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Discrete approximations to local times for reflected diffusions. (English) Zbl 1336.60068

Summary: We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [Z.-Q. Chen and the author, “Hydrodynamic limits and propagation of chaos for interacting random walks in domains”, Preprint, arXiv:1311.2325]. Applications of the join convergence result to PDE problems are illustrated.

MSC:

60F17 Functional limit theorems; invariance principles
60J55 Local time and additive functionals
35K10 Second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
49M25 Discrete approximations in optimal control
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