Fast-diffusion limit with large noise for systems of stochastic reaction-diffusion equations

We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.     

Stabilization of evolution equations by noise

We consider a deterministic equation of evolution

in a separable, real Hilbert space. We prove that if generates a -semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise.     

Random attractors for stochastic reaction-diffusion equations on unbounded domains

The existence of a pullback attractor is established for a stochastic reaction-diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction-diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions.     

Large deviations and approximations for slow-fast stochastic reaction-diffusion equations

A large deviation principle is derived for a class of stochastic reaction-diffusion partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.     

Bounds for a class of stochastic recursive equations

In this note we develop a framework for computing upper and lower bounds of an exponential form for a class of stochastic recursive equations with uniformly recurrent Markov modulated inputs. These bounds generalize Kingman's bounds for queues with renewal inputs.     

Optimal time delays in a class of reaction-diffusion equations

ABSTRACT

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.     

Stabilization of a class of nonlinear stochastic systems

The aim of this paper is to study the stabilizability for a class of multi-inputs nonlinear stochastic systems. We prove under general assumptions the existence of a linear feedback law which stabilizes in probability the system at its equilibrium.     

Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise

In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.     

Asymptotic behavior of a class of reaction-diffusion equations with delays

The authors analyze asymptotic behavior of the partial functional differential equations, and the sufficient conditions on existence of global attractor of a class of reaction-diffusion equations with delays are given.     

Stabilization of a class of nonlinear composite stochastic systems

This paper deals with the stability for a class of nonlinear composite stochastic systems by feedback laws.Firstly,we give sufficient conditions for the existence of feedback laws which render the equilibrium solution of the stochastic system globally asymptotically stable in probability.Secondly,for stochastic systems of the same type,we prove that there exists a linear feedback law which exponentially stabilizes in mean square the closed–loop stochastic system at its equilibrium.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains

In this paper we study the asymptotic dynamics for stochastic reaction-diffusion equation with multiplicative noise defined on unbounded domains. We investigate the existence of a random attractor for the random dynamical system associated with the equation. The asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far-field values of solutions and a cut-off technique.     

A limit theorem for a class of stochastic integral equations

We obtain an existence and uniqueness theorem for a measurable solution of an Itô-Volterra stochastic integral equation and a limit theorem on convergence of sequences of solutions of such equations.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 47–53, 1987.