BV functions in a Gelfand triple for differentiable measure and its applications

TL;DR

This paper extends the concept of BV functions to non-Gaussian differentiable measures within a Gelfand triple using Dirichlet form theory, enabling analysis of reflected stochastic quantization problems.

Contribution

It introduces a new BV function definition for non-Gaussian measures in a Gelfand triple and applies it to establish solutions for reflected stochastic quantization.

Findings

Established existence of martingale solutions for the quantization problem

Extended BV function theory to non-Gaussian measures in Gelfand triples

Applied Dirichlet form theory to stochastic quantization analysis

Abstract

In this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [RZZ12], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $\Gamma$ in a Banach space $E$. We prove the existence of a martingale solution of this problem if $\Gamma$ is a regular convex set.