Prefactor estimates for probability density functions in stochastic dynamical systems via the Gel'fand-Yaglom method
Timo Schorlepp (University of Bochum, Germany)
The quantification of probabilities of rare events in stochastic dynamical systems is an important problem with applications in various fields of science. In this talk, I will show how, in the limit of small noise, precise asymptotic estimates of probability density functions (PDFs) of general observable functions of diffusion processes with additive noise can be obtained. Applying a functional version of Laplace's method to a path integral representation of the PDF under consideration results in a method to estimate the PDF that consists of two steps: First, the minimizer of the action functional of the process under suitable boundary conditions, termed "instanton", has to be computed, which already yields a log-asymptotic estimate for the PDF. Afterwards, this estimate is refined by expanding the action to second order around the instanton and computing the functional determinant of the second variation of the action by means of the Gel'fand-Yaglom method. The resulting prefactor term can efficiently be computed by solving a matrix Riccati equation as an initial value problem along the instanton trajectory. In order to demonstrate the numerical feasibility of this approach, I will present several examples motivated from turbulence theory.