Parameters of the fractional Fokker-Planck equation

We study the connection between the parameters of the fractional Fokker-Planck equation, which is associated with the overdamped Langevin equation driven by noise with heavytailed increments, and the transition probability density of the noise generating process. Explicit expressions for these parameters are derived both for finite and infinite variance of the rescaled transition probability density. Copyright c © EPLA, 2009 Introduction. – Heavy-tailed distributions, i.e., probability distributions with power tails and infinite second moments, are an important tool for studying a number of physical, biological, economical and other systems whose behavior is determined by rare but large events [1–4]. In many cases the continuous-time dynamics of these systems can be effectively described by the (dimensionless) overdamped Langevin equation ẋ(t) = f(x(t), t)+ ξ(t), (1) where x(t) [x(0) = 0] is a state parameter of the system, f(x, t) is a deterministic function, and ξ(t) is a random noise defined by the infinitesimal increments ∆η(t) = ∫ t+τ t dtξ(t) (τ → 0) that are assumed to be independent on non-overlapping intervals and distributed with a heavy-tailed distribution. Since the tails of these distributions cannot be neglected, the classical stochastic theory, which is based on the ordinary central-limit theorem, is not applicable to eq. (1). Specifically, if the increments are distributed according to a Lévy stable distribution [5], i.e., ξ(t) is a Lévy stable noise, then the probability density P (x, t) that x(t) = x satisfies the fractional Fokker-Planck equation [6–11] which can be written as ∂ ∂t P (x, t) = − ∂ ∂x f(x, t)P (x, t)+ γ ∂ ∂|x|α P (x, t) + γβ tan πα 2 ∂ ∂x ∂ ∂|x|α−1 P (x, t). (2) (a)E-mail: [email protected] Here, the Riesz derivative, ∂/∂|x|, is defined as [12] ∂h(x)/∂|x| =−F{|k|hk}, a pair F{h(x)} ≡ hk = ∫∞ −∞ dx eh(x) and F{hk} ≡ h(x) = (1/2π)× ∫∞ −∞ dk ehk represents the Fourier transforms, and α, β and γ are the parameters of the stable distribution. Because of the generalized central limit theorem [13], the Lévy stable distributions constitute an important but a particular class of heavy-tailed distributions. In this letter, we show that the fractional Fokker-Planck equation (2) is valid also for all noises ξ(t) whose increments have heavy-tailed distributions. Explicit expressions for the parameters of eq. (2) are derived in terms of the asymptotic characteristics of these distributions. Definitions and basic equations. – Our starting point is the generalized Fokker-Planck equation [14] ∂ ∂t P (x, t) =− ∂ ∂x f(x, t)P (x, t)+F{Pk(t) lnSk}, (3) which corresponds to the Langevin equation (1) driven by an arbitrary noise. The term “arbitrary” means that the independent increments ∆η(jτ) = η(jτ + τ)− η(jτ) = ∫ jτ+τ jτ dtξ(t) (τ → 0, j = 0, 1, . . .) of the discrete-time noise generating process η(nτ) = ∑n−1 j=0 ∆η(jτ) (n= 1, 2, . . .) are distributed according to an arbitrary probability density function p(∆η, τ). In other words, p(∆η, τ) is the transition probability density of the process η(nτ). It is assumed that i) p(∆η, τ) is properly normalized, i.e., ∫∞ −∞ d(∆η) p(∆η, τ) = 1, ii) the first moment, if it exists, equals zero, i.e., ∫∞ −∞ d(∆η) p(∆η, τ)∆η= 0, and iii) limτ→0p(∆η, τ) = δ(∆η), where δ(·) is the Dirac δ