Kernel estimates for a class of Kolmogorov semigroups
Let \(A:=\sum_{i,j=1}^N a_{ij} D_{ij}+\sum_{i=1}^NF_i D_i\) be a second order partial differential operator with bounded coefficients \((a_{ij})_{i,j=1}^N\), but unbounded drift. Under some mild restriction on the smoothness of the coefficients and ellipticity of \((a_{ij})_{i,j=1}^N\), it is proved that the heat kernel associated with \(A\) can be estimated from above as follows: \[ p(t,x,y)\leq c(t) w(y,t), \] where \(w(t,y)\) is a Lyapunov function for the operator \(\partial_t+A\). Such results are related to [\textit{G.\,Metafune, D.\,Pallara} and \textit{A.\,Rhandi}, ``Global properties of transition probabilities of singular diffusions'', Theor.\ Probab.\ Appl.\ (to appear)].
scientific article; zbMATH DE number 5500565