Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

A AppendixIn addition to the deterministic drift h(x,t), D (1) contains a term which is called the spurious driftor the noise-induced driftnoise−ind = ∂g(x,t) g(x,t) = 1 ∂∂x 2 ∂x D(2) (x,t)D (1)(A.51)It stems from the fact that during a change of Γ(t) also X(t) changes and therefore 〈g(X(t),t)Γ(t)〉is no longer zero.So finally it is clear that for SDEs of Langevin-type like equation (A.37) with Gaussian noise theKramers-Moyal forward expansion turns to the Fokker-Planck equationwhere∂ρ(x,t)∂t= L FP ∗ ρ(x,t) (A.52)L FP = − ∂ ∂x D(1) (x,t) + ∂ 2∂x 2 D(2) (x,t)(A.53)is the Fokker-Planck operator.118