Well-posedness of Fractional Stochastic p-Laplace Equations Driven by Superlinear Transport Noise

TL;DR

This paper establishes the existence and uniqueness of solutions for fractional stochastic p-Laplace equations with superlinear transport noise, using advanced probabilistic and analytical techniques.

Contribution

It extends the theory of stochastic partial differential equations by handling fractional p-Laplace equations with superlinear noise and develops new methods for tightness and convergence.

Findings

Proved existence and uniqueness of solutions.

Developed a new approach for uniform estimates.

Applied Skorokhod-Jakubowski theorem in a novel context.

Abstract

In this paper, we prove the existence and uniqueness of solutions of the fractional p-Laplace equation with a polynomial drift of arbitrary order driven by superlinear transport noise. By the monotone argument, we first prove the existence and uniqueness of solutions of an abstract stochastic differential equation satisfying a fully local monotonicity condition. We then apply the abstract result to the fractional stochastic p-Laplace equation defined in a bounded domain. The main difficulty is to establish the tightness as well as the uniform integrability of a sequence of approximate solutions defined by the Galerkin method. To obtain the necessary uniform estimates, we employ the Skorokhod-Jakubowski representation theorem on a topological space instead of a metric space. Since the strong Skorokhod representation theorem is incorrect even in a complete separable metric space, we pass to the limit of stochastic integrals with respect to a sequence of Wiener processes by a weak convergence argument.