Study of Brownian functionals in physically motivated model with purely time dependent drift and diffusion

In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and difusion by suggesting and examining several Brownian functionals which characterize the lifetime and reactivity of such stochastic processes. We introduce several probability distribution functions (PDFs) associated with such time dependent BMs. For instance, for a BM with initial starting point $x_0$, we derive analytical expressions for : (i) the PDF $P(t_f|x_0)$ of the first passage time $t_f$ which specify the lifetime of such stochastic process, (ii) the PDF $P(A|x_0)$ of the area A till the first passage time and it provides us numerous valuable information about the effective reactivity of the process, (iii) the PDF $P(M)$ associated with the maximum size M of the BM process before the first passage time, and (iv)the joint PDF $P(M; t_m)$ of the maximum size M and its occurrence time $t_m$ before the first passage time. These distributions are examined for the power law time time dependent drift and diffusion. A simple illustrative example for the stochastic model of water resources availability in snowmelt dominated regions with power law time dependent drift and diffusion is demonstrated in details. We motivate our study with approximate calculation of an unsolved problem of Brownian functionals including inertia.