Lower bounds for heat kernels of elliptic operators with unbounded diffusion, drift, and potential terms

Abstract

We establish a pointwise lower bound for the heat kernel of the elliptic operator $\Lambda =(1+|x{|}^{\alpha })\Delta +b|x{|}^{\alpha -2}x\cdot \nabla -|x{|}^{\beta }$, where $d\ge 3$, $\alpha >2$, $\beta >\alpha -2$, and $b\in \mathbb{R}$. For every $\tau >0$, we prove that $$\begin{array}{rl}p(t,x,y)& \ge C_{\tau }{e}^{{\lambda }_{0}t}{\left(\frac{1+|y{|}^{\alpha }}{1+|x{|}^{\alpha }}\right)}^{\frac{b}{2\alpha }}\frac{(|x||y|{)}^{-\frac{d-1}{2}-\frac{\beta -\alpha }{4}}}{1+|y{|}^{\alpha }}\\ & \times \exp \left[-{\int }_1^{|x|}\sqrt{\frac{s^{\beta }}{1+s^{\alpha }}}\mathrm{d}s-{\int }_1^{|y|}\sqrt{\frac{s^{\beta }}{1+s^{\alpha }}}\mathrm{d}s\right]\end{array}$$ for all $t\ge \tau$ and $|x|,|y|\ge 1$, where ${\lambda }_0<0$ is the largest eigenvalue of $\Lambda$ and $C_{\tau }>0$ is independent of $t,x,y$. The proof applies the classical Davies - Simon argument in a weighted symmetric setting.