PUB - Publikationen an der Universität Bielefeld

For diffusion processes in R-d with locally unbounded drift coefficients we obtain a sufficient condition for the strict positivity of transition probabilities. To this end, we consider parabolic equations of the form L* mu = 0 with respect to measures on R-d x ( 0, 1) with the operator Lu := partial derivative(t)u+partial derivative(xi) (a(ij)partial derivative(xj)u) + b(i)partial derivative(xi)u. It is shown that if the diffusion coefficient A = (a(ij)) is sufficiently regular and the drift coefficient b = (b(i)) satisfies the condition exp(kappa vertical bar b vertical bar(2)) is an element of L-loc(1)(mu), where the measure mu is nonnegative, then mu has a continuous density rho(x, t) which is strictly positive for t > tau provided that it is not identically zero for t <= tau. Applications are obtained to finite-dimensional projections of stationary distributions and transition probabilities of infinite-dimensional diffusions.