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TWO SIDED TRANSITION DENSITY BOUNDS TECHNOLOGY

Here is Chen-Kumagai two sided bound for jump diffusion with some assumption on the jump kernel similar to that of $\alpha$-stable jump $1/|x-y|^{-n-\alpha}$.  We use $\sim$ to denote two-sided bound.  Suppose $J(x,y) \sim \frac{1}{|x-y|^n \varphi(|x-y|)}$ is the jump kernel where $\varphi(0)=0$ and $\varphi(1)=1$.  Then

$p(t,x,y) \sim (t^{-n/2} \wedge \varphi^{-1}(t)^{-n}) \wedge (p^c(t,c(|x-y|))+p^j(t,|x-y|)$

where

$p^c(t,r) = t^{-n/2} \exp(-r^2/t)$

$p^j(t,r) = \varphi^{-1}(t)^{-n}\wedge \frac{t}{r^n\varphi(r)}$