Yosida approximations of stochastic differential equations in infinite dimensions and applications, by T. E. Govindan, Probability Theory and Stochastic Modelling

Stochastic partial differential equations are simply partial differential equations in the presence of uncertainty. Uncertainty, in its simplest form, is modeled by (or taken as) the time derivative (in the sense of distributions) of a Wiener process, known commonly as white noise. The introduction of a random force in a partial differential equation (PDE) arises from the need to explain the fluctuations observed in physical phenomena and to account for external disturbances and measurement errors. Randomness is also introduced in order to take advantage of certain special methods and tools in stochastic analysis, such as the Stroock–Varadhan martingale problems and the Girsanov transformation, that do not have a counterpart in the theory of partial differential equations. This opens up the possibility of solving partial differential equations when perturbed by a noise term that are otherwise unsolvable. The addition of a noise term also allows one to study problems, such as the existence of an invariant measure, ergodic behavior of solutions, and the large deviation principle, which do not have meaning in a deterministic setting. A stochastic partial differential equation, in an abstract evolution setup, is an infinite-dimensional stochastic differential equation. To explain the various terminologies and stochastic analysis of such equations, we start with a complete probability space (Ω,F , P ), where Ω is an abstract space, F is a σ-field of subsets of Ω, and P is a probability measure on F . Let (Ft)t≥0 be an increasing family of sub-σ-fields of F , such that F0 contains all P -null sets in F , and let Ft = ⋂ r>tFr for all t ≥ 0. We will call (Ω,F , {Ft}, P ) a stochastic basis. Let X and Y be two real separable Hilbert spaces. Consider the following basic X-valued stochastic differential equation on [0, T ]: