  One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear
Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0,
\ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$
$D$ and $b$ the convergence in $L^1_\textrm{loc}(\mathbb{R}^d)$,
$L^1(\mathbb{R}^d)$, respectively, for some $t_n\to\infty$ of the solution
$u(t_n)$ to an equilibrium state of the equation for a large set of nonnegative
initial data in $L^1$. These results are new in the literature on nonlinear
Fokker-Planck equations arising in the mean field theory and are also relevant
to the theory of stochastic differential equations. As a matter of fact, by the
above convergence result, it follows that the solution to the McKean-Vlasov
stochastic differential equation corresponding to (1), which is a nonlinear
distorted Brownian motion, has this equilibrium state as its unique invariant
measure. Keywords: Fokker-Planck equation, $m$-accretive operator, probability
density, Lyapunov function, $H$-theorem, McKean-Vlasov stochastic differential
equation, nonlinear distorted Brownian motion. 2010 Mathematics Subject
Classification: 35B40, 35Q84, 60H10.

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