PUB - Publikationen an der Universität Bielefeld

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 is an element of 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 measurelt satisfying the equation L*mu = 0 and that the closure of L in L-1 (mu) generates a Markov semigroup {T-t}(tgreater than or equal to0) with the resolvent {G(lambda)}(lambda>0). We prove that, for any Lipschitzian function f is an element of L-1 (mu) and all t, lambda > 0, the functions T-t f and G(lambda) f are Lipschitzian and sup(x,t) \delT(t) f(x)\ less than or equal to sup(x) \del f (x)\ and sup(x) \delG(lambda)f (x)\ less than or equal to (1)/(lambda) sup(x) \del f (x)\. In addition, we show that for every bounded Lipschitzian function g, the function G(lambda) g is the unique bounded solution of the equation lambdaf - Lf = g in the Sobolev class H-loc(2,2)(R-d).