Positivity of transition probabilities of infinite-dimensional diffusion processes on ellipsoids

TL;DR

This paper proves that certain infinite-dimensional diffusion processes have positive transition probabilities on ellipsoids, extending the concept of density positivity to infinite-dimensional spaces, with applications to stochastic PDEs.

Contribution

It establishes the positivity of transition probabilities for infinite-dimensional diffusions on ellipsoids, generalizing finite-dimensional density results to Hilbert spaces.

Findings

Transition probabilities are positive on ellipsoids for a broad class of diffusions.

Results apply to stochastic partial differential equations.

Provides a framework for understanding densities in infinite-dimensional stochastic processes.

Abstract

We consider diffusion processes in Hilbert spaces with constant non-degenerate diffusion operators and show that, under broad assumptions on the drift, the transition probabilities of the process are positive on ellipsoids associated with the diffusion operator. This is an infinite-dimensional analogue of positivity of densities of transition probabilities. Our results apply to diffusions corresponding to stochastic partial differential equations.