We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, polynomial mixing property is established. This done in the setting kinetic taking values an $$L^1$$ -weighted space.

The possible control of competitive invasion by infection of the invader and multiplicative noise is studied. The basic model is the Lotka-Volterra competition system with emergent carrying capacities. Several stationary solutions of the non-infected and infected system are identified as well as parameter ranges of bistability. The latter are used for the numerical study of invasion phenomena. ...

We first prove the second order convergence of the Strang-type splitting scheme for the nonlinear Schrödinger equation. The proof does not require commutator estimates but crucially relies on an integral representation of the scheme. It reveals the connection between Strang-type splitting and the midpoint rule. We then show that the integral representation idea can also be used to study the sto...

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by g(x, t) = D(x)T (t), and the behaviors of probability distributions, for some specific functions of D(x), are analyzed. In particula...

We prove the validity of an averaging principle for a class of systems of slow-fast reactiondiffusion equations with the reaction terms in both equations having polynomial growth, perturbed by a noise of multiplicative type. The models we have in mind are the stochastic Fitzhugh-Nagumo equation arising in neurophysiology and the Ginzburg-Landau equation arising in statistical mechanics.

We consider a slow passage through a point of loss of stability. If the passage is sufficiently slow, the dynamics are controlled by additive random disturbances, even if they are extremely small. We derive expressions for the ‘exit value’ distribution when the parameter is explicitly a function of time and the dynamics are controlled by additive Gaussian noise. We derive a new expression for t...

Edge detection is a fundamental issue in image analysis. The presence of speckle, which can be modelled as a strong multiplicative noise, complicates edge detection in Synthetic Aperture Radar (SAR) images. Several statistical edge detectors have been developed speciically for such images, based on the hypothesis that only one step edge is present in the analyzing window. We here concentrate on...

The Cauchy problem for systems of differential equations with multiplicative random perturbations in the form of infinitedimensional Ito integrals is studied. For the systems correct by Petrovskii, conditionally correct and incorrect we point out Gelfand–Shilov spaces of generalized functions where a generalized solution coincides with a mild solution. AMS Mathematics Subject Classification (20...