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In May 2006, The University of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The goal of this minicourse was to introduce graduate students and recent Ph.D.s to various modern topics in stochastic PDEs, and to bring together several experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic partial differential equations. This monograph contains an up-to-date compilation of many of those lectures. Particular emphasis is paid to showcasing central ideas and displaying some of the many deep connections between the mentioned disciplines, all the time keeping a realistic pace for the student of the subject.

Stochastic partial differential equations --- Stochastic differential equations --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Stochastic partial differential equations. --- Stochastic differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Mathematics. --- Integral equations. --- Partial differential equations. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Partial Differential Equations. --- Integral Equations. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Partial differential equations --- Equations, Integral --- Functional equations --- Functional analysis --- Math --- Science --- Differential equations --- Fokker-Planck equation --- Differential equations, Partial --- Distribution (Probability theory. --- Differential equations, partial. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Distribution (Probability theory)

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This volume presents the lecture notes from two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second Barcelona Summer School on Stochastic Analysis. René Schilling’s notes are an expanded version of his course on Lévy and Lévy-type processes, the purpose of which is two-fold: on the one hand, the course presents in detail selected properties of the Lévy processes, mainly as Markov processes, and their different constructions, eventually leading to the celebrated Lévy-Itô decomposition. On the other, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, making it possible to study the properties of Feller processes as space inhomogeneous processes that locally behave like Lévy processes. The presentation is self-contained, and includes dedicated chapters that review Markov processes, operator semigroups, random measures, etc. In turn, Davar Khoshnevisan’s course investigates selected problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and to deduce, as an application, comparison-type results. The framework in which these problems are addressed goes beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as the existence and uniqueness of a solution for the corresponding equation, become relevant in their own right. These aspects are also developed and supplemented by a wealth of illustrative examples.

Partial differential equations --- Operational research. Game theory --- Probability theory --- Mathematics --- differentiaalvergelijkingen --- waarschijnlijkheidstheorie --- stochastische analyse --- wiskunde --- kansrekening

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Algebra --- Partial differential equations --- Operational research. Game theory --- differentiaalvergelijkingen --- algebra --- stochastische analyse --- kansrekening