Summary
It becomes apparent that stochastic partial differential equations driven by white noise are conceptually flawed for modeling purposes. This is because neighboring infinitesimally small spatial regions receive similar random perturbations, which violates the fundamental assumption of white noise—namely, that these perturbations should be independent. To address this issue, a more natural approach is to employ spatially-correlated or colored noise, which not only rectifies the modeling inconsistencies but also introduces regularization to the solutions. Nevertheless, the utilization of 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.
