Positive random walks and an identity for half-space SPDEs

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 \textit{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 \textit{different} boundary conditions. This identity may be seen as a precursor for proving Gaussian fluctuation behavior of supercritical half-space KPZ at the origin.