  We construct a functional path integral for a stochastic hybrid
reaction-diffusion (RD) equation, in which the reaction term depends on the
discrete state of a randomly switching environment. We proceed by spatially
discretizing the RD system and using operator methods and coherent spin states
to derive a path integral representation of the lattice model. The path
integral specifies the distribution of trajectories in a state-space consisting
of the set of local concentrations and the environmental state. Taking the
continuum limit yields a path integral for the stochastic hybrid RD equation.
We show how in the semi-classical limit, the stochastic dynamics reduces to a
system of Langevin equations. The latter consists of an RD equation coupled to
a stochastic probability function $z$ that evolved according to an
Ornstein-Uhlenbeck process with multiplicative noise. On eliminating the
variable $z$, we obtain an RD equation driven by nonlocal spatiotemporal noise.
This is used to determine leading order corrections to the mean-field RD
equation. We then apply the theory to the example of diffusion with
stochastically gated adsorption under dimerization. Comparison with the
classical problem of diffusion-limited pair-annihilation suggests that the
adsorption model exhibits anomalous large-time behavior in one and two spatial
dimensions. This motivates an alternative path integral formulation based on an
RD master equation that treats diffusion as a set of hopping reactions and
incorporates the effects of a randomly switching environment. A small noise
expansion now yields a stochastic RD equation with both real and complex noise
sources. Finally, diagrammatic and renormalization group methods are used to
determine the anomalous large-time behavior of the mean concentration at the
upper critical dimension $d_c=2$.
