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We study transition semigroups and Kolmogorov equations corresponding to stochastic semilinear equations on a Hilbert space H. It is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on H when endowed with the so-called mixed topology. As a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups. Academic Press

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We study Lyapunov functions for infinite-dimensional dynamical systems governed by general maximal monotone operators. We obtain a characterization of Lyapunov pairs by means of the associated Hamilton-Jacobi partial differential equations, whose solutions are meant in the viscosity sense, as evolved in works of Tataru and Crandall-Lions. Our approach also leads to a new sufficient condition for Lyapunov pairs, generalizing a classical result of Pazy. # 2002 Elsevier Science (USA)