Équations aux dérivées partielles stochastiques avec bruit de Lévy

Abstract

In this thesis, we develop a stochastic calculus for the space-time Lévy white noise introduced in [1] as an alternative for the Gaussian white noise perturbing an stochastic partial differential equation (SPDE). We give a new proof for the Itô formula for some integral processes related to this Lévy white noise. Then, we consider a general non-linear SPDE on R_+* R driven by this Lévy white noise and we show that this equation has a unique random-field solution. Using Rosenthal's inequality, we develop a maximal inequality for the moments of order p≥2 of the stochastic integral with respect to this noise. Based on this inequality, we show that the stochastic wave equation equation has a unique solution, which is weakly intermittent in the sense of [2, 3]. Finally, we develop a Malliavin calculus with respect to the compensated Poisson random measure associated to the Lévy white noise. Under certain conditions, we show that the solution is Malliavin differentiable and its Malliavin derivative satisfies an integral equation. [1] Integration with respect to Lévy colored noise, with application to SPDEs: Stochastics An International Journal of Probability and Stochastic Processes , 87, 363-381. [2] Intermittence and nonlinear parabolic stochastic partial differential equations. Electronic Journal of Probability. Vol 21, 548-568. [3] Analysis of stochastic partial differential equations. CBMS Regional Conference Series in Mathematics, Vol 119. American Mathematical Society.