Stochastic evolution equations with rough boundary noise

TL;DR

This paper establishes the well-posedness and global solutions for stochastic evolution equations with rough boundary noise, including fractional Brownian motion, using rough path theory and extrapolation operators.

Contribution

It introduces a novel combination of rough path techniques and extrapolation operators to analyze stochastic PDEs with boundary noise.

Findings

Global existence of solutions with Neumann boundary noise

Results extend to Dirichlet boundary noise in the Young regime

Framework applies to fractional Brownian motion with Hurst parameter in (1/3,1/2]

Abstract

We investigate the pathwise well-posedness of stochastic evolution equations perturbed by multiplicative Neumann boundary noise, such as fractional Brownian motion for $H\in(1/3,1/2]$. Combining the controlled rough path approach with the theory of extrapolation operators, we establish global existence of solutions and flows for such equations. For Dirichlet boundary noise we obtain similar results in the Young regime.