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

A AppendixWith (n ≥ 0)12πˆ ∞−∞((iu) n e −iu(x−x′) du = −∂x) ∂ nδ(x − x ′ ) (A.28)andδ(x − x ′ ) f (x ′ ) = δ(x − x ′ ) f (x)(A.29)we haveP(x,t + τ|x ′ ,t) =[1 +∞∑n=11n!(]−∂x) ∂ nM n (x ′ ,t,τ) δ(x − x ′ )(A.30)Inserting (A.30) in (A.24) leads toρ(x,t + τ) − ρ(x,t)= ∂ρ(x,t) τ + o(τ 2 )∂t∞(1= ∑ − ∂ ) n ˆM n (x ′ ,t,τ)δ(x − x ′ )ρ(x ′ ,t)dx ′n=1n! ∂x∞(= ∑ − ∂ ) n [ ]Mn (x,t,τ)ρ(x,t)∂x n!n=1(A.31)Let us further assume that the moments M n can be expanded into a Taylor series with respect to τ(n ≥ 1):M n (x,t,τ)n!= D (n) (x,t)τ + o ( τ 2) (A.32)The term with τ 0 must vanish, because for τ = 0 the transition probability P has the initial valueP(x,t|x ′ ,t) = δ(x − x ′ ),(A.33)which leads to vanishing moments in (A.25).By taking into account only the linear terms in τ we finally get the ”Kramers-Moyal forwardexpansion” after using equation (A.32) in (A.31):with∂ρ(x,t)∂t= L KM ∗ ρ(x,t), (A.34)114