Onsager-Machlup functional for stochastic differential equations with additive noise

Abstract

This paper is devoted to studying the Onsager-Machlup functional for stochastic differential equations (SDEs) with additive noise of the α-Hölder, 0 < α < 1/4, dXt = f (t, Xt)dt + g(t)dWt. When the diffusion term of SDEs is not a fixed constant, we obtain the Onsager-Machlup functional. This function is the Lagrangian giving the most probable transition path connecting metastable states for stochastic processes driven by additive noise. This is done by introducing new measurable norms and applying an appropriate version of the Girsanov transformation. Moreover, we provide examples of numerical simulations, including a one-dimensional SDE and a fast-slow SDE for multiscale stochastic volatility models. MSC2020 subject classifications: Primary 82C35, 60H10; secondary 37H10