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In this paper we extend the derivation of mezoscopic partial differential equations (or stochastic partial differential equations (SPDE's)) from particle systems with finite conserved mass to infinite conserved mass. We also sketch the history of stochastic and deterministic (i.e., mezoscopic and macroscopic) reaction-diffusion models initiated by Arnold's work as well as some results of Dawson's measure processes approach to SPDE's. At the end of the paper we show how to include creation and annihilation through a fractional step method into the mezoscopic PDE's. [ABSTRACT FROM AUTHOR]

In terms of the characteristic exponents, we provide conditions under which the second fundamental form describing the second order approximation of the image of a submanifold of the Euclidean space behaves asymptotically under the effect of a random dynamical system with independent increments as a positive recurrent Markov process. [ABSTRACT FROM AUTHOR]

Methods of classical fractional calculus are applied to generalized Stieltjes and stochastic integration theory. Under these aspects we also consider stochastic differential equations driven by processes with generalized quadratic variations. The paper gives a survey on this approach. [ABSTRACT FROM AUTHOR]

We discuss various methods and problems pertaining to that part of Random Dynamical Systems which deals with real noise processes. We consider the theory of random orthogonal polynomials, and show how methods applied to the study of the random Schrödinger operator can be generalized to study them. We further discuss certain questions in the area of random bifurcation theory; we formulate and illustrate a "robustness" criterion pertaining to bifurcation in the presence of real noise. Finally, we give a brief survey of some other recent advances in the field. [ABSTRACT FROM AUTHOR]

We give a review of our results related to stochastic analysis on product manifolds (infinite products of compact Riemannian manifolds). We introduce differentiable structures on product manifolds and prove the existence and uniqueness theorem for stochastic differential equations on them. This result is applied to the construction of Glauber dynamics for classical lattice models with compact spin spaces. We discuss the relations between ergodicity of the dynamics and extremality of the corresponding Gibbs measures. Further, we construct the associated stochastic dynamics in the space of macroscopic fluctuations of our system. [ABSTRACT FROM AUTHOR]

Random dynamical systems are intrinsically nonautonomous and are formulated in terms of cocycles rather than semigroups. They consequently require generalizations of the commonly used dynamical systems concepts such as attractors and invariance of sets. This cocycle formalism is reviewed here and then the approximation of such dynamical behaviour by time discretized numerical schemes is discussed, outlining results that have been obtained and those that remain to be resolved. [ABSTRACT FROM AUTHOR]

This article exploits a thermodynamic formalism and the mathematics of diffusion processes to investigate the evolutionary stability of structured populations, that is the invulnerability of the populations to invasion by rare mutants. Growth in structured populations is described in terms of random dynamical systems, and random transfer operators are used to obtain and characterize the steady state in terms of a set of macroscopic parameters. We analyze the extinction dynamics of interacting populations via coupled diffusion equations and show that evolutionary stability is characterized by the extremal states of entropy. [ABSTRACT FROM AUTHOR]

In this paper we review some results about the approximation of the upper Lyapunov exponents λ of linear and nonlinear diffusion processes X. The stochastic differential system solved by X is discretized by the Euler scheme. Under appropriate assumptions, the upper Lyapunov exponent $$ \bar \lambda $$ of the resulting approximate process $$ \bar X $$ is well defined and can be efficiently computed by simulating one single trajectory of $$ \bar X $$ during a time long enough. We describe the mathematical technique which leads to estimates on the convergence rate of $$ \bar \lambda $$ to λ. We start by an elementary example, then we deal with linear systems, and finally we consider nonlinear systems. [ABSTRACT FROM AUTHOR]

In order to investigate the long term behavior of a linear dynamical system under the impact of multiplicative mean zero noise, the top Lyapunov exponent associated with the system is studied with its dependence upon the noise intensity and other parameters of the (mostly 2-dimensional) systems. The perturbation methods, most singular, are surveyed systematically, which yield (a) asymptotic expansions of the Lyapunov exponents in terms of small and large intensities of different kinds of noise; (b) a comparison of white and real noise and (c) a characterization of stabilizing noise. [ABSTRACT FROM AUTHOR]

The theory of dynamical systems has become a center piece in the systematic study of systems with deterministic or stochastic perturbations, based on measurable, topological, and smooth dynamics. Recent developments also forge a close connection between control theory and topological and smooth dynamics. On the other hand, the support theorem of Stroock and Varadhan shows how control theoretic techniques may aid in the Markovian analysis of systems perturbed by diffusion processes. This paper presents an overview of topological, smooth, and control techniques and their interrelations, as they can be used in the study of perturbed systems. We concentrate on global analysis and parameter dependent perturbation systems, where we emphasize comparison of the Markovian and the dynamical structure of systems with Markovian diffusion perturbation process. A series of op en problems highlights the areas in which the interconnections between different techniques and system classes are not (yet) well understood. [ABSTRACT FROM AUTHOR]

We study a canonical stochastic differential equation (SDE) with jumps driven by a Lévy process. The equation is defined through the canonical stochastic integral, which is different from the Itô integral or the Stratonovich one. A feature of the canonical SDE is that it is a coordinate free representation of an SDE with jumps and the solution admits a nice geometrical interpretation, making use of the integral curves of vector fields appearing in the equation. A main result of this paper is stated in Theorem 3.3, where we determine the support of the probability distribution of the solution of the canonical SDE. It is characterized through solutions of control systems with jumps for ordinary integro-differential equations. It can be regarded as an extension of the support theory for continuous SDE due to Stroock and Varadhan. [ABSTRACT FROM AUTHOR]

We review definitions of random hyperbolic sets and introduce a characterization using random cones. Moreover we discuss problems connected with symbolic representations and the thermodynamic formalism for random hyperbolic systems both in discrete and continuous time cases. In the discrete time case we prove the existence of Markov partitions to guarantee symbolic dynamics and the existence of SRB-measures, while in the continuous time case we explain why a respective method does not work. We illustrate the theory with a number of examples. [ABSTRACT FROM AUTHOR]