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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)

We consider a two-player, zero-sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite-dimensional space. We prove that the value function of the game is the unique viscosity solution of the corresponding Hamilton᎐Jacobi᎐Isaacs equation in the sense of M. G. Crandall Ž and P. L. Lions ''Evolution Equations, Control Theory and Biomathematics,'' .

This paper provides a numb e r o f w orking tools for the discussion of fully nonlinear parabolic equations. These include: a proof that the maximum principle which provides L 1 estimates of strong" solutions of extremal equations by L n+1 norms of the forcing term over the contact set" remains valid for viscosity solutions in an L n+1 sense, a gradient estimate in L p for p n + 1 n + 2 for solutions of extremal equations with forcing terms in L n+1 , the use of this estimate in improving the range of p for which the maximum principle rst alluded to holds obtaining some p n + 1 -but without the contact set , a proof of the strong solvability of Dirichlet problems for extremal equations with forcing terms in L p for some p n + 1 , and the twice parabolic di erentiability a.e. of W 2;1;p functions for n + 2 = 2 p . 0. Introduction. In this work we provide a number of tools for the discussion of nonlinear parabolic equations under appropriate structure conditions. In particular, we ll gaps in the current literature and thereby prove a full generalization of the maximum principle" see below to viscosity solutions of certain extremal fully nonlinear equations with measurable forcing terms. While going about this task, we m ust resolve certain existence questions, and this is done as well. The results obtained are formulated in terms of standard extremal equations so that they apply to many other equations. In addition, some new proofs of known to varying degrees results are given.Let 0 be constants and de ne P , X = , trace X + + trace X , for X 2 S n , the set of real symmetric n n matrices. Here trace X + respectively, trace X , is the sum of the positive eigenvalues of X respectively, ,X .Let 0 be another constant. We discuss several results concerning solutions of the parabolic inequality 0:1 L u : = u t + P , D 2 u , jDuj f as well as the equation L u = f in Q = 0; T , where is an open, bounded domain in IR n , n 2. Here Ducorresponds to the spatial gradient u x 1 ; : : : ; u x n , D 2 u corresponds to the spatial Hessian matrix u xi;xj and jpj is the Euclidean 1 length of p 2 IR n . The notation P , " re ects the fact that this function was introduced by Pucci as well as the property that P , X is the minimum of ,trace AX o v er A 2 S n with I A I. The extremal parabolic equations L u = f w ere rst studied by Astesiano 1 ; for the modern theory of nonlinear parabolic equations we refer to Krylov 24 and Lieberman 26 .The rst version of the maximum principle of concern to us is stated below for easy reference. It follows immediately from Tso's work 32 which re ned a result of Krylov 23 see also the contemporaneous works of Reye 30 and Nazarov and Ural'tseva 29 , and a recent paper of Cabre 3 . In the statement and hereafter, the parabolic boundary of Q is denoted by @ p Q, the diameter of b y d , and the diameter of Q by d Q : where C = C 1= ; n; jQj n+1 =d depends only on the indicated quantities and is bounded for bounded a r guments.Hereafter we refer to the inequality 0.4 as the maximum principle". Ge...

We consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions, where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous. We prove that the value function is a viscosity solution of the Bellman equation and is in fact the minimal nonnegative supersolution.