Some regularity results on the Ventcel-Freidlin quasi-potential function

Abstract

We consider regularity properties of the quasipotential function V defined by A. D. Ventcel and M. I. Freidlin in their work on asymptotically small random perturbations of stable dynamical systems. The regularity properties of V are important for the success of various asymptotic calculations carried out in the literature. Employing classical techniques from the calculus of variations and differential equations, we prove various results about the smoothness of V and its level sets. Among other things, there exists a dense connected open set, containing the stable point for the underlying dynamical system, in which V is continuously differentiable to the same degree as the Lagrangian involved in the defining variational problem.

Regularity of boundary quasi-potentials for planar systems

“…A number of facts relevant to our discussion of W can be gleaned from [5] and [6]. We take advantage of this where possible.…”

“…subject to the fixed endpoint conditions, x(0) = q~(0) and x(T) = q~(T). The characterization of these extremals was discussed in the proof of Theorem 1 of [5]. As explained there, the Euler-Lagrange equations may be cast in Hamiltonian form: x(t) and p(t) = Lv(x(t), ~c(t)) satisfy 5c = Hp(x, p),…”

“…The idea of our approach, taken from [5], is easily explained. We have already pointed out that the minimizing paths x(') form, together with their associated adjoint components p(-), a family of solutions to the Hamiltonian system (2.4), with (2.5).…”

“…The detailed approach to the exit problem of [2] required regularity properties of V beyond those implied directly by its variational characterization. Day and Darden [5] provided a study of these properties using analytical techniques including the stable manifold theorem. Our recent approach I-3] to the exit problem for regions with "characteristic boundary" requires similar smoothness properties of the boundary quasi-potential function W, defined by (1.4) below.…”

The exterior sphere condition for the Ventcel-Freidlin quasi-potential function

“…In a recent paper with Tom Darden, [3], we studied the Ventcel-Freidlin quasipotential function V(x) which plays an important role in the study of small random perturbations of stable dynamical systems. We developed a number of regularity results concerning V(x).…”