Semilinear Stochastic Equations Research Articles

Ergodic Control of Semilinear Stochastic Equations and the Hamilton–Jacobi Equation

In this paper we consider optimal control of stochastic semilinear equations with Lipschitz continuous drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton–Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we find the optimizing controls given in the feedback form. To obtain these results we prove also some new results on the transition semigroups of semilinear diffusion acting in the spaces of continuous function with the weighted sup norms and on the optimal control of semilinear diffusions for the discounted cost functional.

On a class of stochastic equations in hilbert spaces: solvability and smoothing properties

A semilinear stochastic equation in a Hilbert space is considered, with nonlinearity of gradient type and constant but possibly degenerate diffusion term. Under suitable assumptions, existence and uniqueness of the solution and some smoothing properties for the associated transition semigroup are proved. In particular, strong Feller property and irreducibility are deduced. The main tools are the Malliavin calculus and the Girsanov Theorem

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Existence of weak solutions for stochastic differential equations and martingale solutions for stochastic semilinear equations

Almost sure exponential stability for stochastic partial functional differential equations*

In this paper we consider the almost sure a symptotic behavior of mild solutions of the semilinear stochastic evolution equation with finite delays: where f, g have the Lipschitz condition and the linear growth condition. That is, we present the existence theorem, and the estimates of moment and almost sure Lyapunov exponent of the above equation. For illustrating the theorem we discuss a semilinear stochastic heat equation with finite delays

Semilinear stochastic equations for symmetric diffusions

New sufficient conditions are provided for the existence of weak solutions to a class of semilinear stochastic equations, which at the same time guarantee that the solution is a symmetric ergodic right process. The invariant measure (which is also the symmetrizing measure) is given explicitly. The emphasis is on the fact that due to the symmetry in our situation, the assumptions on the nonlinear term can be relaxed considerably

Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures

Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures

A semilinear Mckean-Vlasov stochastic evolution equation in Hilbert space

This paper deals with a semilinear stochastic equation in a real Hilbert space and formulates a related McKean-Vlasov type measure-valued evolution equation. It is shown that the stochastic equation has a unique mild solution such that the corresponding probability law is the unique measure-valued solution of McKean-Vlasov evolution equation.

Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupRt are obtained. We show thatRt is a compactC0-semigroup in all Sobolev spacesWn,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inLp(μ) spaces and spacesW1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inLp(μ) for allp≧1.

Asymptotic stability theorems of semilinear stochastic evolution equations in hilbert spaces

In this paper we consider exponentially asymptotic stability in the p-th mean of the equilibrium mild solution to a semilinear stochastic evolution equation in a separable Hilbert space H and discuss under what conditions any mild solution approaches exponentially the equilibrium mild solution, almost surely. We shall generalize Haussmann's results [J. Math. Anal. Appl., 65 (1978), 219-235]. For illustrating the theorems we consider a semilinear stochastic heat equation

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.

Strong feller property for stochastic semilinear equations

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 Fréchet derivative of Ptf is given, exhibiting this smoothing property

Non-explosion, boundedness, and ergodicity for stochastic semilinear equations

Sufficient conditions for non-explosion, boundedness, and ergodicity of solutions of a class of semilinear stochastic equations are given. Applications of the results are based on a new regularity theorem for solutions of the linear problem.

Invariant measures for semilinear stochastic equations

Invariant measures for semilinear stochastic equations

A note on semilinear stochastic equations

A note on semilinear stochastic equations