The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling

TL;DR

This paper explores the use of Malliavin calculus to establish compactness and prove the existence of solutions for a complex degenerate PDE-SDE system, also demonstrating convergence of a numerical scheme.

Contribution

It introduces a novel application of Malliavin derivative estimates to analyze nonlinear stochastic PDEs and their numerical approximations.

Findings

Established compactness for the PDE-SDE system.

Proved existence of global solutions.

Demonstrated convergence of a semi-discretisation scheme.

Abstract

Compactness is one of the most versatile tools in the analysis of nonlinear PDEs and systems. Usually, compactness is established by means of some embedding theorem between functional spaces. Such theorems, in turn, rely on appropriate estimates for a function and its derivatives. While a similar result based on simultaneous estimates for the Malliavin and weak Sobolev derivatives is available for the Wiener-Sobolev spaces, it seems that it has not yet been widely used in the analysis of highly nonlinear parabolic problems with stochasticity. In the present work we apply this result in order to study compactness, existence of global solutions, and, as a by-product, the convergence of a semi-discretisation scheme for a prototypical degenerate PDE-SDE coupling.