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We consider diffusion processes in Hilbert spaces with constantnon-degenerate diffusion operators and show that, under broad assumptions onthe drift, the transition probabilities of the process are positive onellipsoids associated with the diffusion operator. This is aninfinite-dimensional analogue of positivity of densities of transitionprobabilities. Our results apply to diffusions corresponding to stochasticpartial differential equations.

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