Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized Solutions

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 Itô 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)−ρ(H). Basic examples
Analysis of (S)−ρ(H)-valued processes
S-transform and Wick product. Hitsuda–Skorohod integral. Main properties. Connection with Itô integral
Generalized solutions to stochastic Cauchy problems in spaces of abstract stochastic distributions