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 It\^o, Stratonovich and anti-It\^o prescriptions). We obtain its stationary state pₛₜ(x) for a class of constitutive relations between drift and diffusion and show that it has a q-exponential form, pₛₜ(x)=Nq[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.