Probability Theory and Related Fields

, Volume 128, Issue 3, pp 441–466 | Cite as

Random perturbations of dynamical systems and diffusion processes with conservation laws

  • Mark Freidlin
  • Matthias Weber


In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow component of the perturbed system in an appropriate time scale for a general class of perturbations. The phase space of the slow motion is a graph defined by the first integral. This is a natural generalization of the results concerning random perturbations of Hamiltonian systems. Considering diffusion processes as the unperturbed system allows to study the multidimensional case and leads to a new effect: the limiting slow motion can spend non-zero time at some points of the graph. In particular, such delay at the vertices leads to more general gluing conditions. Our approach allows one to obtain new results on singular perturbations of PDE’s.


Averaging principle Random Perturbations 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mark Freidlin
    • 1
  • Matthias Weber
    • 2
  1. 1.Department of MathematicsUniversity of MarylandMarylandU.S.A
  2. 2.Department of Information Technology and MathematicsUniversity of Applied SciencesDresdenGermany

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