Positive Densities of Transition Probabilities of Diffusion Processes

For diffusion processes in ${\bf R}^d$ 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 ${\cal L}^*\mu=0$ with respect to measures on ${\bf R}^d\times (0,1)$ with the operator ${\cal L} u:=\partial_t u +\partial_{x_i}(a^{ij}\partial_{x_j}u)+ b^i\partial_{x_i}u.$ It is shown that if the diffusion coefficient $A=(a^{ij})$ is sufficiently regular and the drift coefficient $b=(b^i)$ satisfies the condition $\exp(\kappa |b|^2)\in L_{\rm loc}^1(\mu)$, where the measure $\mu$ is nonnegative, then $\mu$ has a continuous density $\varrho(x,t)$ which is strictly positive for $t>\tau$ provided that it is not identically zero for $t\le\tau$. Applications are obtained to finite-dimensional projections of stationary distributions and transition probabilities of infinite-dimensional diffusions.