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On harmonic functions of symmetric Lévy processes. (English. French summary) Zbl 1298.60054

The author considers subordinate Brownian motions \(X_t:= B(S_t)\) such that:
(i)
The Brownian motion \(B\) and the subordinator \(S\) are independent.
(ii)
The Lévy and potential measures of \(S\) have decreasing densities.
(iii)
The Laplace exponent \(\phi(\lambda):=-\log(\operatorname{E} e^{-\lambda S_1})\) of \(S\) satisfies \[ \lim_{\lambda\to +\infty} \frac{\phi'(\lambda x)}{\phi'(\lambda)} = x^{\alpha/2 -1} \quad (x>0) \] for some \(\alpha\in [0,2]\).
For this class of Lévy processes, the author proves the existence of a constant \(c>0\) such that for any \(r\in (0,\frac{1}{4})\) and any bounded function \(f:\mathbb{R}^d \rightarrow \mathbb{R}\) which is harmonic in \(B_{4r}(0):=\{x: |x|<4r\}\), \[ |f(x)-f(y)|\leq c \|f\|_{\infty} \frac{\phi(r^{-2})}{\phi(|x-y|^{-2})} \quad (x,y\in B_{4r}(0)). \] With such a priori regularity estimates for harmonic functions, the author extends, in particular, results of Krylov and Safonov (as in [R. F. Bass and D. A. Levin, Potential Anal. 17, No. 4, 375–388 (2002; Zbl 0997.60089)]) and H. Šikić et al. [Probab. Theory Relat. Fields 135, No. 4, 547–575 (2006; Zbl 1099.60051)].

MSC:

60G51 Processes with independent increments; Lévy processes
60J45 Probabilistic potential theory
60J75 Jump processes (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
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References:

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