  We consider a countable system of interacting (possibly non-Markovian)
stochastic differential equations driven by independent Brownian motions and
indexed by the vertices of a locally finite graph $G = (V,E)$. The drift of the
process at each vertex is influenced by the states of that vertex and its
neighbors, and the diffusion coefficient depends on the state of only that
vertex. Such processes arise in a variety of applications including statistical
physics, neuroscience, engineering and math finance. Under general conditions
on the coefficients, we show that if the initial conditions form a second-order
Markov random field on $d$-dimensional Euclidean space, then at any positive
time, the collection of histories of the processes at different vertices forms
a second-order Markov random field on path space. We also establish a bijection
between (second-order) Gibbs measures on $(\R^d)^V$ (with finite second
moments) and a set of space-time (second-order) Gibbs measures on path space,
corresponding respectively to the initial law and the law of the solution to
the stochastic differential equation. As a corollary, we establish a Gibbs
uniqueness property that shows that for infinite graphs the joint distribution
of the paths is completely determined by the initial condition and the
specifications, namely the family of conditional distributions on finite vertex
sets given the configuration on the complement. Along the way, we establish
various approximation and projection results for Markov random fields on
locally finite graphs that may be of independent interest.

Probability