Understanding Correlated Noise in Stochastic Gradient Systems

Stochastic Partial Differential Equations (SPDE's) are continuum models of spatially-extended noisy physical systems. In practice, such models are often explored numerically through problem-specific discretizations. At a critical correlation length scale, the assumption of independent (white) noise fails, and a spatial correlation must be considered. Under certain assumptions, correlated systems obey the same invariant measure as their independent counterparts; however, spatial correlations affect the dynamic behavior and the exploration of the energy landscape. Of particular interest are energy landscapes with multiple minima that exhibit metastability. We therefore study the mean time to transition between local energy minimums, and the expected path between these states. We begin with a two-dimensional correlated system amenable to mathematical analysis, verifying results through numerical simulations.