  Starting from a continuous time random walk (CTRW) model of particles that
may evanesce as they walk, our goal is to arrive at macroscopic
integro-differential equations for the probability density for a particle to be
found at point r at time t given that it started its walk from r_0 at time t=0.
The passage from the CTRW to an integro-differential equation is well
understood when the particles are not evanescent. Depending on the distribution
of stepping times and distances, one arrives at standard macroscopic equations
that may be "normal" (diffusion) or "anomalous" (subdiffusion and/or
superdiffusion). The macroscopic description becomes considerably more
complicated and not particularly intuitive if the particles can die during
their walk. While such equations have been derived for specific cases, e.g.,
for location-independent exponential evanescence, we present a more general
derivation valid under less stringent constraints than those found in the
current literature.
