Propagation of Chaos for reflecting diffusions with local-time dependent noise

Abstract

We prove existence and uniqueness of a reaction-diffusion equation whose diffusivity is a non-linear functional of the boundary temperature. We do this by studying systems of one-dimensional reflecting diffusions whose noise is a function of the reflection local-time of the system, and by characterizing the large-scale (hydrodynamic) behavior by showing propagation of chaos. In addition, we analyze the one-particle case by computing the distribution of the hitting times of its reflection local-time. This work is the noise analog of work done by Frank Knight (2001).