  This paper is about nonequilibrium steady states (NESS) of a class of
stochastic models in which particles exchange energy with their "local
environments" rather than directly with one another. The physical domain of the
system can be a bounded region of $\mathbb R^d$ for any $d \ge 1$. We assume
that the temperature at the boundary of the domain is prescribed and is
nonconstant, so that the system is forced out of equilibrium. Our main result
is local thermal equilibrium in the infinite volume limit. In the Hamiltonian
context, this would mean that at any location $x$ in the domain, local marginal
distributions of NESS tend to a probability with density $\frac{1}{Z} e^{-\beta
(x) H}$, permitting one to define the local temperature at $x$ to be
$\beta(x)^{-1}$. We prove also that in the infinite volume limit, the mean
energy profile of NESS satisfies Laplace's equation for the prescribed boundary
condition. Our method of proof is duality: by reversing the sample paths of
particle movements, we convert the problem of studying local marginal energy
distributions at $x$ to that of joint hitting distributions of certain random
walks starting from $x$, and prove that the walks in question become
increasingly independent as system size tends to infinity.
