Dissertation

Geometric approach to evolution problems in metric spaces

Promotor: S.M. Verduyn Lunel, Co-promotor: O.W. van Gaans

Author

Igor Stojković

Date

19 April 2011

Links

Thesis in Leiden Repository

This PhD thesis contains four chapters where research material is presented. In the second chapter the extension of the product formulas for semigroups induced by convex functionals, from the classical Hilbert space setting to the setting of general CAT(0) spaces. In the third chapter, the non-symmetric Fokker-Planck equation is treated as a flow on the Wasserstein-2 space of probability measures, and it is proven that its semigroup of solutions possesses similar properties to those of the gradient flow semigroups. In the forth chapter, a general theory of maximal monotone operators and the induced flows on Wasserstein-2 spaces is developed. This theory generalizes the theory of gradient flows by Ambrosio-Gigli-Savare. In the final fifth chapter the problem of the existence of an invariant measure for stochastic delay equations is proven. The diffusion coefficient has delay, and is assumed to be locally Lipschitz and bounded.

• approximation theory
• cat(0) spaces
• convex functionals
• fokker-planck equations
• gradient flows
• invariant measure
• levy processes
• maximal monotone operators
• metric spaces
• product formulas
• regularizing effect
• resolvents
• semigroups
• stochastic processes
• variation-of-constants formula