Concentration estimates for SPDEs driven by fractional Brownian motion

Nils Berglund and Alexandra Blessing (Neamțu)
Electron. Commun. Probab. 30:1-13 (2025)

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 > ¼. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any H ∈ (0,1).

Mathematical Subject Classification: 60G15, 60G17, 60H15.

Keywords and phrases: Concentration estimates, slow-fast systems, fractional Brownian motion, SPDEs.

 

Journal Homepage Published article: 10.1214/25-ECP664 MR4873043 Zbl8055062

PDF file (160 Kb) hal-04560006 arXiv/2404.16485