Preserving equilibrium in stochastic models with spatially-correlated noise

In the continuum limit, spatially-correlated noise-driven stochastic partial differential equation (SPDE) models arise from considering that neighboring infinitesimally small spatial regions receive similar random perturbations. An additional benefit over white-noise-driven SPDEs is the regularization of solutions, especially in more than one spatial dimension. Nevertheless, using colored noise introduces a complication, as it disrupts the fluctuation-dissipation relationship of thermal equilibrium. In this presentation, I will discuss how the continuous spatiotemporal limit of a Metropolis-Hastings random walk can be employed to derive a stochastic partial differential equation driven by colored noise that preserves its ability to sample the equilibrium distribution, even for a system of magnetic spins. This introduces an additional geometric constraint and yields non-trivial interactions with the correlated noise, further enhancing our understanding of these intricate systems.