Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription

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 Itô, Stratonovich and anti-Itô 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.

arXiv comment6 pages

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• On the integration of Ito equations with a random or a W-symmetry
• Non-linear Fokker-Planck equations from conformal metrics and scalar curvature