  We study a 12-parameter stochastic process involving particles with two-site
interaction and hard-core repulsion on a $d$-dimensional lattice. In this
model, which includes the asymmetric exclusion process, contact processes and
other processes, the stochastic variables are particle occupation numbers
taking values $n_{\vec{x}}=0,1$. We show that on a 10-parameter submanifold the
$k$-point equal-time correlation functions $\exval{n_{\vec{x}_1} \cdots
n_{\vec{x}_k}}$ satisfy linear differential- difference equations involving no
higher correlators. In particular, the average density $\exval{n_{\vec{x}}} $
satisfies an integrable diffusion-type equation. These properties are explained
in terms of dual processes and various duality relations are derived. By
defining the time evolution of the stochastic process in terms of a quantum
Hamiltonian $H$, the model becomes equivalent to a lattice model in thermal
equilibrium in $d+1$ dimensions. We show that the spectrum of $H$ is identical
to the spectrum of the quantum Hamiltonian of a $d$-dimensional, anisotropic
spin-1/2 Heisenberg model. In one dimension our results hint at some new
algebraic structure behind the integrability of the system.
