Ergodic Properties of SPDEs driven by Levy Noise

Stochastic partial differential equations (SPDEs) arise in the description on of spatially distributed systems in the same way as do ordinary partial differential equations (PDEs) with the additional feature that the systems described have an intrinsically random nature or are subjected to random perturbations. SPDEs are usually treated as stochastic differential equations in infinite dimensional spaces, e.g. a parabolic SPDE corresponds to a stochastic equation of the following form dXt =(AX t +F(Xt ))dt + B(Xt )d W(t), X(0)= x0 , where x0 belongs to E, where E is a suitable Banach space. A is a generator semigroup and F and B are generally discontinuous mappings acting on appropriate spaces. Moreover, W stands for a Wiener process on a suitable Hilbert space H, defined on a stochastic basis, i.e. an appropriate probability space. SPDEs are used for example in neurophysiology, mathematical finance, chemical reaction-diffusion, population dynamics, environmental pollution and nonlinear filtering. In the example the intrinsically randomness is described by Gaussian noise. But - for instance - in neurophysiology the driving noise of the cable equation is basically impulsive, e.g.\ of a Poisson type. Thus, random variables of Poisson type provide often a better description of real life phenomena than their Gaussian counterparts and from the point of view of applications one might feel that the restriction on Gaussian noise is unsatisfactory; to handle such cases one can replace the Gaussian noise by a Lévy process. In recent years Lévy randomness began to draw much attention. Nevertheless, Lévy randomness needs other techniques, is quite intricate and far from amenable to mathematical analysis. Moreover, the dynamical behaviour of SPDEs changes essentially, if the Brownian noise is replaced by Lévy noise. Ergodic theory is a field of mathematics which studies the long time behaviour of dynamical systems, i.e. the problem of qualitative properties of systems, as time tends to infinity, is addressed. There is a wide range of possible applications, as can be demonstrated by applying results that have been obtained so far, e.g., to problems of mathematical physics and biology (stochastic hydrodynamics, reaction and diffusion kinetics, spin systems, or population dynamics). In particular, the problem of long time dynamics for climate models could be investigated, as well as its dependence on parameters, like the concentration of carbon and nitrogen. In the project we will consider the Ergodic Properties of Stochastic Partial Differential Equations driven by Lévy noise.

Stochastic partial differential equations (SPDEs) arise in the description on of spatially distributed systems in the same way as do ordinary partial differential equations (PDEs) with the additional feature that the systems described have an intrinsically random nature or are subjected to random perturbations. SPDEs are usually treated as stochastic differential equations in infinite dimensional spaces, e.g. a parabolic SPDE corresponds to a stochastic equation of the following form dXt =(AXt +F(Xt ))dt + B(Xt )d W(t), X(0)= xo, where x0 belongs to a suitable Banach space. The operator A is a infinitesimal generator of a semigroup and F and B are generally discontinuous mappings acting on appropriate spaces. Moreover, W stands for a Wiener process on a suitable Hilbert space H, defined on a stochastic basis, i.e. an appropriate probability space. SPDEs are used for example in neurophysiology, mathematical finance, chemical reaction-diffusion, population dynamics, environmental pollution and nonlinear filtering. In the example the intrinsically randomness is described by Gaussian noise. But - for instance - in neurophysiology the driving noise of the cable equation is basically impulsive, e.g. of a Poisson type. Thus, random variables of Poisson type provide often a better description of real life phenomena than their Gaussian counterparts and from the point of view of applications one might feel that the restriction on Gaussian noise is unsatisfactory; to handle such cases one can replace the Gaussian noise by a Levy process. In recent years Levy randomness began to draw much attention. Nevertheless, Levy randomness needs other techniques, is quite intricate and far from amenable to mathematical analysis. Moreover, the dynamical behaviour of SPDEs changes essentially, if the Brownian noise is replaced by Levy noise. Ergodic theory is a field of mathematics which studies the long time behaviour of dynamical systems, i.e. the problem of qualitative properties of systems, as time tends to infinity, is addressed. There is a wide range of possible applications, as can be demonstrated by applying results that have been obtained so far, e.g., to problems of mathematical physics and biology (stochastic hydrodynamics, reaction and diffusion kinetics, spin systems, or population dynamics). In particular, the problem of long time dynamics for climate models could be investigated, as well as its dependence on parameters, like the concentration of carbon and nitrogen. In the project we will consider the Ergodic Properties of Stochastic Partial Differential Equations driven by Levy noise.