The Onsager-Machlup function as Lagrangian for the most probable path of a diffusion process

Abstract

By application of the Girsanov formula for measures induced by diffusion processes with constant diffusion coefficients it is possible to define the Onsager-Machlup function as the Lagrangian for the most probable tube around a differentiable function. The absolute continuity of a measure induced by a process with process depending diffusion w.r.t. a quasi translation invariant measure is investigated. The orthogonality of these measures w.r.t. quasi translation invariant measures is shown. It is concluded that the Onsager-Machlup function cannot be defined as a Lagrangian for processes with process depending diffusion coefficients.

Transition paths in molecules at finite temperature

“…11 Studying the properties of these minimizers, for large inverse temperature ␤ and for various choices of the transition time U, is the subject of this paper. For large U it is intuitive that minimization will tend to occur via solutions, which are approximately constant on minimizers of G and exhibiting rapid transitions ͑relative to time-scale U͒ between different such minimizers; we refer to this phenomena as segmenting.…”

“…Further iteration of the minimization algorithm leads to paths that make still larger excursions in x 2 and consequently lead to continued decrease in I. Numerically this appears to continue indefinitely, and thus we encounter yet another unphysical result, which results from the minimization's over-reliance on the minima of G. Note however that we expect entropic effects to eliminate this behavior when the effect of noise is included. In particular, if one looks at a ball of probability centered on a path 11 instead of a path itself ͑which is a line and hence has zero probability͒, these long excursions would be seen as extremely unlikely despite the fact that they are minimizers of the OnsagerMachlup potential.…”

“…͑5͒ must hold. As indicated by Durr and Bach, 11 one must look for the maximum probabilities of a ball in pathspace. However taking the limit and looking at the line of maximal probability are fraught with pitfalls.…”

“…An analog for finite temperature transitions, over finite time intervals, is given by the Onsager-Machlup functional. 11 The purpose of this work is to investigate properties of Onsager-Machlup minimizers. We refer to the paths found this way as most probable paths, MPPs for short.…”

Characterization of Rare Events in Molecular Dynamics

“…More precisely, (ϕ) is the limiting ratio between the probability that the solution of Equation (2) remains in a small tubular neighborhood of a smooth path ϕ(·) and the probability that √ 2 B t remains in a small neighborhood of the initial value x = ϕ(0), as the size of the neighborhoods go to zero [25].…”