Theoretical analysis derives sample-path concentration estimates for slowly time-dependent SPDEs, establishing bounds when the Hurst index exceeds one-fourth.
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>14. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any H∈(0,1).
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Berglund et al. (2025) studied this question.
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