Diffusion semigroups on abstract Wiener space

Piech, M. Ann

doi:10.1090/S0002-9947-1972-0295141-3

Diffusion semigroups on abstract Wiener space
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Trans. Amer. Math. Soc. 166 (1972), 411-430 Request permission

Abstract:

The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form $\partial u/\partial t = {L_x}u$ was previously established by the author. The present paper investigates the relationship between ${L_x}$ and the infinitesimal generator $\mathcal {U}$ of the semigroup. In particular, it is shown that $\mathcal {U}$ is the closure of ${L_x}$ in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.

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