Positive random walks and an identity for half-space SPDEs

2019/01/27 by Shalin Parekh, Parekh, Shalin · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1901.09449

50 pages, 1 figure

arxiv created 2019/01/27 · arxiv updated 2019/01/29

Abstract

The purpose of this article is threefold. First, we introduce a new type of boundary condition for the multiplicative-noise stochastic heat equation on the half space. This is essentially a Dirichlet boundary condition but with a nontrivial normalization near the boundary which leads to inhomogeneous transition densities (roughly, those of a Brownian meander) within the associated chaos series. Secondly, we prove a new convergence result of the directed-polymer partition function in an octant to the multiplicative stochastic heat equation with this type of boundary condition, which in turn involves a detailed analysis of the aforementioned inhomogeneous Markov process. Thirdly, as a corollary, we prove a surprising equality-in-distribution for multiplicative-noise stochastic heat equations on the half space with different boundary conditions. This identity may be seen as a precursor for proving Gaussian fluctuation behavior of supercritical half-space KPZ at the origin.

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