A weak overdamped limit theorem for Langevin processes

Abstract

In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.Date: December 1, 2018. 1991

Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials

“…Our results are complementary to those in [19] in the following sense: First, there the authors have to assume the interaction term ∇Φ 1 to be continuous. Second, there the state space is assumed to be the d−dimensional torus T d .…”

“…the invariant measure µ Φ . This aspect is more restrictive than in [19]. Additionally, the Φ 1 in [19] may also depend on ε > 0.…”

“…Physically this corresponds to large friction forces and an appropriate time-scaling (see [16][Chapter 2.2.4] for a physical interpretation). The authors of [19] prove convergence in law of (X ε t ) t≥0 as ε tends to zero to a solution of the overdamped Langevin equation…”

Uniform strong and weak error estimates for numerical schemes applied to multiscale SDEs in a Smoluchowski–Kramers diffusion approximation regime

“…which is more precise than (26). It suffices to combine (28) and Assumption 2 to obtain (26). Observe that if p 0 " 0, the strong error estimate (28) yields the following refinement of ( 26 where the error estimate is uniform with respect to both the time-scale separation parameter P p0, 0 q and to the temporal index n P t0, .…”

“…The convergence result q Ñ q 0 is often called a Smoluchowski-Kramers diffusion approximation result in the literature. If σ is constant and equal to the identity, and if f " ´∇V for some potential energy function V : R d Ñ R, the SDE system (1) describes the Langevin dynamics, whereas the SDE (2) describes the overdamped Langevin dynamics, see for instance [17,Section 6.3.4], [19, Sections 2.2.3 and 2.2.4], [23,Section 6.5], and also the recent article [26] and references therein. We also refer to the monographs [17], [19] and [23] for analysis and applications of (overdamped) Langevin dynamics.…”