G. Da Prato

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.

A stochastic filippov theorem

We prove a generalization of the Filippov Theorem, [4], for stochastic differential inclusions, and present an application to linearization of differential inclusions and to infinitesimal behaviour of solutions

Periodic and almost periodic solutions for semilinear stochastic equations

We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space. Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. Similar results hold in the affine case under mean square stability of the linear part of the equation.

Stochastic nagumo's viability theorem

This paper is devoted to viability of random set-valued variables by stochastic differential equations, characterized in terms of stochastic tangent sets to random set-valued variables

Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov’s fundamental result on Rd to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin–Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

On Parabolic Equations for Measures

A new existence result is established for weak parabolic equations for probability measures on ℝ d . A priori estimates for solutions of such equations are obtained.

Existence and Regularity for a Class of Integrodifferential Equations of Parabolic Type

The method we use rests on the inversion of the formal Laplace transform of R and allows one to analyze the regularity of the solution U. Problem (1.1) has been studied by several authors using various methods (see, for instance, [l-18, 211). The Laplace transform approach presented here has been used in [S, 7-9, 14, 153. In our previous papers [S, 7-93 we have in fact analyzed a more general situation while here we take full 36 0022-247X/85

STRONG FELLER PROPERTY FOR STOCHASTIC SEMILINEAR EQUATIONS.

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Convergence to equilibrium for classical and quantum spin systems

A method from stochastic flow theory is used to obtain smoothing properties of the transition semigroup Pt of a class of stochastic differential equations on Hilbert space. The equations considered may have unbounded coefficients and include such stochastic partial differential equations as for Xt in L2 (0,π) In certain cases a formula for the Frechet derivative of Ptf is given, exhibiting this smoothing property

Parabolic equations for measures on infinite-dimensional spaces

SummaryThe paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].