  The diffusion equation is the primary tool to study the movement dynamics of
a free Brownian particle, but when spatial heterogeneities in the form of
permeable interfaces are present, no fundamental equation has been derived.
Here we obtain such an equation from a microscopic description using a lattice
random walk model. The sought after Fokker-Planck description and the
corresponding backward Kolmogorov equation are employed to investigate
first-passage and local time statistics and gain new insights. Among them a
surprising phenomenon, in the case of a semibounded domain, is the appearance
of a regime of dependence and independence on the location of the permeable
barrier in the mean first-passage time. The new formalism is completely
general: it allows to study the dynamics in the presence of multiple permeable
barriers as well as reactive heterogeneities in bounded or unbounded domains
and under the influence of external forces.
