Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Introduction: Introduction An overview The general theory of large deviations: Large deviations and exponential tightness Large deviations for stochastic processes Large deviations for Markov processes and semigroup convergence: Large deviations for Markov processes and nonlinear semigroup convergence Large deviations and nonlinear semigroup convergence using viscosity solutions Extensions of viscosity solution methods The Nisio semigroup and a control representation of the rate function Examples of large deviations and the comparison principle: The comparison principle Nearly deterministic processes in $R^d$ Random evolutions Occupation measures Stochastic equations in infinite dimensions Appendix: Operators and convergence in function spaces Variational constants, rate of growth and spectral theory for the semigroup of positive linear operators Spectral properties for discrete and continuous Laplacians Results from mass transport theory Bibliography Index.

Large Deviations for Stochastic Processes

Stochastic Processes.- Diffusion Processes.- Introduction to Stochastic Differential Equations.- The Fokker-Planck Equation.- Modelling with Stochastic Differential Equations.- The Langevin Equation.- Exit Problems for Diffusions.- Derivation of the Langevin Equation.- Linear Response Theory.- Appendix A Frequently Used Notations.- Appendix B Elements of Probability Theory.

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Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations

A brief survey on graph models for wave propagation in thin structures is presented. Such models arise in many areas of mathematics, physics, chemistry and engineering (dynamical systems, nanotechnology, mesoscopic systems, photonic crystals etc). Considerations are limited to spectral problems, although references to works with other studies are provided.

Graph models for waves in thin structures

This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

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On the constructions of the skew Brownian motion

A number of asymptotic problems for "classical" stochastic processes leads to diffusion processes on graphs. In this paper we study several such examples and develop a general technique for these problems. Diffusion in narrow tubes, processes with fast transmutations and small random perturbations of Hamiltonian systems are studied.

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Diffusion Processes on Graphs and the Averaging Principle

Consider the dynamical system in R r defined by a smooth vector field b(x): 

$$ {\dot x_t} = b({x_t}),\quad {x_0} = x \in {R^r}.$$

(1.1)

Random perturbations of Hamiltonian systems

Diffusion processes on an open book and the averaging principle

We consider small perturbations of a simple completely integrable system with many degrees of freedom: a collection of independent one-degree-of-freedom oscillators (in the perturbed system the individual oscillators are no longer independent). We show that the long-time behavior of such a system, even in the case of purely deterministic perturbations, should, in general, be described as a stochastic process. The limiting stochastic process is a Markov process on an open book space corresponding to the collection of first integrals of the non-perturbed system.

Long-Time Behavior of Weakly Coupled Oscillators

In this chapter we shall consider perturbations of a dynamical system 

$$ {\dot x_t} = b({x_t}),\quad {{\text{x}}_{\text{0}}}{\text{ = x,}}\quad b(x) = ({b^1}(x),...,{b^r}(x))$$

(1.1)

by a white noise process or by a Gaussian process in general.

Alexander D. Wentzell

Papers

Diffusion Processes on Graphs and the Averaging Principle

Random perturbations of Hamiltonian systems

Diffusion processes on an open book and the averaging principle

Long-Time Behavior of Weakly Coupled Oscillators

Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point