Global gradient bounds for dissipative diffusion operators

Abstract

Let L be a second order elliptic operator on R d with a constant diffusion matrix and a dissipative (in a weak sense) drift b ∈ L loc p with some p > d . We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measure μ satisfying the equation L * μ = 0 and that the closure of L in L 1 ( μ ) generates a Markov semigroup { T t } t ⩾ 0 with the resolvent { G λ } λ > 0 . We prove that, for any Lipschitzian function f ∈ L 1 ( μ ) and all t , λ > 0 , the functions T t f and G λ f are Lipschitzian and sup x , t | ∇ T t f ( x ) | ⩽ sup x | ∇ f ( x ) | and sup x | ∇ G λ f ( x ) | ⩽ 1 λ sup x | ∇ f ( x ) | . In addition, we show that for every bounded Lipschitzian function g, the function G λ g is the unique bounded solution of the equation λ f − L f = g in the Sobolev class H loc 2 , 2 ( R d ) . To cite this article: V.I. Bogachev et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).