In this paper we study the effect on a dynamical system of small random perturbations type white noise:

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

The equation $u_t = u_{xx} + u(1 - u)$ is the simplest reaction-diffusion equation. Introduction of a small parameter allows construction geometric optics approximations for solutions such equations; these are approximated by step-functions with values 0 and 1. region where solution close to 1 propagates according Huygens principle corresponding velocity field $v(x, e)$ which calculated via New effects may emerge, as stops jumps wave front. Feynman-Kac formula implies that certain Cauchy...

ContentsIntroduction § 1. Null approximation and normal deviations 2. Large from the averaged system 3. deviations. Continuation 4. Moderate 5. The behaviour of over large time intervals 6. Examples. Remarks 7. averaging principle for stochastic differential equations 8. Inequalities probabilities deviationsReferences

1 Stochastic Processes Defined by ODE's.- 2 Small Parameter in Higher Derivatives: Levinson's Case.- 3 The Large Deviation 4 Averaging Principle for and Partial Differential Equations.- 5 Principle: Continuation.- 6 Remarks Generalizations.- 7 Diffusion PDE's Narrow Branching Tubes.- 8 Wave Fronts Reaction-Diffusion 9 Slowly Changing Media.- 10 Scale Approximation 11 Homogenization Processes.- References.

Article DataHistorySubmitted: 25 May 1963Published online: 17 July 2006Publication DataISSN (print): 0040-585XISSN (online): 1095-7219Publisher: Society for Industrial and Applied MathematicsCODEN: tprbau

Random fields ue(t,x) = (u\(t,x),... ,uen(t,x)), defined as the solutions of a system PDE due.-^ Lku% + h(x;u\,...,v?n) ee.k(t,x) are considered.Here L/t second-order linear elliptic operators, ç/t Gaussian white-noise fields, independent for different k, and e is small parameter.The most attention given to problem determining behavior invariant measure /x£ Markov process u\ (u\(t,-),... ,u^(t,-)) in space continuous functions -► 0, also describing transitions Uj between stable stationary...

Previous article Next On the Factorization of Non-Negative Definite MatricesM. I. FreidlinM. Freidlinhttps://doi.org/10.1137/1113046PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Kiyosi Ito, stochastic differential equations, Mem. Amer. Math. Soc., 1951 (1951), 51– MR0040618 (12,724a) 0054.05803 Google Scholar[2A] Itô, Stochastic equations in a differentiable manifold, Nagoya J., 1 (1950), 35–47 MR0038596 (12,425g) 0039.35103 CrossrefGoogle...