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Zbl 1203.60125
Husseini, Ryad; Kassmann, Moritz
Jump processes, $\cal L$-harmonic functions, continuity estimates and the Feller property. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 45, No. 4, 1099-1115 (2009). ISSN 0246-0203

In the paper [{\it R.~F.~Bass} and {\it M.~Kassmann}, Commun. Partial Differ. Equations 30, No.~8, 1249--1259 (2005; Zbl 1087.45004)], the authors proved the Hölder continuity of harmonic functions of an integro-differential operator of variable order satisfying certain conditions. Their approach was probabilistic, the main object being the corresponding Markov procees. The key to obtaining regularity results is the Krylov-Safonov-type estimate of hitting probabilities: There exists a constant $c>0$ such that for all $r\in (0,1/2)$, $A\subset B(x_0,r)$ satisfying $|A|\ge \frac12 |B(x_0,r)|$ and all $y\in B(x_0,r/2)$ it holds that $\mathbb P^y(T_A<\tau_{B(x_0,r)})\ge c$. Here $T$ denotes the hitting time, and $\tau$ the exit time from a set. In the paper under review, the authors discuss continuity a priori estimates of harmonic functions for similar integro-differential operators but with one of the conditions relaxed. The first consequence of this relaxation is that the Krylov-Safonov-type estimate need not hold -- the lower bound $c$ may depend on the radius $r$ and thus converges to zero as $r\to 0$. The main continuity result for bounded functions $u$ harmonic in the ball $B(x_0,R)$ gives the following modulus of continuity: $$ \sup\Sb x,y\in B(x_0,R/2)\\ |x-y|<t\endSb |u(x)-u(y)|\le c ||u||_{\infty}|\log t|^{-\rho}\, ,\quad \forall t\in (0,1/2)\, , $$ where the constant $\rho >0$ depends on the constants appearing in the conditions.
[Zoran Vondraček (Zagreb)]

MSC 2000:: *60J75 Jump processes
35B45 A priori estimates
31C05 Generalizations of harmonic (etc.) functions
47D07 Markov semigroups of linear operators

Keywords: jump processes; Lévy measure; Feller property; martingale problem; integro-differential operators; harmonic functions; a priori estimates

Citations: Zbl 1087.45004

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