Random perturbations of Hamiltonian systems

Abstract

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Random perturbations of invariant Lagrangian tori of Hamiltonian vector fields

“…Problems closely related in their statements and in the approach based on the trajectories of the limit (as e ~ +0) Hamiltonian vector field were studied in the monograph [7] for Eq. (5) in case 1) and in [8,9] for Eq.…”

“…(5) in case 2). The asymptotics of the transition probability from one closed trajectory of a one-dimensional Hamiltonian system to another closed trajectory (the connected component) of this system under the action of random perturbations was studied in [7]. The multiplicative asymptotics of the fundamental solution of a multidimensional equation (5) with a degenerate diffusion operator was obtained in [8,9].…”

“…The results obtained in [7][8][9] are very fine and hold only under some restrictive conditions. In this paper we derive "coarse" asymptotic formulas that are simple and explicit and hold under rather general weak assumptions.…”

Dirichlet forms for singular one-dimensional operators and on graphs

“…In [10] it is explained how diffusion processes on graphs can arise from various limiting or averaging procedures of diffusion processes on higher dimensional sets. This theory has been worked out in more specific situations in [6][7][8][9]11]. To close the introduction we describe the contents of the paper in more detail.…”

On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes

“…Solutions of boundary-value and initial-boundary-value problems with the operator L ε and Dirichlet-type boundary conditions and x changing in a region G are expressed as expectations of functionals of the trajectory X ε t for t up to the first time τ ε (G) at which the process leaves the region G. This was the way in which many new results about the limiting behavior of solutions of problems with Dirichlet-type boundary conditions were obtained in the recent decades: [4,7,9].…”