Stochastic fields on a lattice

A Caldeira-Leggett-type system-bath Hamiltonian is used to construct a new class of local stochastic equations on a lattice and to determine the conditions under which the lattice field evolves to thermal equilibrium. Both scalar and two-dimensional vector fields driven by multiplicative noise are considered. The latter model is developed to describe magnetization dynamics with spatial dispersion of relaxation. A systematic method of constructing stochastic field equations from a complex dispersion relation is proposed. The corresponding lattice Fokker-Planck equation is written down and it is shown that thermal equilibrium is its stationary state