Maximal dissipativity of class of elliptic degenerate operators in weighted L 2 spaces Giuseppe Da Prato Scuola Normale Superiore Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail: daprato@sns.it Alessandra Lunardi Dipartimento di Matematica, Universit`a di Parma Parco Area delle Scienze 53, 43100 Parma, Italy E-mail: lunardi@unipr.it www: http://math.unipr.it/∼lunardi April 19, 2005 Abstract We consider a degenerate elliptic Kolmogorov–type operator arising from second order stochastic differential equations in R n perturbed by noise. We study a realization of such an operator in L 2 spaces with respect to an explicit invariant measure, and we prove that it is m-dissipative. AMS Subject Classification: 35J70, 60H10, 37L40. Key words: Second order stochastic equations, degenerate elliptic operators, Kolmogorov operators, invariant measures. 1 Introduction We are concerned with a Kolmogorov operator in R 2n = R n x × R n y , Kϕ(x,y)= 1 2 Δ x ϕ(x,y) −〈My + x + D y U (y),D x ϕ(x,y)〉 + 〈x,D y ϕ(x,y)〉, (1.1) where M is a symmetric positive definite matrix, and U ∈ C 1 (R n , R) is a nonnegative function satisfying suitable assumptions. We stress that U and its derivatives may be unbounded, and even grow exponentially as |y|→ +∞. The operator K arises in the study of the second order stochastic initial value problem in R n ,    Y ′′ (t)= −MY (t) − Y ′ (t) − DU (Y (t)) + W ′ (t), Y (0) = y, Y ′ (0) = x. (1.2) 1