Elementary pathwise methods for nonlinear parabolic and transport type SPDE with fractal noise

TL;DR

This paper reviews recent pathwise methods for analyzing nonlinear parabolic and transport SPDEs driven by fractal noise, focusing on existence, uniqueness, and regularity of solutions using semigroup theory.

Contribution

It introduces a pathwise approach to SPDEs driven by fractional noises, extending classical PDE techniques to stochastic settings with fractal noise.

Findings

Established existence and uniqueness of solutions for certain nonlinear SPDEs.

Provided regularity results for solutions driven by fractional noise.

Applied semigroup theory to analyze stochastic PDEs with fractal noise.

Abstract

We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.