Stochastic partial differential equations driven by cylindrical Lévy processes

Abstract

Cylindrical Lévy processes provide a unified framework for different kinds of infinite-dimensional Lévy noise considered in the literature. In the recent papers by Jakubowski and Riedle, the integral with respect to cylindrical Lévy processes was defined. The aim of the thesis is to prove the existence and uniqueness of solutions to the stochastic differential equations driven by such processes. Equations with specific kinds of noises have been considered for many years, motivating the study of this generalised equation.To be precise, we are considering the equation dX(t) = F (X(t)) dt + G(X(t)) dL(t). Here L is a cylindrical Lévy process from U to L0(Ω). Typically, F contains some differential operator. We assume that F and G satisfy monotonicity and coercivity assumptions and solve this equation in the so-called variational approach. We cover the case of non-square-integrable noise of diagonal structure. We derive conditions when the behaviour of jumps of the cylindrical Lévy process enables the use of the interlacing construction.In another approach, we construct an integral with respect to cylindrical Lévy process integrable with some power p, 1 ≤ p < 2, in the Banach space setting. We show existence of solutions in the semigroup approach.Thirdly, we consider a canonical stable cylindrical Lévy process. Assuming that the functions appearing in the equation map between domains of certain powers of a generator of a strongly continuous semigroup we prove existence and uniqueness of solutions using tightness arguments and the Yamada–Watanabe theorem. In the proofs we make use of tail and moment inequalities, which are new in the case of cylindrical noise.

Date of Award	1 Mar 2020
Original language	English
Awarding Institution	King's College London
Supervisor	Markus Riedle (Supervisor)