Global properties of invariant measures

Abstract

We study global regularity properties of invariant measures associated with second order differential operators in R N . Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.

Kernel estimates for Schrödinger operators

“…We give several examples and show the optimality of such estimates with respect to t. In particular, in the case of polynomial potentials the estimate is similar to that obtained in [2,Corollary 4.5.5]. Similar techniques have been employed in [8] in the elliptic case, to obtain upper and lower bounds for the density of invariant measures of certain diffusion processes.…”

Invariant measures for semilinear SPDE’s with local Lipschitz drift coefficients and applications

“…In the case of stochastic reaction diffusion equations we thus obtain the square integrability of the nonlinear drift part F with respect to the invariant measure (see Theorem 4.4) which seemed not to be realizable following more classical approaches when dealing with non-gradient systems. Note that in finite dimensions the square integrability of the nonlinear drift F with respect to the invariant measure has been obtained also for non-gradient systems if the Kolmogorov operator corresponding to (1.1) has a specific Lyapunov function in the sense of Hasminskii's criterion, see [16]. The square-integrability of the drift F enables us to present in Theorem 3.4 wellposedness of the Kolmogorov equation associated with the SPDE (1.1) in the space L 1 (H, µ) by using analytic techniques presented in [19].…”

Global properties of transition pProbabilities of singular diffusions

“…Then p satisfies ∂ t p(x, y, t) = A * y p(x, y, t), t > 0, (1.3) where A * y denotes the adjoint operator of A, which acts on the variable y (see Lemma 2.1). The great amount of work devoted to these equations (see, e.g., [1]- [7], [12]- [14], [19], [20] and the references there) witnesses the interest towards global properties of solutions. Beside the effort to extend as far as possible the classical results on uniformly elliptic and parabolic equations, solution measures are important in stochastics, being stationary distributions in the elliptic case and transition probabilities in the parabolic one.…”

“…For global boundedness and Sobolev regularity, as well as Harnack inequalities and pointwise estimates in the elliptic case, we refer to [19] and [4]. Pointwise bounds on kernels of Schrödinger operators, which can be treated with methods similar to those of the present paper, are proved in [20].…”

“…Observe that estimates in this space are known for invariant measures, that is, for the limit, as t → ∞, of the transition kernels p(x, •, t), see [2], [5], [19], [4].…”