Study of Brownian functionals for a Brownian process model of snow melt dynamics with purely time dependent drift and diffusion

Abstract

In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and diffusion by suggesting and examining several Brownian functionals, which characterize the stochastic model of water resources availability in snowmelt dominated regions with power law time dependent drift and diffusion. Snow melt process is modelled by a overdamped Langevin equation for a Brownian process with power law time dependent drift (μ(t) ~ qktα) and diffusion (D(t) ~ ktα) where they are proportional to each other. We introduce several probability distribution functions (PDFs) associated with such time dependent BMs. For instance, with initial starting value of snow amount H0, we derive analytical expressions for: (i) the PDF P(tf|H0) of the first passage time tf which specify the lifetime of such stochastic process, (ii) the PDF P(A|H0) of the area A till the first passage time and it provides us numerous valuable information about the average available water resources, (iii) the PDF P(M) associated with the maximum amount of available water M of the BM process before the complete melting of snow, and (iv) the joint PDF P(M;tm) of the maximum amount of available water M and its occurrence time tm before the first passage time. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equation. We obtain a very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviour in the asymptotic limits for the above mentioned PDFs are predicted, which can be verified further from experimental observation.