Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription

TL;DR

This paper derives a nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich framework, revealing that its stationary state has a q-exponential form with an index independent of the stochastic prescription in nonlinear cases.

Contribution

It introduces a generalized Fokker-Planck equation for nonlinear, inhomogeneous systems and shows the stationary state's q-exponential form with an index unaffected by the stochastic prescription.

Findings

Stationary state p_{st}(x) has a q-exponential form.

The q-index is independent of the stochastic prescription α in nonlinear cases.

Contrasts with the linear case where q depends on α.

Abstract

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic $\alpha$-, prescription ($\alpha=0$, $1/2$ and $1$ respectively correspond to the It\^o, Stratonovich and anti-It\^o prescriptions). We obtain its stationary state $p_{st}(x)$ for a class of constitutive relations between drift and diffusion and show that it has a $q$-exponential form, $p_{st}(x) = N_q[1 - (1-q)\beta V(x)]^{1/(1-q)}$, with an index $q$ which does not depend on $\alpha$ in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index $q$ is $\alpha$-dependent.