Regularity conditions on parabolic measures (Q1842625)

The author considers the transition probability measure \(P(t,x;s, \cdot)\) of the diffusion process generated by the parabolic operator of the form \((1/2) \sum^ n_{i,j = 1} a_{ij} (t,x) \partial^ 2/ \partial x_ i \partial x_ j - \partial/ \partial t\). The polynomial upper and lower bounds for the probability \(P(t,x; 0,U_ \rho (y))\) for small \(\rho\) and fixed \(t,x\) and \(y\), where \(U_ \rho (y)\) denotes the open ball with radius \(\rho\) and center \(y\), are obtained. Appropriate counterexamples show that the exponents which occur in the upper and lower bounds cannot be replaced by larger or smaller numbers, respectively.