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Archive for September, 2016

chenjumpkernelboundscarlenkusuokastroock-upperboundssymmetricjumpkernelsknopova-schilling-densityestimateslevy-2010approximatedensitylevy

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)}

 

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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.

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