Stochastic PDEs, random fields and exact mean-values

Abstract

Introducing projector-operator technique and algebra of Terwiel's cumulants we study stochastic linear partial differential equations with global and local disorder. We present the evolution equation for the mean-value of the field as a series in terms of Terwiel's cumulant operators. Then, we prove that if we use binary disorder with time exponential-correlated structure, as source of the stochastic perturbation, this series cuts leading to a treatable evolution equation. We apply this approach to find the exact mean-value solution of electromagnetic waves with stochastic absorption of energy in conducting media. This model shows the occurrence of novel time-scale separation phenomena. Local disorder in telegrapher's equation is also presented. Thus we show that strong disorder leads to anomalous behavior at short and long time regimes. In addition, other physical systems with global disorder are worked out to find exact mean-value solutions: finite-velocity diffusion in the presence of a deterministic force (Smoluchoswki-like process generalizing, in this way, Feynman–Kac's formula for its numerical solution); Lorentz' force on a fluctuating charge model (we calculate the diffusion coefficient transverse to the applied magnetic field); and a generalized non-Maxwellian velocity distribution (Ornstein–Uhlenbeck like process showing a noise-induced transition in the stationary distribution).