First passage time statistics of Brownian motion with purely time dependent drift and diffusion

TL;DR

This paper derives analytical first passage time statistics for Brownian motion with time-dependent drift and diffusion, addressing systems near critical thresholds and applying results to snowmelt water availability modeling.

Contribution

It provides new analytical solutions for first passage times in systems with time-dependent parameters, extending the applicability of the method of images.

Findings

Analytical expressions for survival probabilities and first passage time densities.

Limitations of the method of images for state-dependent coefficients.

Application to snowmelt water resource modeling.

Abstract

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model for water availability from surface runoff in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.