We investigate the hydrodynamical behavior of a system of random walks with zero-range interactions moving in a common 'Sinai-type' random environment on a one dimensional torus. The hydrodynamic equation found is a quasilinear SPDE with a 'rough' random drift term coming from a scaling of the random environment and a homogenization of the particle interaction. Part of the motivation for this work is to understand how the space-time limit of the particle mass relates to that of the known single particle Brox diffusion limit. In this respect, given the hydrodynamic limit shown, we describe formal connections through a two scale limit.
We find explicitly the Green kernel of a random Schrödinger operator with Brownian white noise. To do this, we first handle the random operator by defining it weakly using the inner product of a Hilbert space. Then, using classic Sturm-Liouville theory, we can build the Green kernel with linearly independent solutions of a homogeneous problem. As a corollary we have that the random operator has a discrete spectra.
The reflected Brownian motion is being used in areas such as physiology, electrochemistry and nuclear magnetic resonance. We study the first passage time problem of this process which is relevant in applications; specifically, we find a Volterra integral equation for the distribution of the first time that a reflected Brownian motion reaches a non-decreasing barrier. Additionally, we make mention of how a numerical procedure can be used to handle the integral equation.
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