Positive random walks and an identity for halfspace SPDEs
Abstract
The purpose of this article is threefold. First, we introduce a new type of boundary condition for the multiplicativenoise 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 \textit{meander}) within the associated chaos series. Secondly, we prove a new convergence result of the directedpolymer 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 equalityindistribution for multiplicativenoise stochastic heat equations on the half space with \textit{different} boundary conditions. This identity may be seen as a precursor for proving Gaussian fluctuation behavior of supercritical halfspace KPZ at the origin.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 arXiv:
 arXiv:1901.09449
 Bibcode:
 2019arXiv190109449P
 Keywords:

 Mathematics  Probability
 EPrint:
 50 pages, 1 figure