EVOLUTION EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 60 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2003 GENERALIZED FOKKER-PLANCK EQUATIONS AND CONVERGENCE TO THEIR EQUILIBRIA PIOTR BILER and GRZEGORZ KARCH Instytut Matematyczny, Uniwersytet Wroclawski pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland E-mail: Piotr.Biler@math.uni.wroc.pl, Grzegorz.Karch@math.uni.wroc.pl Institute of Mathematics of the Polish Academy of Sciences (2002–2003) In memory of Adam Rybarski (1930–2001) Abstract. We consider extensions of the classical Fokker-Planck equation u t + Lu = ∇· (u∇V (x)) on R d with L = −Δ and V (x)= 1 2 |x| 2 , where L is a general operator describing the diffusion and V is a suitable potential. 1. Introduction. We consider the initial value problem for the generalized Fokker- Planck equation u t + Lu = ∇· (u∇V ), x ∈ R d , t> 0, (1.1) u(x, 0) = u 0 (x) ∈ L 1 (R d ), (1.2) with a sufficiently regular potential V = V (x), u = u(x, t) and u 0 (x) ≥ 0. This equation generalizes the classical Fokker-Planck equation u t − Δu = ∇· (ux) (1.3) in two ways. First, the second order elliptic differential operator −Δ is replaced by a Markov diffusion operator L so that −L generates a positivity and mass preserving semigroup e −tL on L 1 (R d ). Second, the potential V (x)= 1 2 |x| 2 with ∇V (x)= x is re- placed by a more general potential V which is large enough as |x|→∞ so that V is confining. Our assumptions below will guarantee the existence of the unique steady state u ∞ = 2000 Mathematics Subject Classification : Primary 35K90; Secondary 35B40. Key words and phrases : Fokker-Planck equation, steady states, asymptotics of solutions, relative entropy. The paper is in final form and no version of it will be published elsewhere. [307]