Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions
I. V. Melnikova · CERN Document Server (European Organization for Nuclear Research) · 2016
Well-Posed and Ill-Posed Abstract Cauchy Problems. The Concept of Regularization Semi-group methods for construction of exact, approximated, and regularized solutions The Cauchy problem and strongly continuous semi-groups of solution operators The Cauchy problem with generators of regularized semigroups: integrated, convoluted, and R-semi-groups R-semi-groups and regularizing operators in the construction of approximated solutions to ill-posed problems Distribution methods for construction of generalized solutions to ill-posed Cauchy problems Solutions in spaces of abstract distributions Solutions in spaces of abstract ultra-distributions Solutions to the Cauchy problem for differential systems in Gelfand-Shilov spaces Examples. Supplements Examples of regularized semi-groups and their generators Examples of solutions to Petrovsky correct, conditionally correct and incorrect systems Definitions and properties of spaces of test functions Generalized Fourier and Laplace transforms. Structure theorems Infinite-Dimensional Stochastic Cauchy Problems Weak, regularized, and mild solutions to Ito integrated stochastic Cauchy problems in Hilbert spaces Hilbert space valued variables, processes, and stochastic integrals. Main properties and results Solutions to Cauchy problems for equations with additive noise and generators of regularized semi-groups Solutions to Cauchy problems for semi-linear equations with multiplicative noise Extension of the Feynman-Kac theorem to the case of relations between stochastic equations and PDEs in Hilbert spaces Infinite-dimensional stochastic Cauchy problems with white noise processes in spaces of distributions Generalized solutions to linear stochastic Cauchy problems with generators of regularized semi-groups Quasi-linear stochastic Cauchy problem in abstract Colombeau spaces Infinite-dimensional extension of white noise calculus with application to stochastic problems Spaces of Hilbert space valued generalized random variables: (S)-rho(H). Basic examples Analysis of (S)-rho(H)-valued processes S-transform and Wick product. Hitsuda-Skorohod integral. Main properties. Connection with Ito integral Generalized solutions to stochastic Cauchy problems in spaces of abstract stochastic distributions