  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.
