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Concentration estimates for SPDEs driven by fractional Brownian motion

Nils Berglund, Alexandra Blessing

TL;DR

The paper develops concentration (sample-path) estimates for slowly time-dependent SDEs perturbed by additive fractional Brownian motion and for semilinear SPDEs driven by cylindrical fractional Brownian motion. It relies on Gaussian-process supremum bounds via Slepian's lemma and mean-square Hölder control (with exponent $\gamma=2H$), avoiding martingale methods and enabling results for all $H\in(0,1)$ in the SDE case and $H\in(\tfrac14,1)$ in the SPDE case. For one-dimensional SDEs, explicit variance bounds for non-autonomous fractional Ornstein--Uhlenbeck processes yield concentration around slow solutions in the linear case and around stable slow manifolds in the nonlinear case. For SPDEs on the torus, linear and nonlinear analyses give concentration in time-dependent fractional Sobolev norms $\|\cdot\|_{s,t}$ with $s\in(0,2H-\tfrac12)$, requiring $H>\tfrac14$, and use Schauder-type estimates to handle nonlinearities. Overall, the results extend Brownian ($H=\tfrac12$) concentration phenomena to general fractional noise, providing quantitative bounds with explicit $\varepsilon$-dependent rates and paving the way for applications to bifurcations and slow-fast dynamics under memory effects.

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 $H\in(0,1)$.

Concentration estimates for SPDEs driven by fractional Brownian motion

TL;DR

The paper develops concentration (sample-path) estimates for slowly time-dependent SDEs perturbed by additive fractional Brownian motion and for semilinear SPDEs driven by cylindrical fractional Brownian motion. It relies on Gaussian-process supremum bounds via Slepian's lemma and mean-square Hölder control (with exponent ), avoiding martingale methods and enabling results for all in the SDE case and in the SPDE case. For one-dimensional SDEs, explicit variance bounds for non-autonomous fractional Ornstein--Uhlenbeck processes yield concentration around slow solutions in the linear case and around stable slow manifolds in the nonlinear case. For SPDEs on the torus, linear and nonlinear analyses give concentration in time-dependent fractional Sobolev norms with , requiring , and use Schauder-type estimates to handle nonlinearities. Overall, the results extend Brownian () concentration phenomena to general fractional noise, providing quantitative bounds with explicit -dependent rates and paving the way for applications to bifurcations and slow-fast dynamics under memory effects.

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 of the fractional Brownian motion satisfies . As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any .
Paper Structure (8 sections, 9 theorems, 62 equations)

This paper contains 8 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Fractional Brownian motion
  3. The one-dimensional SDE case
  4. Linear case
  5. Nonlinear case
  6. The SPDE case
  7. Linear case
  8. Nonlinear case

Key Result

Theorem 2.2

Let $(X_t)_t$ be a continuous Gaussian process with zero mean on $[0,T]$ for $T>0$. Assume that $(X_t)_t$ is mean-square Hölder continuous, i.e., there are constants $G$ and $\gamma$ such that Then there exists a constant $K := K(G,\gamma)$ such that for $c > 0$ and $A \subset [0,T]$, one has where $\sigma^2(A) := \sup_{t \in A} \text{\em Var} \left( X_t\right)$.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4: Concentration estimate for the linear SDE
  • proof
  • ...and 14 more