j ∈ {1, . . . , N }; otherwise we keep it and set ξ i n := ξ i n .

• Mutation: We move the selected walker ξ i n = x to a new location ξ i n+1 = y using the transition kernel P n+1 (x, dy).

The selection transition associated with the choice ϵ n (η N n ) = 0 coincides with the so-called proportional selection/reconfiguration. Note that the walker with the highest potential value is always selected when

ϵ n (η N n ) is the inverse of the η N n -essential supre- mum of G n .
For [0, 1]-valued potential functions G n we can also choose ϵ n (η N n ) = 1. In this situation, particle are killed at a geometric clock that depends on the potential function.

In the context of the discrete time approximating Feynman-Kac models discussed in [START_REF] Del | Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups[END_REF], when the time step δ tends to 0, these geometric killing-rates converges to an exponential killing rate. The limiting DMC scheme in continuous time consists of a system of N walkers. Between killing times, walkers explore the space with a free evolution with generator L; at rate U the walkers are killed and instantly another walker in the pool duplicates (see for instance [START_REF] Del | Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups[END_REF][START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF][START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF][START_REF] Arnaudon | A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces[END_REF][START_REF] Rousset | On the control of an interacting particle estimation of Schrödinger ground states[END_REF]). We underline that without the geometric killing rates, the variance of the proportional reconfiguration associated with the choice ϵ n (η N n ) = 0 blows up when the time step tends to 0. We expect that the occupation measures of the system approximate the solution of the measure-valued process [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF]; that is, in a sense to be given, for any time horizon n ≥ 0 we have

η N n = 1 N N i=1 δ ξ i n -→ N →∞ η n ,
as well as

ψ Gn η N n = N i=1 G n (ξ i n ) 1≤j≤N G n (ξ j n ) δ ξ i n -→ N →∞ ψ Gn (η n ) = η n .
The Lyapunov drift condition (1) combined with the local minorization condition (2) ensures that the free evolution of the walkers is stable, in the sense that it forgets exponentially fast its initial condition (see [START_REF] Malvin | Monte Carlo calculations of the ground state of three-and fourbody nuclei[END_REF]). This condition is not satisfied for linear gaussian processes with an unstable drift matrix. In this context, as shown in Proposition 1 and Proposition 2 (see also Section 5.3) the DMC method diverges for any fixed number of walkers, even if the asymptotic variance of the Central Limit Theorem stated in Lemma 7 is uniformly bounded w.r.t. the time horizon.

Note that the stability regions and the Lyapunov functions are connected to the potential function by [START_REF] Reynolds | Fixed-node quantum Monte Carlo for molecules[END_REF]. Importance sampling techniques and twisted guiding waves can be used to design stable-like free evolutions. For instance, using the decomposition

Q n+1 (f )(x) = G n P n+1 (f ),
with the potential function

G n := P n+1 (G n+1 ) and P n+1 (f ) := P n+1 (G n+1 f ) P n+1 (G n+1 ) ,
we readily check the evolution equation

η n+1 = ϕ n+1 ( η n ) := ψ Gn ( η n ) P n+1 .
This shows that the updated measures evolve as in [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF] by replacing (G n , P n ) by ( G n , P n ).

The DMC associated with these objects is a genetic-type Monte Carlo sampler with selection fitness functions G n and mutation transitions P n . From the mathematical viewpoint, this model coincides with the one discussed in [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF]. Nevertheless, the free evolution of the walkers associated with ( G n , P n ) is now driven by the potential function.

This local conditioning importance sampling strategy if often used to turn infinite energy absorbing wells (a.k.a. hard obstacles) into soft ones [START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF][START_REF] Del | A note on random walks with absorbing barriers and sequential monte carlo methods[END_REF]. More generally, for any given time mesh k n ≤ k n+1 we have

Q kn,k n+1 (f ) = G kn P kn,k n+1 (f ),
with the potential function

G kn := Q kn,k n+1 (1) and P kn,k n+1 (f ) := Q kn,k n+1 (f ) Q kn,k n+1 (1)
.

This yields the formula

η k n+1 (f ) = ϕ kn,k n+1 ( η kn ) = ψ G kn ( η kn ) P kn,k n+1 .
This shows that the updated measures η kn evolve as in [START_REF] Caffarel | Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman-Kac formula. i. formalism[END_REF] by replacing (G n , P n ) by ( G kn , P kn,k n+1 ). The DMC associated with these objects is a genetic-type Monte Carlo sampler with selection fitness functions G kn and mutation transitions P kn,k n+1 . As shown in Section 5.2 (see also Corollary 4) in the context of coupled harmonic oscillators there exists a time mesh for which the mutation transitions P kn,k n+1 and the potential functions G kn satisfy the required stability properties. For more general models, these objects do not have an analytic form. In this context, we can use the unbiased Monte Carlo methodologies discussed in [START_REF] Del | Measure-valued processes and interacting particle systems. application to nonlinear filtering problems[END_REF], see also Section 2.3.2 in [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF], and Section 11.5 in [START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF].

3 Statement of the main results

Some regularity conditions

For f ∈ C b (E), time-uniform L p -convergence of the error made by the DMC method in estimating η n (f ) have been obtained (see for example [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF][START_REF] Del | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF], as well as Chapter 4 in [START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF], Chapter 12 in [START_REF] Del | Mean field simulation for Monte Carlo integration[END_REF] and the more recent article [START_REF] Del Moral | On the stability of positive semigroups[END_REF]) under the strong mixing assumption that there exist

ϵ P ∈ R * + and ϵ G ∈ R * + such that ∀(x 1 , x 2 , n) ∈ E 2 × N, G n (x 1 ) ≥ ϵ G G n (x 2
) and P n (x 1 , dy) ≥ ϵ P P n (x 2 , dy).

A significant consequence of this assumption is a time uniform bound on the potential function defined for some γ ∈ P V (E) and k ∈ N by

G γ k,k+n : x ∈ E → G γ k,k+n (x) := 1/H γ k,k+n (x)
. Unfortunately, these uniform minorization and majorization conditions are rarely satisfied when E is non-compact.

In order to guarantee a time-uniform L p -convergence for more general models including coupled harmonic oscillators, our framework requires to estimate uniformly in time the inverse moments of η N q (H γ q,n ). To do this, we first assume that there exists

γ ∈ P V (E) such that sup n∈N ϕ 0,n (γ) (G n ) < +∞. ( 17
)
For time-homogeneous models, without any further conditions on the potential, condition ( 17) is easily checked with γ = η ∞ . For this scenario, we will then consider in the rest that γ = η ∞ . Moreover, this hypothesis trivially holds if the functions G n are bounded by some constant independent of n.

We assume that there exists

W ∈ C V (E), α ∈]0, 1] and a Q-Lyapunov function W ∈ C V (E) such that Q n (W ) ≥ χ × W and W -α ≤ W , (18)
where χ := sup

(n,x)∈(N * ×E) P n (G n )(x).
For time homogeneous models, this condition can be relaxed into the following

Q(W ) ≥ min {χ, E 0 } × W and W -α ≤ W . (19)
Note that, for time-homogeneous models, the set of functions W ∈ P V (E) such that Q(W ) ≥ E 0 W is non-empty as it contains at least the ground state h. It is also worth noting that it is not necessary to know the exact value of χ nor the one of E 0 in order to prove that ( 18) or ( 19) hold. Indeed, if one of these constants is less than some C ∈ R ∪ {+∞}, then it is sufficient to prove that for any c < C, there exist W c ∈ C V (E) and α c ∈ (0, 1] such that

Q(W c ) ≥ c W c and W -αc c ≤ V.
Finally, we assume that there exists a Q-Lyapunov function V ∈ C(E) and λ ∈ 2N such that

V λ ≤ V . (20)
Without further mention, we assume that V, V and W are integrable with respect to η 0 , i.e., η 0 ∈ P V (E) ∩ P W (E) ∩ P V (E). Under conditions [START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF] and [START_REF] Rousset | On the control of an interacting particle estimation of schrödinger ground states[END_REF] or [START_REF] Cancés | Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation[END_REF], it is then possible to obtain a time uniform bound on the random potential function G γ k,n .

Lemma 1. Let γ and (η, µ) be defined as in [START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF]. There exists β ∈ R * + such that for any β ≤ β we have sup (q,n,N )∈N 3 q≤n E ϕ q (η N q-1 )(H γ q,n ) -β < +∞ and sup

(q,n,N )∈N 3 q≤n E η N q (H γ q,n ) -β < +∞. ( 21
)
The proof of this pivotal Lemma is postponed to the appendix.

A time-uniform convergence Theorem

The main goal of this paper is to establish that in the context we described, the L p -norm of the error made by the DMC method in approximating the Feynman-Kac measures η n remains bounded in time and converges to zero as the number of particles increases.

Our main result can be stated as follows Theorem 1. For any p ∈ N * , there exists c ∈ R and β ∈ (0, 1] such that for any

f ∈ C V λ 4p
(E) we have

sup n∈N E |η n (f ) -η N n (f )| p 1 p ≤ cN -β 2 .
The proof of this theorem is provided in subsection 4.1. For time-homogeneous models, a direct consequence of Theorem 1 is a control over the estimation of the limiting quasi-invariant measure η ∞ .

Corollary 1. For any p ∈ N * , there exists (a, b, c) ∈ R * 3 + and β ∈ (0, 1] such that for any f ∈ C

V λ 4p (E) we have sup n≥a+b ln(N ) E |η ∞ (f ) -η N n (f )| p 1 p ≤ cN -β 2 .
The proof the above Corollary is provided in subsection 4.2 .

Assuming that there exists a measure µ ∈ P(E) that is reversible for P , it becomes possible to obtain a re-normalized weak form of the ground state h and its associated eigenvalue from the limit measures η ∞ and η ∞ of η n and η n .

Indeed, referring to Section 9.5.5 in [START_REF] Del | Stochastic Processes: From Applications to Theory[END_REF] (see also [START_REF] Del | Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups[END_REF] as well as Section 12.4 in [START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF]), we have

η ∞ (G) = E 0 and η ∞ (f ) = µ(P (h)f ) µ(P (h)) = µ(hP (f )) µ(h) = ψ h (µ)P (f ), Note that η ∞ (f ) = µ(Q(h)f /G) µ(h) = E 0 ψ h (µ)(f /G).
In the reverse angle, we have the updated limiting measures

η ∞ (f ) = ψ G (η ∞ )(f ) = ψ h (µ)(f ) = µ(hf ) µ(h) =⇒ η ∞ = η ∞ P
The existence of a reversible measure is actually not required to express the groundstate energy using the limit measure; indeed, we always have

η ∞ (h) = ϕ(η ∞ )(h) = η ∞ (GP (h)) η ∞ (G) = E 0 η ∞ (h) η ∞ (G) ⇒ η ∞ (G) = E 0 .
Those equalities, combined with the convergence stated in Corollary 1, guarantee the efficiency of the DMC method for approximating the ground-state energy and wave function of quantum systems. (E) we have

               sup n≥a+b ln(N ) E |E 0 -η N n (G)| p 1 p ≤ cN -β 2 sup n≥a+b ln(N ) E ψ h (µ)(P (f )) -η N n (f ) p 1 p ≤ cN -β 2 sup n≥a+b ln(N ) E E 0 ψ h (µ)(f /G) -η N n (f ) p 1 p ≤ cN -β 2 .

Coupled harmonic oscillators

To illustrate the practical applications of Theorem 1, we carry out an in-depth study of the generalized coupled harmonic oscillator [START_REF] Del | Coupled quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles[END_REF]. First, we demonstrate its relevance by establishing easily verifiable sufficient conditions for time-uniform control of the DMC method in a framework that includes the harmonic oscillator.

We consider E = R d for some d ∈ N * . For any real definite positive d × d matrices B and S and any a real d × d matrix A, we denote by P A,B and G S the Markov kernel and the potential function defined as follows

P A,B (x, dy) = 1 (2π) k/2 |B| 1/2 e -1 2 (Ax-y) ⊤ B -1 (Ax-y) and G S (x) := exp - x T Sx 2 .
Note that the matrix A may not be symmetric nor a stable (also called Hurwitz) matrix. In addition, the transition P A,B is not necessarily reversible, unless AB = BA T , where A T stands for the transposition of the matrix A.

Consider a family of Feller Markov transitions (P n ) n∈N with positive densities p n , an initial distribution η 0 ∈ P(E) and a family of positive functions (G n ) n∈N ∈ C 0 (R d ) N that is uniformly bounded in time.

We assume the following, where (A, A ′ ) are real matrices, (B, B ′ , S) real positive definite matrices, p A,B the density of P A,B and E A,B,S the ground-state energy associated to the operators P A,B and G S .

• There exists c 1 ∈ R such that for any (n, x, y) ∈ N × R d 2 we have

p n (x, y) ≤ c 1 p A,B (x, y). (22)
• For any (n, x, y) ∈ N × R d 2 we have

G n (x) p n+1 (x, y) ≥ χ E A ′ ,B ′ ,S × G S (x) p A ′ ,B ′ (x, y). (23)
• There exists a compact K ⊂ R d such that for any (n, x, y)

∈ N × R d 2 we have [G S (x) -G S (y)][G -1 n (x)1 R d \K (x) -G -1 n (y)1 R d \K (y)] ≥ 0. ( 24
)
• There exists

c 3 ∈ R * + such that G -c 3 S
is integrable with respect to η 0 .

If P n and G n are time-independent, it is possible to replace χ in ( 23) by the ground state energy associated with P and G.

These conditions hold trivially for the coupled harmonic oscillator, i.e if P n = P A,B , G n = G S and η 0 is a normal distribution.

In this context, Theorem 1 leads to a simple sufficient matrix condition which guarantees the uniform convergence of the DMC method. The proof of the following corollary can be found in subsection 5.1.

Corollary 3.

Assume that A T SA < S. In this situation, for any p ∈ N * , there exist

(β, α, c) ∈ (0, 1] × R * 2 + such that for any function f ∈ C V (R d ) we have sup n∈N E |η n (f ) -η N n (f )| p 1 p ≤ cN -β 2 , with V : x ∈ R d → exp α 2 x T Sx .
Shifting the focus to the approximation of the measures ( η n ) n∈N within the coupled harmonic oscillator framework, the convergence condition given in the previous corollary can be overcome with a change of transition and selection in the DMC method. Considering P = P A,B , G = G S as well as η 0 ∼ N (X 0 , P 0 ) and assuming that A, B and S can be diagonalized in the same basis, we recursively define for any k ≥ 1 the function G (k) ∈ C V (E) and the Markov kernel P (k) on E such that for all f ∈ C V (E) we have

G (k) = P (G G (k-1) ) and P (k) (f ) = P (k-1) (P (f G)) P (k-1) (P (G)) ,
with the convention G (0) = 1 and P (0) = Id for k = 0. Equivalently, for any f ∈ C V (E) and k ≥ 1 we have the formula [START_REF] Metropolis | The Monte Carlo method[END_REF] and

Q 0,k (f ) = G (k) P (k) (f ) with G (k) := Q 0,k (
P (k) (f ) := Q 0,k (f )/ Q 0,k (1)
. ( 25) For k ∈ N * , consider a system of walkers ξ

(k) n = ξ (k),i n 1≤i≤N
associated to the DMC method with initial distribution ψ G (η 0 ), transitions P (k) and selection function G (k) as well as the empirical measures

η (k),N n := 1 N 1≤i≤N δ ξ (k),i n . ( 26
)
This system of walkers offers an approximation of the measures η n for any n ∈ kN. This type of change in the approximation, based on an importance sampling transformation is analogous to using a guiding waves function to direct the Monte Carlo moves. Without any additional condition, Theorem 1 ensures the uniform convergence of the model. The details of the proof can be found in Subsection 5.1.

Corollary 4. Let p ∈ N * , there exists k ∈ N such that for any k ≥ k, there exist

(β, α, c) ∈ (0, 1] × R * 2 + satisfying for any f ∈ C V (R) sup n∈N E | η nk (f ) -η (k),N n (f )| p 1 p ≤ cN -β 2 , with V : x ∈ R d → exp α 2 x T Sx .
Although the method does not provide an approximation for every time step, several strategies can be used to fill the gaps left by the approximation. A simple approach, though more computationally intensive, is to run independent systems of walkers for each time step in the interval 0, k -1 . This method not only fills in the gaps, but also maintains the convergence property.

Our study concludes with a focus on the divergence of the DMC method when approximating the one-dimensional harmonic oscillator. This confirms that the stability condition stated in Corollary 3 is necessary and that, in some cases, the set of assumptions presented can closely approximate a sufficient and necessary condition. Additionally, it emphasizes the significance of the importance sampling method introduced in the previous corollary. Specifically, in the one-dimensional context, the sufficient condition for uniform convergence of the DMC method is expressed by A 2 < 1. Proposition 1 establishes the divergence of the error made by the DMC method when A 2 > 1, leaving open only the case A = 1.

Proposition 1. Assume that A 2 > 1 and P 0 > 0. For any p ∈ N * we have

sup n∈N E |η n (I) -η N n (I)| p 1 p = +∞.
The proof of this proposition can be found in Subsection 5.3. Note that all corollaries in this subsection can be extended to a control on the estimation of the limit measures, ground state, and eigenvalue using the same approach as presented in Corollary 2.

Stochastic interpolation 4.1 Time varying semigroups

In this subsection, we focus on proving Theorem 1. To take advantage of the conditional independence of the walkers, we structure our approach around the following decomposition of the difference between the Feynman-Kac measure and its empirical approximation, using the convention η N -1 = η 0 . Following [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF][START_REF] Del | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF], we use the following stochastic interpolation formula

η N n -η n = n q=0 [ϕ q,n (η N q ) -ϕ q,n (ϕ q (η N q-1 ))]. ( 27
)
Each term on the right-hand side represents the error that occurs when using the DMC approximation instead of the real propagator for a single extra time step. Combining the uniform bound given in Lemma 1 with the contraction property [START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem[END_REF], the following Lemma establishes an exponentially decreasing control for these local errors.

Lemma 2. For any p ∈ N * , there exists (c, ρ, β) ∈ R * 2 + × (0, 1] such that for any function

f ∈ C V λ 4p
(E) and any (N, q, n) ∈ N 3 with q ≤ n we have

E [ϕ q,n (η N q ) -ϕ q,n (ϕ q (η N q-1 ))](f ) p 1 p ≤ ce -(n-q)ρ N -β 2 . ( 28
)
Proof :

Let (η, µ) ∈ P(E) and let γ ∈ P V (E) be defined as in [START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF]. Consider H q,n := H γ q,n as defined in [START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration[END_REF]. Applying the updating formula (8), we obtain

ϕ q,n (η)(f ) -ϕ q,n (µ)(f ) = (ψ Hq,n (η) Qq,n -ψ Hq,n (µ) Qq,n )(f ) = 1 η(H q,n ) (η -µ) H q,n Qq,n [f -ψ Hq,n (µ) Qq,n (f )] .
This yields the formula

ϕ q,n (η)(f ) -ϕ q,n (µ)(f ) = 1 η(H q,n ) (η -µ)(F µ q,n ), (29)
with the function

F µ q,n (x) := H q,n (x) E ψ Hq,n (µ)(dy)[ Qq,n (f )(x) -Qq,n (f )(y)].
Then, applying Hölder's inequality for any β ∈ [0, 1), we obtain the estimate

E ϕ q,n (η N q )(f ) -ϕ q,n (ϕ q (η N q-1 ))(f ) p 1 p ≤ E η N q (H q,n ) -2p (η N q -ϕ q (η N q-1 ))(F N q,n ) 2p(1-β) 1 2p E (η N q -ϕ q (η N q-1 ))(F N q,n ) 2βp 1 2p , with F N q,n := F ϕq(η N q-1 ) q,n
. Using [START_REF] Whiteley | Sequential Monte Carlo samplers: error bounds and insensitivity to initial conditions[END_REF] this yields the estimate

E ϕ q,n (η N q )(f ) -ϕ q,n (ϕ q (η N q-1 ))(f ) p 1 p ≤ E η N q (H q,n ) -2βp ϕ q,n (η N q )(f ) -ϕ q,n (ϕ q (η N q-1 ))(f ) 2p(1-β) 1 2p × E (η N q -ϕ q (η N q-1 ))(F N q,n ) 2βp 1 2p
.

Recalling that f ∈ C V λ 4p (E), this implies that E ϕ q,n (η N q )(f ) -ϕ q,n (ϕ q (η N q-1 ))(f ) p 1 p ≤ E ϕ q,n (η N q )(V λ ) 1 4p + E ϕ q,n (ϕ q (η N q-1 ))(V λ ) 1 4p ×E η N q (H q,n ) -4βp 1 4p E (η N q -ϕ q (η N q-1 ))(F N q,n ) 2βp 1 2p .
From Lemmas 1 and 9, we deduce that, to conclude, it is enough to prove that, for some constant c ∈ R * + independent of n, q and N , we have

E η N q -ϕ q (η N q-1 ) F N q,n 2βp 1 2p < ce -c 2 (n-q) N -β 2 . ( 30
)
Let β ′ = 2pβ λ and assume β small enough so that β ′ < 1/4. For q > 0, the walkers (ξ i q ) 1≤i≤N are independent conditionally to η N q-1 we have

[η N q -ϕ q (η N q-1 )](f ) = η N q   1 N 1≤i≤N h i   with h i := f -S q-1,η N q-1 P q (f )(ξ i q-1 ).
Moreover, we have

ϕ q (η N q-1 ) = 1 N 1≤i≤n µ i with µ i = δ ξ i q-1 S q-1,η N q-1 P q .
Since for any i ∈ 1, N , we have µ i (h i ) = 0, we can apply Lemma 7.3.3 from [START_REF] Del | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems With Applications[END_REF], and deduce that there exists C ∈ R such that

E η N q -ϕ q (η N q-1 ) F N q,n 2βp 1 2p = E E η N q -ϕ q (η N q-1 ) F N q,n λβ ′ | ξ q-1 1 2p ≤ E E η N q -ϕ q (η N q-1 ) F N q,n λ | ξ q-1 β ′ 1 2p ≤ C N β/2 E ϕ q (η N q-1 ) |F N q,n | λ β ′ 1/2p .
For q = 0, the walkers are iid with common distribution η 0 . The previous reasoning therefore holds using the convention E(X | ξ -1 ) = E(X).

Applying the contraction property [START_REF] Sorella | Green function Monte Carlo with stochastic reconfiguration: An effective remedy for the sign problem[END_REF] with µ = δ x and η = δ y we get the existence of (a, ρ) ∈ R 2 + such that

| Qq,n (f )(x) -Qq,n (f )(y)| ≤ ae -ρ(n-q) 1 + V (x) H q,n (x) 1 + V (y) H q,n (y) . (31)
By substituting [START_REF] Arnaudon | A lyapunov approach to stability of positive semigroups: An overview with illustrations[END_REF] into the definition of F q,n and applying Hölder's inequality along with Jensen's inequality, we obtain, for some (a

′ , ρ ′ ) ∈ R * 2 + E ϕ q (η N q-1 ) |F N q,n | λ β ′ 1/2p ≤ a ′ e -ρ ′ (n-q) E ϕ q (η N q-1 ){(H q,n + V ) λ } β ′ ϕ q (η N q-1 ){(H q,n + V )} λβ ′ ϕ q (η N q-1 )(H q,n ) -λβ ′ 1 2p ≤ a ′ e -ρ ′ (n-q) E ϕ q (η N q-1 ){(H q,n + V ) λ } 1 4p E ϕ q (η N q-1 )(H q,n ) -2λβ ′ 1 4p .
From our hypothesis on Q n , we deduce from Lemma 3.2 in [START_REF] Del Moral | On the stability of positive semigroups[END_REF] that there exists a constant c such that for any (q ′ , n ′ ) ∈ N 2 , H q ′ ,n ′ ≤ cV . We can then conclude by choosing a small enough β ′ and using Lemmas 1 and 9.

The proof of Theorem 1 is now relatively straightforward.

Proof of Theorem 1:

Let f ∈ C V λ 4p
(E). From the sub-additivity of the L p -norm applied in [START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF], we have

E(|η N n (f ) -η n (f )| p ) 1 p ≤ n q=0 E ϕ q,n (η N q )(f ) -ϕ q,n (ϕ q (η N q-1 ))(f ) p 1 p .
By applying Lemma 2, we deduce that there exists (C, ρ, β) ∈ R × R * + × (0, 1] such that for any N ∈ N * we have

E(|η N n (f ) -η n (f )| p ) 1 p ≤ C N β 2 0≤l≤n e -(n-l)ρ ≤ C N β 2 (1 -e -ρ )
.

This ends the proof of the theorem.

Ground state estimates

This subsection concentrates on proving Corollary 1. We consider thus the time-homogeneous model.

Let f ∈ C V λ 4p
(E). Notice that we can decompose the error made by the DMC method in the following way:

E(|η N n (f ) -η ∞ (f )| p ) 1 p ≤ E(|η N n (f ) -η n (f )| p ) 1 p + |η n (f ) -η ∞ (f )|. Theorem 1 implies that there exists (C 1 , β) ∈ R * × (0, 1] such that sup n∈N E(|η N n (f ) -η n (f )| p ) 1 p ≤ C 1 N β/2 .

20

According to Theorem 4.3 in [START_REF] Del Moral | On the stability of positive semigroups[END_REF], there exists (C 2 , ω) ∈ R * 2 + such that

|η n (f ) -η ∞ (f )| ≤ C 2 e -ωn .
Hence

E(|η N n (f ) -η ∞ (f )| p ) 1 p ≤ C 1 N β/2 + C 2 e -ωn .
Thus, letting a = 1 ω ln(C 2 /C 1 ) and b = β 2ω , we deduce that there exists C ∈ R such that sup n≥a+b ln(N )

E(|η N n (f ) -η ∞ (f )| p ) 1 p ≤ C N β/2 .
This concludes the proof.

5 Coupled harmonic oscillators

Lyapunov functions

This subsection is dedicated to the proof of Corollary 3. Therefore we place ourselves within the framework associated with this corollary. We only consider the general case where P and G depend on a time parameter. If this is not the case, and χ is replaced by the ground state energy in [START_REF] Del | Large deviations for interacting particle systems: applications to non-linear filtering[END_REF], then the demonstration is completely analogous. The Markov transition kernels P n considered are Feller. Moreover, it is clear from Subsection 2.2 that proving the existence of a continuous P -Lyapunov function V ∈ C ∞ (R d ) makes ( 22) and ( 23) sufficient condition for (2) to hold. To guarantee the existence of an appropriate Q-Lyapunov function, we need a result obtained by Kato in [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]. We present it here using the formulation provided in [START_REF] Li | Eigenvalue continuity and gersgorin's theorem[END_REF].

Lemma 3. Suppose that D ⊂ R is an interval, and let A be a continuous function from D to the space of real d × d matrices. In this case, there exist d eigenvalues (counted with algebraic multiplicities) of A(t) which can be parameterized as continuous functions λ 1 (t), ..., λ d (t) from D to R.

We can now ensure the existence of a Q-Lyapunov function under a simple matrix condition.

Lemma 4. Assume that A T SA < S. There exists α ∈ R * + such that the function

V : x ∈ R d → exp α 2 x T Sx , (32)
is a Q-Lyapunov function and it is integrable w.r.t η 0 .

Proof :

From (24), we deduce that the r.h.s of (5) holds for V and G n . Then, together with [START_REF] Del | Mean field simulation for Monte Carlo integration[END_REF], we deduce that it is enough to prove that V is a Lyapunov function for P A,B with ϵ < 1/c. Let's then compute P A,B (f ) for any function of the form

f : x ∈ R d → exp 1 2 x T F x ,
where F is an invertible matrix such that B -1 -F is positive definite.

In this setting, the Woodbury matrix identity provides the following equality:

(B -F -1 ) -1 = B -1 -B -1 B -1 -F -1 B -1 .
We have then for any x ∈ R d , with BF := (B -1 -F ) -1 :

P A,B (f )(x) = 1 (2π) d/2 det(B) 1/2 R d e -1 2 [(Ax-y) ⊤ B -1 (Ax-y)-y T F y] dy = f (x) exp( 1 2 x T (A T B -1 BF B -1 A -A T B -1 A -F )x) (2π) d/2 det(B) 1/2 × R d exp - 1 2 ( BF B -1 Ax -y) ⊤ B-1 F ( BF B -1 Ax -y) dy.
This yields the formulae

P A,B (f )(x) = det( BF ) det(B) f (x) exp 1 2 x T (A T [B -1 (B -1 -F ) -1 B -1 -B -1 ]A -F )x = 1 det(I d -BF ) f (x) exp - 1 2 x T (F + A T (F B -I d ) -1 F A)x . (33)
From those calculations, we deduce that V is a Lyapunov function for P A,B if the matrices B -1 -αS and S -A T (I d -αSB) -1 SA are positive definite.

Let λ B be the greatest eigenvalue of B and λ S be the greatest eigenvalue of S. It is clear that for α ∈ (0, 1 λ B λ S ), B -1 -αS is positive definite. Consider now the function

ψ : α ∈ 0, 1 λ B λ S → sp(S -A T (I d -αSB) -1 SA) ∈ R d .
Here, sp(M ) represents the spectrum of a matrix M with multiplicity taken into account.

Given the hypotheses on A and S, we can conclude that ψ(0) ⊂ R * d + . Furthermore, by Lemma 2, it is clear that ψ is a continuous function. Since R * d

+ is an open set, there exists ᾱ ∈ R such that for any α ∈ (0, ᾱ), ψ(α) ⊂ R * d + . By choosing a sufficiently small value for α to ensure that V is integrable w.r.t η 0 , we can conclude.

From this Lemma and the hypothesis on G n , we deduce that the l.h.s of (5) holds as well. We can now focus on verifying that [START_REF] Cancés | Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation[END_REF] holds by proving the following Lemma.

Lemma 5. There exists a positive definite matrix H such that

∀n ∈ N, Q n (W ) ≥ χ × W with W (x) := exp(- 1 2
x T Hx).

Proof :

From ( 23), we have

Q n (W ) ≥ χ E A ′ ,B ′ ,S × G S P A ′ ,B ′ (W ).
Using [START_REF] Malvin | Monte Carlo calculations of the ground state of three-and fourbody nuclei[END_REF], we derive the following expression for G S (x

)P A ′ ,B ′ (W )(x) with x ∈ R d : 1 det(I d + B ′ H) W (x) exp 1 2 x T (H -A ′ T (HB ′ + I d ) -1 HA ′ -S)x .
Hence, chosing H as the solution to the Riccati equation

H -A ′ T (HB ′ + I d ) -1 HA ′ -S = 0, we deduce that E A ′ ,B ′ ,S = 1 det(I d + B ′ H) .
Thus:

Q n (W ) ≥ χ × W.
For α sufficiently small, W -α is lower than V . The right-hand side of ( 19) is then also verified.

Under the condition A T SA < S, we have confirmed that all the assumptions concerning P n , G n , and η 0 in Theorem 1 are satisfied. We can therefore apply it to conclude on the proof of Corollary 3.

Conditional free evolutions

This subsection focuses on the study of the importance sampling described in Section 2.4, Section 3.3 and on the proof of Corollary 4.

We consider the coupled harmonic oscillator, i.e, for some matrices A, B and S, with B and S symmetric definite positive, we consider P = P A,B and G = G S . Up to this point, we have established that the L p -norm of the error made by the DMC method is uniformly bounded in time, with a convergence rate of 1 N β/2 for some β ∈ (0, 1] when A T SA < A.

Our aim is now to use Theorem 1 to prove that the approximation of the measures η n made by the DMC method -enhanced by the importance sampling scheme described in (26) -remains uniformly bounded in time, regardless of the value of A and S. However, there's a trade-off involved: we will only have access to the measures at specific times. Indeed, despite the converging property that we are about to prove, the sequence of empirical measures η (k) n only approximates the measures η l for l ∈ kN. To proceed with our proof, we make the necessary assumption that the matrices A, B, and S can all be diagonalized in the same basis.

Before presenting the central corollary of this subsection, we lay the foundation with a lemma that, using relation [START_REF] Del | Central limit theorem for nonlinear filtering and interacting particle systems[END_REF], proves that this scenario can be interpreted as another instance of the coupled harmonic oscillator approximated by the usual DMC method. We then proceed to compute the specific constants of this scenario. Consider the parameters

λ 1 := 1 det(I + BS) & S 1 := A T (S -1 + B) -1 A,
as well as

A 1 := (I + BS) -1 A & B 1 := (B -1 + S) -1 .
For any n ≥ 0 we also set

λ n+1 := λ n det(I + BS + BS n ) & S n+1 := A T (B + (S + S n ) -1 ) -1 A,
as well as

A n+1 := (I + AB k A T S + BS) -1 AA n B n+1 := (I + AB n A T S + BS) -1 (AB n A T + B).
Lemma 6. For any k ≥ 0 we have

G (k) (x) = λ k exp(- 1 2 x T S k x) and δ x P (k) ∼ N (A k x, B k ).
The proof of this Lemma is relatively straightforward. However, it requires some technical calculations. It is therefore postponed to the Appendix.

We can now proceed to prove the central result of this subsection.

Within this framework, we can break down the evolution of the walkers into two distinct steps, a mutation transition and a selection transition

ξ i n i∈ 1,N ∈ R N selection -----→ ξ i n i∈ 1,N ∈ R N mutation ------→ ξ i n+1 i∈ 1,N .
The initial configuration ξ i 0 i∈ 1,N is determined by sampling N independent random variables from the distribution η 0 . The selection transition involves the sampling of N independent random variables ξ i n i∈ 1,N using the weighted distributions

ϵ n (η N n )G S (ξ i n )δ ξ i n + (1 -ϵ n (η N n )G S (ξ i n )) k∈ 1,N e -S 2 ξ k 2 n j∈ 1,N e -S 2 ξ j 2 n δ ξ k n .
The mutation transition is defined using a family of Gaussian random variables with zero-mean and unit variance (V i n ) i∈ 1,N such that

ξ i n = A ξ i n-1 + √ BV i n .
The measures (η n ) n∈N can be described exhaustively using the Kalman filter equations. It provides us with the mean and variances (m n , σ 2 n ) of the Gaussian random variables η n with the recurrent equations

           m n+1 = A 1 + Sσ 2 n m n σ 2 n+1 = A 2 σ 2 n 1 + Sσ 2 n + B . (36)
In this scenario, when the condition A 2 > 1 is met, it is possible to prove that the DMC's error in approximating the Feynman-Kac measure does not admit a uniform-intime bound. It is properly stated in Property 1, and we can now conduct its proof.

Proof of Proposition 1:

For any n ≥ 2, we know from (36) that

η n (I) := m n = A 2 (1 + Sσ 2 n-1 )(1 + Sσ 2 n-2 ) m n-2 ≤ A 2 m n-2 .
For any n ∈ N * , let ξ * n = min i∈ 1,N ξ i n and define the random variables V * n in the following way:

V * n = -max i∈ 1,N V i n .
By definition of the evolution of the walkers, there exits (i, j) ∈ 1, N such that

ξ * 2n = A 2 ξ j 2n-2 + √ BV i 2n-1 + A √ BV j 2n-2 ≥ A 2 ξ * 2n-2 + √ BV * 2n-1 + |A| √ BV * 2n-2 . Thus η N 2n (I) -η 2n (I) ≥ A 2 (ξ * 2(n-1) -m 2(n-1) ) + √ BV * 2n-1 + |A| √ BV * 2n-2 .
Iterating the process, we obtain

η N 2n (I) -η 2n (I) A 2n ≥ (ξ * 0 -m 0 ) + √ B 1≤k≤n V * 2k-1 A 2k + |A| √ B 1≤k≤n V * 2(k-1) A 2k .
For any sequence of N independent centred Gaussian random variables U i with unit variance, we have

E max 1≤i≤N |U i | ≤ 2 log(2N ).
This inequality is obtained by using Jensen's inequality as follows, with t = 2 log(2N )

exp tE max 1≤i≤N |U i | ≤ E exp t max 1≤i≤N |U i | ≤ N i=1 E [exp (t|U i |)] ,
and noticing that

E [exp (t|U i |)] = 2 +∞ 0 exp - (x -t) 2 + t 2 2 dx ≤ 2 exp(t 2 /2).
Then, on the event

Ω ϵ := ξ * 0 ≥ ϵ + m 0 + 2 √ B(1 + |A|) A 2 -1 2 log(2N ) , (37)
with ϵ ∈ R * + , we have

E[η N 2n (I) -η 2n (I)|ξ * 0 ] ≥ ϵA 2n n→+∞ -----→ +∞. (38)
Integrating over ξ * 0 we deduce

E[|η N 2n (I) -η 2n (I)||] ≥ ϵA 2n P(Ω ϵ ). (39)
We can then conclude by noticing

P(Ω ϵ ) = η 0 ϵ + X 0 + 2 √ B(1 + |A|) A 2 -1 2 log(2N ), +∞ N > 0. ( 40
)
η n ∼ N (m n , σ 2 n ) with            m n+1 = A 1 + Sσ 2 n m n σ 2 n+1 = A 2 1 + Sσ 2 n + B .
The limit measure is given by η ∞ = N (0, σ 2 ∞ ) where σ 2 ∞ is the fixed point of the function x ∈ R + → A 2 1+Sx + B. From now on, we assume that η 0 = η ∞ . We have then

σ 2 n = n p=0 η ∞ Q p,n (I) 2 (η ∞ Q p,n (1)) 2 .
Using the calculations and notations from Lemma 10, we have then

             η ∞ (Q p,n (1)) = λ n-p 2πσ 2 ∞ R e -y 2 2σ 2
∞ e -y 2 2 q n-p dy = λ n-p q n-p σ 2 ∞ + 1 As a solution to a Riccati equation, q n converges towards some q ∞ ∈ R * + such that

η ∞ Q p,n (I) 2 = µ 2 n-p 2πσ 2 ∞ R y 2 e -y 2
q ∞ = A 2 q ∞ 1 + q ∞ B + S.
Since A > 1, we obtain that 1+q ∞ B = A 2 +S/q ∞ > A. Thus, there exists C ∈ (0, 1) such that, for n large enough, we have

µ n+1 ≤ C µ n √ 1 + q n B
Thus, comparing the defintion of λ n and the previous bound, we deduce that there exists α ∈ R such that we have µ 2 n-p λ 2 n-p (q n-p σ 2 ∞ + 1) 3/2 (2q n-p σ 2 ∞ + 1) 3/2 ≤ αC 2(n-p) . We can thus conclude on the right-hand side. of [START_REF] Del | Non linear filtering: Interacting particle solution[END_REF]. The left-hand side is a direct application of Proposition 1.

Proof: First, let's prove that for any γ ∈ P V (E), we have, with (ϵ, c) ∈ [0, 1) × R defined in (1), sup (q,n)∈N 2 q≤n ϕ q,n (γ

)(V ) ≤ γ(V ) + c 1 -ϵ . ( 43
)
To do so, we begin by using the Q-Lyapunov property of V as well as Lemma 8 for some l ∈ N * to deduce that ϕ q,n (γ)(V ) = ϕ q,n-1 (γ)(G n P n (V )) ϕ q,n-1 (γ)(G n ) ≤ ϵ ϕ q,n-1 (γ)(V ) + c ′ .

By iterating the process, we obtain [START_REF] Del | Stochastic Processes: From Applications to Theory[END_REF]. We now prove that

sup n∈N E η N n (V ) ≤ η 0 (V ) + c ′ 1 -ϵ . ( 44
)
Notice first that

E η N n (V ) | η N n-1 = 1 N N i=1 η N n-1 S n-1,η N n-1 P n (V ) = ψ G n-1 (η N n-1 )(P n (V )).
Then, using as previously the Q-Lyapunov property of V and Lemma 8 we obtain ψ G n-1 (η N n-1 )(P n (V )) ≤ ϵ η N n-1 (V ) + c ′ .

By iterating the process, we obtain

E η N n (V ) ≤ ϵ n η 0 (V ) + c ′ n-1 i=0 ϵ i ≤ η 0 (V ) + c ′ 1 -ϵ .
Thus, by combining ( 43) and ( 44), we can conclude regarding the first part of (42). The second part is obtained by proceeding in a strictly analogous way.

Proof of Lemma 1

We first consider the case where only [START_REF] Rousset | On the control of an interacting particle estimation of schrödinger ground states[END_REF] holds and prove that ϕ 0,q (γ)(Q q,n (1)) ≤ Cχ n-q-1 .

(

) 45
For q ≤ n + 2, we have

Q q,n (1)(x) = Q q,n-2 (G n-1 P n (G n ))(x)
≤ χ Q q,n-2 (G n-1 )(x) ≤ χ Q q,n-1 (1)(x).

Iterating the process, we deduce Q q,n (1) ≤ χ n-q-1 G q-1 .

Using [START_REF] Del | Particle motions in absorbing medium with hard and soft obstacles[END_REF] we deduce [START_REF] Del Moral | A Moran particle system approximation of Feynman-Kac formulae[END_REF].

For q ≥ 1 and µ ∈ {η N q , ϕ q (η N q-1 )}, we have then 1 µ(H γ q,n ) = ϕ q,n (µ)(W ) × ϕ 0,q (γ)Q q,n (1) µ(Q q,n (W )) ≤ C χ n-q-1 ϕ q,n (µ)(W ) µ(Q q,n (W )) .

From Holder's inequality, Jensen's inequality and the hypothesis on W , we get :

E µ(H q,n ) -β ≤ C β E ϕ q,n (µ)(W ) 2β
1 2 E χ 2β(n-q-1) µ(Q q,n (W )) -2β

1 2 ≤ C β χ β E ϕ q,n (µ)(V ) 2β 1 2 E µ(W -2β ) 1 2 .
Then, by choosing β small enough such that W -2β ≤ W and V 2β ≤ V , where W and V are Q-Lyapunov functions, we obtain sup (q,n,N )∈N 3 q≤n E µ(H γ q,n ) -β ≤ C β χ β sup (q,n,N )∈N 3 q≤n E µ( V ) We can conclude using Lemma 9.

The demostration for time-homogeneous models with E 0 < χ is analogous. Indeed the equivalent of ( 45) is obtained by noticing that

ϕ 0,q (η ∞ )Q q,n (1) = η ∞ Q q,n-1 (G) = η ∞ Q q,n-1 (1)η ∞ (G) = E 0 × η ∞ Q q,n-1 (1) = E n-q 0 .
The rest of the proof follows the same arguments, thus it is skipped. This ends the proof of the lemma.

Proof of Lemma 6

Let us begin by proving the result for k = 1. Take x ∈ R d :

G (1) (x) = 1 det(I + BS) exp - 1 2
x T A T (S -1 + B) -1 Ax , with the parameters (q 0 , λ 0 , µ 0 ) = (S, 1, A) and

           q n+1 = A 2 q n 1 + q n B + S λ n+1 = λ n √
1 + q n B and µ n+1 = Aµ n (1 + q n B) 3/2 . Proof : For n = 0 the result is immediate. Assume that it holds for some n ∈ N. In this situation, we have

Q n+1 (1)(x) = Q(Q n (1))(x) = λ n Q y → e -

Corollary 2 . 3 +

 23 Let p ∈ N * , and assume that G ∈ C V λ 4p (E). There exists (a, b, c) ∈ R * and β ∈ (0, 1] such that for any f ∈ C V λ 4p

2σ 2 ∞ 2 n-p σ 2 ∞(2q n-p σ 2 ∞ + 1 )(q n-p σ 2 ∞

 222212 e -y 2 q n-p dy = µ + 1) 3/2 (2q n-p σ 2 ∞ + 1) 3/2 .

Q.

  n+1 (I)(x) = Q(Q n (I))(x) = µ n Q y → ye -This ends the proof.

Acknowledgments

MC thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 863481) for financial support. Luc de Montella would like to thank Naval Group for its support as part of a CIFRE fellowship.

Proof of Corollary 4: Let k ∈ N. From Corollary 3, to prove that the DMC method associated with G (k) and P (k) is uniformly converging toward the Feynman-Kac measures, it is enough to prove that A T k S k A k < S k . Since all these matrices can be diagonalized in the same basis, proving that this criterion holds for the matrices A k and S k is equivalent to proving that all eigenvalues of A k are in the interval (-1, 1).

Let B a basis in which the matrices A, B and S are diagonal. We want to prove that for i ∈ 1, d , the i-th eigenvalue of A is in the right interval. In the rest of the proof, we denote by M (i) the i-th eigenvalue of a matrix M that can be diagonalized in B.

Using the expression derived in Lemma 6, we obtain:

We will establish that as n tends towards infinity, the limit of

1+S (i) tn is strictly less than 1. This result will consequently imply the convergence of the sequence A (i) n towards 0. For n ∈ N * :

Here, φ represents a Riccati operator defined as described in [START_REF] Del | A theoretical analysis of one-dimensional discrete generation Ensemble Kalman particle filters[END_REF]. Using Equation (51) from the same article, we can derive that

Hence lim n→+∞

1+S (i) tn < 1. For n large enough, the approximation made by the DMC method enhanced by importance sampling is then the same as the usual DMC approximation of an harmonic oscillator with a stable Markov transition. We can thus conclude using Theorem 1.

Divergence and fluctuation estimates

In the previous subsections, we presented a simple sufficient condition for controlling the DMC method and introduced an importance sampling technique that satisfies this criterion. However, it is natural to question the robustness of this condition and whether it is necessary to use importance sampling. Specifically, for the uni-dimensional harmonic oscillator, the convergence condition reduced to A 2 < 1, and we will prove divergence of the DMC method when this stability condition is not met.

For the case A = 1, we are not able to assert whether or not a uniform bound exists. To the best of our knowledge, the best divergence-type result proved to date is a linear bound on the variance of the unnormalized measure when R = S = 1 [START_REF] Whiteley | Linear variance bounds for particle approximations of time-homogeneous Feynman-Kac formulae[END_REF].

The divergence result in Proposition 1 highlights the importance of studying nonasymptotic uniform convergence results rather than relying solely on central limit theorems (CLTs). Extensive research has been devoted to CLTs and, under appropriate assumptions. We quote the first studies in this field [START_REF] Del | Central limit theorem for nonlinear filtering and interacting particle systems[END_REF][START_REF] Del Moral | Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering[END_REF] mainly based on uniformly bounded potential and test functions.

More general fluctuation theorems that apply to more general models including diffusion-type processes with Lipschitz drift and diffusion functions as well as test functions with at most polynomial growth are discussed in [START_REF] Del | The Monte-Carlo method for filtering with discrete time observations. central limit theorems[END_REF][START_REF] Del | Interacting particle filtering with discrete-time observations: asymptotic behaviour in the gaussian case[END_REF]. We can formulate the following result : Lemma 7. In the context of proportional selection/reconfiguration, for any n ≥ 1, we have the following convergence in law as N tends toward +∞

with the asymptotic variance

Qp,n (I -η n (I))

2

.

Related asymptotic variance formulae and comparisons are discussed in [START_REF] Del | A note on random walks with absorbing barriers and sequential monte carlo methods[END_REF] in the context of random walks with absorbing barriers, including geometric killing rates and local reflection moves.

In [START_REF] Whiteley | Stability properties of some particle filters[END_REF], Nick Withley also obtains a uniform time bound on the asymptotic variance. However, our next proposition shows that, for a given system, the DMC method can have a uniform time bound on its asymptotic variance, despite the fact that its nonasymptotic variance is unbounded.

2 S , and P such that δ x P ∼ N (Ax, B) for some

+ . There exists an intial distibution η 0 such that

where σ 2 n is defined as in Lemma 7.

Proof :

Considering η 0 ∼ N (m 0 , σ 2 0 ) and using the Kalman filter's equations, we are able to fully describe the measures η n . For n ∈ N * we have Appendix Some technical Lemmas Lemma 8. Let (G, V ) ∈ C(E)×C ∞ (E) be positive functions and K ⊂ E be such that the right-hand side of (5) holds. There exists c ∈ R + such that for any probability measure µ on E :

Integrating with respect to the probability measure µ over both x and y, we obtain:

Applying the Woodbury matrix identity we get:

Assume the property true for k ∈ N * . We let Sk := S k + S. In this notation, we have

This implies that

from which we check that

k ) we also have

According the Gaussian update formula (see Proposition 4.5.2 in [START_REF] Del | Stochastic Processes: From Applications to Theory[END_REF] for example), if η = N (m, Σ), then ψ G S η) = N [(I + ΣS) -1 m , (I + ΣS) -1 Σ . Then we have ψ G δ x P (k+1) ∼ N (I + AB k A T S + BS) -1 AA k x, (I + AB k A T S + BS) -1 (AB k A T + B) .

This concludes the proof.