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

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 of water resources availability in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated. ► Brownian motion with an absorbing barrier mimics resources at a critical threshold. ► Its time-dependent coefficients μt and σt2/2 resemble external drivers like climate. ► Associated Fokker–Planck equation is solved via the method of images (MoI). ► MoI is applicable to time-dependent problems only if μt and σt2/2 are proportional. ► The results are used to model water availability in snowmelt dominated regions.