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

We establish a pointwise lower bound for the heat kernel of the elliptic operator $ Λ=(1+|x|^α)Δ+b|x|^{α-2}x\cdot\nabla-|x|^β$, where $d\geq3$, $α>2$, $β>α-2$, and $b\in\mathbb R$. For every $τ>0$, we prove that $$ \begin{aligned} p(t,x,y) &\geq C_τe^{λ_0t} \left(\frac{1+|y|^α}{1+|x|^α}\right)^{\fra