On Skorokhod Problems for Reflected and Singular Stochastic Heat Equations

Abstract

We prove a Skorokhod decomposition for the Markov processes $X^a$ and $X$ associated to the gradient Dirichlet forms with respect to the measures $ρ^aμ^β$ and $ρμ^β$, respectively. Here, $μ^β$ is the law of the standard Brownian bridge $β$, while $ρ^a$ and $ρ$ denote densities which are given by $ρ^a(z) := \mathbf{1}_{[0,\infty)}(\bar{z}_a)$ and $ρ(z) := \int_0^1 \mathbf{1}_{[0,\infty)}(\bar{z}_x) \, dx$, respectively, for all $z\in L^2(0,1)$ which have a (unique) continuous representative $\bar{z}$ which vanishes at zero and one. To this end, we derive infinite-dimensional integration by parts formulas (IbPFs) w.r.t. $ρ^aμ^β$ and $ρμ^β$, which contain Hida distributions alongside the usual drift terms. We represent these Hida distributions by integration w.r.t. vector measures of bounded variation. The vector measures in question are constructed via an approximation argument, making use of a generalization of Prokhorov's theorem for vector measures. We further prove that, almost surely, the sample paths of $X^a$ and $X$ take values in the equivalence class of continuous functions vanishing at zero and one for all and $dt$-almost all times, respectively. The main motivation for studying $ρ^aμ^β$ and $ρμ^β$ lies in the fact that the distributional terms in their IbPFs are simplifications of the distributional term in the IbPF w.r.t. the law of the reflected Brownian bridge on the unit interval $μ^{|β|}$. Representing the latter by integration w.r.t. a vector measure of bounded variation is still an open problem.