Particle approximation for Lagrangian Stochastic Models with specular boundary condition

In this paper, we prove a particle approximation, in the sense of the propagation of chaos, of a Lagrangian stochastic model submitted to specular boundary condition and satisfying the mean no-permeability condition.


Introduction
In this paper, we construct a particle approximation of the following Lagrangian stochastic model (X, U ) on a finite time interval [0, T ], submitted to specular reflections at the boundary of a domain D of R d : (U s − · n D (X s )) n D (X s )1 {Xs∈∂D} , ρ(t) is the Lebesgue density of the law of (X t , U t ) for t ∈ (0, T ]. (1.1) The initial condition (X 0 , U 0 ) is distributed according to a given probability measure µ 0 , and is independent to the R d -Brownian motion (W t ; t ∈ [0, T ]), n D is the outward normal unit vector of the smooth boundary ∂D. We are considering Lagrangian stochastic model, this means that the dependencies in x of the coefficients in the velocity equation (1.1) are expressed as a conditional expectation with respect to the event {X t = x}. Here the drift component B[x; ρ(t)] is a version of the conditional expectation E [b(U t ) | X t = x]. Thus given a kernel b, B is defined for (x, γ) ∈ D × L 1 (D × R d ) as is well-defined and almost-surely grows to T as n goes to ∞.
Numerical algorithms for (SLM) are based on particle approximation methods (see e.g. [1] and the references therein). Here we give a first convergence result of a particle approximation of (1.1). We study the limit behavior of the interacting particle system {(X i,ǫ,N , U i,ǫ,N , K i,ǫ,N ), i = 1, . . . , N }, on a given probability space (Ω, F , (F t ; t ≥ 0), Q) endowed with independent copies {(X i 0 , U i 0 , (W i t ; t ∈ [0, T ])), i = 1, . . . , N } of (X 0 , U 0 , (W t ; t ∈ [0, T ])), defined as the solution to the following SDE system: is defined for all x ∈ D and all γ in the set of probability measure on D × R d as where β ǫ (y) = 1 {dist(y,∂D)>ǫ} cutoffs the support of γ from a distance ǫ to ∂D. The existence and uniqueness in law for the solution of (1.3) simply follow from Girsanov's transformation and from the wellposedness of the confined Langevin process (i.e. the case b = 0, see Theorem 2.1 in [3]). This step only requires that D has a C 3 -boundary and that the support of µ 0 is included in D × R d .
Our main result is stated in Theorem 2.1: as the number of particle grows to infinity and the mollifiers parameter ǫ goes to 0, we prove that the particles (1.3) propagate the initial chaos with a limit law given by the solution to (1.1). Remark 1.1 (About the mean no-permeability condition on ∂D). In [3], we prove that the solution to (1.1) satisfies the so-called mean no-permeability condition: for x ∈ ∂D, Stochastic Lagrangian models have been introduced for complex simulation in Computational Fluid Dynamic (CFD). The mean-Dirichlet boundary condition (1.5) grounds the stochastic particle algorithm used to downscale simulations in CFD applications (we refer to [4] [1] and the references therein for further details). Notice that the particle approximation of (1.5) (with the kernel b(u, x) = u · n D (x)) in a neighborhood of ∂D is still an issue, that seems to require the continuity of the density of (X t , U t ) over D. Except in the 1D case studied in [2], and to the best of our knowledge, such regularity result is unknown in the PDE literature on trace problems. Remark 1.2 (About the sequence of passage times on ∂D). When D = (0, +∞) and b = 0, the explicit expression of the joint law of (τ n , U τn , n ≥ 1) enables to control uniformly the confinement process (K i,ǫ,N t ; t ∈ [0, T ]). For more general domain, we compensate the lack of such control by studying the trace problem for the density ρ.
Notice that the estimate (3.6) in [2] on the upper-bound of P(τ n ≤ T ) contains a mistake, claiming that this probability decreases with n uniformly in T . This shall be reformulated as follows: when the initial law µ 0 has its support in (R d−1 × (0, +∞)) × R d , there exists a constant C(T, σ, β * ) depending on T , σ and the distance This clarification of the constant in front of 1/2 n in the right hand side does not impact the results in [2], as we were worked with fixed T . For completeness we give a short proof of (1.6) in Appendix A.2.

Notation. E denotes the set of paths
For all t ∈ (0, T ], we introduce the following sets: Q t := (0, t) × D × R d , Denoting by dσ ∂D the surface measure on ∂D, the product measure on Σ T is dλ ΣT := dt ⊗ dσ ∂D (x) ⊗ du. For a given positive weight function ω on R d , we define the following weighted Sobolev spaces M(E) denotes the set of probability measure on E.

Hypotheses.
From now on, we assume that the domain D, the distribution µ 0 of (X 0 , U 0 ), and the kernel b in (1.1) satisfy the following hypotheses (H).
Notice that (H) are slightly more restrictive than the hypotheses in [3] for the existence:here b is assumed continuous to simplify some weak convergence arguments, and the weight function ω is chosen in order to control R d |u| 3 /ω(u)du. Theorem 2.1. Assume (H). Let P be the law on E of (X, U, K) defined in (1.1), and let P ǫ,N be the law of For the convenience of the reader, we summarize the results we need from [3], in the following.

1)
and satisfies the following energy estimate where C > 0 depends only on d, α and b ∞ . In addition, ρ and its traces γ ± (ρ) admit the following Maxwellian bounds where G σ (t) is the centered Gaussian density function with variance σ 2 t, * stands for the convolution product, a ± , ν ± are constants depending only on T, d, α and b ∞ and are such that a + > 0, a − < 0 and ν ± > 0.
Proposition 2.2 is clearly also true when the drift B[x, ·] is replaced by its smoothed version B ǫ [x, ·], or by a linear and bounded drift V (t, x). In the following, we slightly extend Proposition 2.2 for a generic linear drift V (t, x). The proof of Corollary 2.3 is postponed in the appendix. The rest of the paper is devoted to the proof of the propagation of chaos result. Although we give a particle approximation of the confined Lagrangian model, we are not able to use such approximation to construct a solution under lighter hypotheses than (H). In particular, we still have a deep used of the PDE analysis of the Fokker Planck equation. The main difficulty resides in the uniform integrability result of the density traces, that we are able to show only with the strong Mawellian bound tool.

Proof of Theorem 2.1
Equipped with the Skorokhod topology, E is a Polish space. We denote by The proof consists in the study of the double limits, first as N tends to ∞, next as ǫ tends to 0. Mainly, we will detail the two following steps:
Proposition 3.1 has its analog in [2]. But now, the fact that the jump term is a finite variation process is not for free in the proof, as it is no more an increasing process.
Notice also that even if we consider constant diffusion process, the mild-equation tool that we strongly used in [4] and [2] is useless here, as the controls we have on the semigroup derivative of the Lagrangian process are only for the L 2 -norm.

The limit as N tends to ∞
We check that all limit points of {π ǫ,N ; N ≥ 1} have full measure on the set of probability measures under which the canonical process (x(t), u(t), k(t); t ∈ [0, T ]) satisfies (3.1). Let π ǫ,∞ denotes the limit of a converging subsequence of π ǫ,N ; N ≥ 1 that we still index by N for simplicity.
The uniqueness in law for the solution of (3.1) ensures that all converging subsequences of {π ǫ,N , N ≥ 1} tend to δ {P ǫ } , and enables us to conclude on the propagation of chaos property (3.2).
Proof of Lemma 3.3. The scheme of the proof is the same than for Lemma 4.8 in [2]. We only need to take care about points (b) as in the multi-dimensional case, the jump process k is no more and increasing process. For We replicate some arguments of Sznitman [8] and introduce the closed sets Then applying two times the Chebyshev's inequality, and using the exchangeability of the particles, Owing to Lebesgue's monotone convergence theorem, we have And, by the trace representation in Corollary 2.3 Let us first observe that d|k n | converges towards d|k| in the sense of measures. Next, using the change of variable s → λ −1 n (s), and since t → η n (t) := dist(x n (t), ∂D) and t → η(t) := dist(x(t), ∂D) are continuous, Since max t∈[0,T ] |η n (s) − η(s)| tends to 0 and d(|k| n • λ n ) converges weakly to d|k|, the conclusion follows.

The limit as ǫ tends to 0
The tightness of {P ǫ ; ǫ > 0} can be shown again by replicating the verification of the Aldous's criterion given in Lemma 4.4 of [2]. The main concern in that step is for the identification of the limit points. With Lemma 3.2, we easily check that any limit P 0 of a converging subsequence of {P ǫ ; ǫ > 0} is a weak solution to the (1.1), as it satisfies the following martingale problem conditions (i) P 0 • (x(0), u(0), k(0)) −1 = µ 0 ⊗ δ 0 , where δ 0 denotes the Dirac mass at 0 on R d , by hypothesis.
Indeed, as observed in [2], this is a direct consequence of the following convergence lim |h|,|δ|→0 lim sup that can be immediately deduced from Lemma 3.2.

A.1 Proof of Corollary 2.3
We prove that ρ(t) is in L 2 (ω; D × R d ) using the following observation: the couple (y t , v t ) defined by where we set |D| := D dx. From the Riesz Representation Theorem, it is sufficient to check, for all t ∈ (0, T ], there exists some constant C > 0 such that Without loss of generality let us assume that ψ is nonnegative. Using H-(iii), After a Cauchy-Schawrz inequality, we get with C(α) = ωG σ (t, ·) ∞ , that allows to conclude on (A.1). Now we prove the probabilistic interpretation of trace integrals in (2.6). We consider the unique solution ρ in V 1 (ω; Q T ) and γ ± (ρ) in L 2 (ω; Σ ± T ), of the following weak Fokker-Planck equation starting from ρ 0 : for all and such that γ ± (ρ) satisfy the Maxwellian bounds (2.4) (see [3]-Proposition 3.14). Then it i straightforward to check that ρ(t) is also the density of (X t , U t ) using the identification by mild-equation Using the surjectivity of the application φ → φ Σ ± T (see e.g. Brezis [5], p. 315), we conclude on (2.6).

A.2 Proof of the upper bound (1.6) on P(τ n ≤ T )
We consider the Langevin process on the probability measure P y,v , with y = 0, and the sequence of passage times τ n = inf{τ n−1 < t ≤ T ; Y t = 0}, for n ≥ 1, τ 0 = 0.