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In studying $X$ (Banach space) valued stochastic processes, I tend to see two different norms used: $$ \sup_{t\leq T} \mathbb{E}[\|u(t)\|_{X}^p]^{1/p} $$ and $$ \mathbb{E}[\sup_{t\leq T} \|u(t)\|_X^p] …

In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form $$ \int_0^t f(s,X(s))ds. $$ If we …

In the case that I'm working with a separable Hilbert space, $H$, on which I have a trace class operator, $K$, that's coming from a Gaussian (i.e., $K$ is self-adjoint, and for simplicity, has trivial …

For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, equi …

I'm looking at a Martingale (actually a Martingale difference sequence), $$ M_n = \sum \delta M_n, $$ and I'd like to prove something about convergence. If Martingale is Hilbert space valued (infini …

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional Ana …

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form: \begin{equation} dX_t = f(X_ …