This thesis consists of three subsequent parts addressing the applications of stochastic
processes to the analysis and solutions of parabolic equations with discontinuous coefficients
that are of mathematical interest.
The first two parts consist of three manuscripts, in which we analyze solutions
of Fickian convection dispersion equations with discontinuous coefficients and provide
mathematical treatment of solute transport across a sharp interface. We assume the dispersion
coefficient is a piecewise constant across a plane (interface), the drift is constant
and perpendicular to the interface. Also, assume the flux is continuous across the interface
(interface condition). The transition probability density function of the underlying
stochastic process, called skew diffusion with drift, is given. As an application, we obtain
specific results to explain the interesting types of symmetries and asymmetries in the
breakthrough curves of a conservative tracer across a sharp interface. Also, to answer an
unsolved problem we give the first passage time density formula of skew Brownian motion.
The third part is essentially an extension of the first two parts. In this part we
analyze the stochastic processes associated with convection reaction-diffusion equations
with discontinuous coefficients. The geometry of the interface and the condition at the
interface considered here are more general than in the first two parts.