Concentration estimates for SPDEs driven by fractional Brownian motion

TL;DR

This paper derives concentration estimates for solutions of SPDEs driven by fractional Brownian motion, extending existing methods to non-martingale settings and providing new bounds for fractional SDEs across all Hurst indices.

Contribution

It introduces a novel approach to estimate the solution paths of SPDEs driven by fractional Brownian motion without relying on martingale techniques.

Findings

Concentration estimates for SPDE solutions with Hurst index H > 1/4.

New bounds for fractional SDEs valid for all H in (0,1).

Extension of Gaussian supremum estimates to fractional Brownian motion.

Abstract

The main goal of this work is to provide sample-path estimates for the solution of slowly time-dependent SPDEs perturbed by a cylindrical fractional Brownian motion. Our strategy is similar to the approach by Berglund and Nader for space-time white noise. However, the setting of fractional Brownian motion does not allow us to use any martingale methods. Using instead optimal estimates for the probability that the supremum of a Gaussian process exceeds a certain level, we derive concentration estimates for the solution of the SPDE, provided that the Hurst index $H$ of the fractional Brownian motion satisfies $H>\frac14$. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any $H\in(0,1)$.