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Given a function, $c(x,y):\mathbb{R}\times \mathbb{R}\to \mathbb{R}$, what are sufficient conditions for this to be the covariance of some (centered) Gaussian random field $X:\mathbb{R}\to \mathbb{R}$ …

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 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 …

I have (presumably non-degenerate) Gaussian $\mu$ over separable Banach space $X$. I would like to prove that for any ball of radius $r$ centered at $x$, $\mu(B_r(x))$. I know how to prove this in th …

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_ …

I asked this over on cross validated, but thought it might also get an answer here: The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the …