Predict the discretization parameter α that ensures convergence of the short-time series expansion

Determine, for a given one-dimensional Fokker–Planck equation with spatially heterogeneous diffusion g(x,t) and drift h(x,t), the values of the stochastic integral discretization parameter α ∈ [0,1] for which the short-time Taylor series expansion of the regular factor F(x,t;y,t0) in the propagator decomposition K(x,t;y,t0) = K0(x,t;y,t0) F(x,t;y,t0) converges; specifically, establish α-dependent convergence criteria applicable across problem classes and explain the α-sensitivity observed in examples such as exponential diffusion g(x) = G0 exp(γx), where the expansion converges only for α = 1/2.

Background

The paper develops a short-time expansion for the propagator of one-dimensional Fokker–Planck equations with heterogeneous diffusion, writing K = K0 F where the singular term K0 is explicit and the regular term F is expanded in a Taylor series with coefficients obtained from recursive ordinary differential equations. The expansion depends on the discretization parameter α that specifies the stochastic integral interpretation (e.g., Itô α=0, Fisk–Stratonovich α=1/2, Hänggi–Klimontovich α=1).

In the exponential diffusion example g(x) = G0 exp(γx), the authors rigorously derive that the Taylor coefficients Dn vanish for n ≥ 1 only when α = 1/2, yielding a finite expansion; for α ≠ 1/2, the coefficients grow superfactorially and the series cannot converge. They note that beyond this case, a general theoretical criterion to predict, for a given drift h and diffusion g, which α yields convergence is lacking. Clarifying this dependence is important because the practical applicability of the expansion hinges on selecting an interpretation of the stochastic calculus that guarantees convergence.