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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

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

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

For a Feller process with a pseudodifferential generator $-p(x,D)$ with symbol $p(x,\xi)$ satisfying conditions $p(x,\xi) \le C(1+|\xi|^2)$ and $|Im p(x,\xi)| \le c_0 Re p(x,\xi)$ we can bound all moments of the exit time from a ball of radius using the Lemma 4.1 of Schilling’s 1998 paper schilling1998-growthholderregularityfellerprocesses.  Let $\tau_R = \inf \{ t>0: |X_t - x|>R\}$ note that $\{ \sup_{s\le t} |X_s - x| > R\} \subset \{ \tau_{R} < t\} \subset \{ \sup_{s\le t} |X_s-x| \le R\}$ and then use the bound
$P^x(\sup_{s\le t} |X_s-x| < R) \le \frac{c^k}{t^{2k} h^{2k}(x, R\cdot 2^k)}$.
Then recall that the $L^p$ norm can be expressed as $\int |f|^p dx = p \int_0^\infty \mu( |f| > \alpha) \alpha^{p-1} d\alpha$ and integrate the above bound.