In this talk we consider two connected problems:

\nFirst
\, we study the classical problem of the first passage hitting density of
an Ornstein-Uhlenbeck process. We give two complementary (forward and back
ward) formulations of this problem and provide semi-analytical solutions f
or both. The corresponding problems are comparable in complexity. By using
the method of heat potentials\, we show how to reduce these problems to l
inear Volterra integral equations of the second kind. For small values of
t we solve these equations analytically by using Abel equation approximati
on\; for larger t we solve them numerically. We also provide a comparison
with other known methods for finding the hitting density of interest\, and
argue that our method has considerable advantages and provides additional
valuable insights.

\nSecond\, we study the non-linear diffusion equati
on associated with a particle system where the common drift depends on the
rate of absorption of particles at a boundary. We provide an interpretati
on as a structural credit risk model with default contagion in a large int
erconnected banking system. Using the method of heat potentials\, we deriv
e a coupled system of Volterra integral equations for the transition densi
ty and for the loss through absorption. An approximation by expansion is g
iven for a small interaction parameter. We also present a numerical soluti
on algorithm and conduct computational tests.