Applications of a simple but useful technique to stochastic convolution of $\alpha$-stable processes

TL;DR

This paper introduces a simple integration by parts technique to analyze stochastic convolution of $ ext{alpha}$-stable processes, providing new estimates, regularity results, and applications to stochastic PDEs with stable noise.

Contribution

The paper develops a novel integration by parts method for stochastic convolution, enabling uniform estimates and regularity results for $ ext{alpha}$-stable noises, which are not accessible by classical methods.

Findings

Established uniform estimates for $ ext{alpha}$-stable stochastic convolution.

Proved the stochastic convolution stays in small neighborhoods with positive probability.

Applied the technique to stochastic Burgers equation for trajectory regularity.

Abstract

Our simple but useful technique is using an integration by parts to split the stochastic convolution into two terms. We develop five applications for this technique. The first one is getting a uniform estimate of stochastic convolution of $\alpha$-stable processes. Since $\alpha$-stable noises only have $p<\alpha$ moment, unlike the stochastic convolution of Wiener process, the well known Da Prato-Kwapie\'n-Zabczyk's factorization ([5]) is not applicable. Alternatively, combining this technique with Doob's martingale inequality, we obtain a uniform estimate similar to that of stochastic convolution of Wiener process. Using this estimate, we show that the stochastic convolution of $\alpha$-stable noises stays, with positive probability, in arbitrary small ball with zero center. These two results are important for studying ergodicity and regularity of stochastic PDEs forced by $\alpha$-stable noises ([9]). The third application is getting the same results as in [12]. The fourth one gives the trajectory regularity of stochastic Burgers equation forced by $\alpha$-stable noises ([9]). Finally, applying a similar integration by parts to stochastic convolution of Wiener noises, we get the uniform estimate and continuity property originally obtained by Da Prato-Kwapie\'n-Zabczyk's factorization.