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

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.