Sub-diffusion processes in Hilbert space and their associated stochastic differential equations and Fokker-Planck-Kolmogorov equations

Abstract

This paper focuses on the time-changed Q-Wiener process, a Hilbert space-valued sub-diffusion. It is a martingale with respect to an appropriate filtration, hence a stochastic integral with respect to it is definable. For the resulting integral, two change of variables formulas are derived. Via a duality theorem for integrals, existence and uniqueness theorems for stochastic differential equations (SDEs) driven by the time-changed Q-Wiener process are discussed. Associated fractional Fokker-Planck-Kolmogorov equations are derived using either a time-changed It\^o formula or duality. Connections are established between three integrals driven by time-changed versions of the Q-Wiener process, cylindrical Wiener process, and martingale measure.