We analyze a piecewise deterministic PDE consisting of the diffusion
equation on a finite interval Ω with randomly switching boundary conditions
and diffusion coefficient. We proceed by spatially discretizing the diffusion
equation using finite differences and constructing the Chapman-Kolmogorov (CK)
equation for the resulting finite-dimensional stochastic hybrid system. We show
how the CK equation can be used to generate a hierarchy of equations for
the rth moments of the stochastic field, which take the form of r-dimensional
parabolic PDEs on Ωr
that couple to lower order moments at the boundaries.
We explicitly solve the first and second order moment equations (r = 2). We
then describe how the rth moment of the stochastic PDE can be interpreted in
terms of the splitting probability that r non-interacting Brownian particles all
exit at the same boundary; although the particles are non-interacting, statistical
correlations arise due to the fact that they all move in the same randomly
switching environment. Hence the stochastic diffusion equation describes two
levels of randomness; Brownian motion at the individual particle level and a
randomly switching environment. Finally, in the limit of fast switching, we use a
quasi-steady state approximation to reduce the piecewise deterministic PDE to an
SPDE with multiplicative Gaussian noise in the bulk and a stochastically-driven
boundary