Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise

Abstract

We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise. Under additional assumptions, we prove Gâteaux and Fréchet differentiability of solutions with respect to the initial datum. As an application, we obtain gradient estimates for the resolvent associated to the mild solution. Finally, we prove the strong Feller property of the associated semigroup.

Maximal Inequalities of the Itô Integral with Respect to Poisson Random Measures or Lévy Processes on Banach Spaces

“…However, it is straightforward to show that these two inequalities imply two inequalities which are valid for martingales driven by Lévy processes of pure jump type. Related inequalities, are proven in Marinelli et al, Lemma 2.2 [21], Bass and Cranston, Lemma 5.2 [6], and Protter and Talay, Lemma 4.1 [26]. However, the inequalities 1.1 and 1.2 are new in this general form.…”

On the Stability of Stochastic Jump Kinetics

“…In [10] jump stochastic partial differential equations are treated, and existence/uniqueness results as well as ergodic results for the case of a multiplicative noise, are found in [11] [12]. Numerical aspects in a similar setting are discussed in [13].…”

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

“…For the stochastic integral, using the result in ( [26], Corollary 3.1, Remark 3.6 or [40], Lemma 3.1), we have the estimate…”

“…For this purpose, we need to estimate the fourth order moment of the stochastic integral with respect to a compensated Poisson measure. This is achieved by using the result in ( [26], Corollary 3.1, Remark 3.6 or [40], Lemma 3.1) concerning the maximal inequality for stochastic integral with respect to a compensated Poisson measure. In Theorem 2, we prove that the whole sequence of the Galerkin approximation converges in mean square to the strong solution of the stochastic 3D LANS-α model.…”