Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription

We deduce a nonlinear and inhomogeneous Fokker-Planck equation within a generalized Stratonovich, or stochastic $α$-, prescription ($α=0$, $1/2$ 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, $p_{st}(x) = N_q[1 - (1-q)βV(x)]^{1/(1-q)}$, with an index $q$ which does not depend on $α$ in the presence of any nonvanishing nonlinearity. This is in contrast with the linear case, for which the index $q$ is $α$-dependent.

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