A Weak Overdamped Limit Theorem for Langevin Processes

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…”