Jump processes, \(\mathcal L\)-harmonic functions, continuity estimates and the Feller property
In the paper [\textit{R.~F.~Bass} and \textit{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 \(|A|\geq \frac12 |B(x_0,r)|\) and all \(y\in B(x_0,r/2)\) it holds that \(\mathbb P^y(T_A<\tau_{B(x_0,r)})\geq c\). 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: \[ \sup_{\substack{ x,y\in B(x_0,R/2)\\ |x-y|<t}} |u(x)-u(y)|\leq c ||u||_{\infty}|\log t|^{-\rho}\, ,\quad \forall t\in (0,1/2)\, , \] where the constant \(\rho >0\) depends on the constants appearing in the conditions.
scientific article; zbMATH DE number 5758872
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