Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach
Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appeared here for the first time in book form. The authors start with a detailed analysis of Lévy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical Lévy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.
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Why equations with Levy noise?
Poisson random measures
Cylindrical processes and reproducing kernels
Stochastic parabolic problems
Wave and delay equations
Equations driven by a spatially homogeneous noise
Equations with noise on the boundary
Stochastic Burgers equation
Environmental pollution model
Assume Banach space boundary conditions bounded linear operator cadlag Co-semigroup compound Poisson process Consequently consider constant continuous functions continuous trajectories converges convolution semigroup covariance operator cylindrical Wiener process defined Deﬁnition denote Dirichlet domain existence Feller family finite following result formula Gaussian given Hence Hilbert space Hilbert-Schmidt inequality inﬁnite invariant measure jump L´evy noise Laplace operator Lemma Levy process Lipschitz continuous mapping Math Moreover non-negative non-negative-definite norm Note orthonormal basis Partial Differential Equations Peszat Poisson random measure Prato and Zabczyk problem Proof Let proof of Theorem Proposition random variables real-valued RKHS satisfying Section semigroup semigroup of measures sequence square integrable square integrable martingale square integrable mean-zero stochastic integral stochastic partial differential stochastic process submartingale subsection taking values unique solution v(dy wave equation Wiener process