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Restricted mean value property for positive functions. (English) Zbl 0837.31001

Let \(U\) be an open subset of \(\mathbb{R}^d\) and let \(r: U\to ]0,\infty [\) be such that for every \(x\in U\) the ball \(B(x, r(x))\) with center \(x\) and radius \(r(x)\) is contained in \(U\) if \(U\neq \mathbb{R}^d\) and \(r\leq |\cdot |+ M\) if \(U= \mathbb{R}^d\). Suppose that for some \(\alpha\in ]0,1 [\) the inequality \(r(y)\geq \alpha (r (x)- |x- y|)\) holds for every \(x\in U\) and \(y\in B (x, r(x))\). It is shown that every real function \(f\geq 0\) on \(U\) having the (restricted) mean value property \[ f(x)= (\lambda (B(x, r(x)))^{-1} \int_{B(x, r(x))} f d\lambda \] for every \(x\in U\) is harmonic.
This improves a result obtained by W. A. Veech (1973), J. R. Baxter (1978) and A. Cornea and J. Veselý [Potential Anal. 4, No. 5, 547-569 (1995)] using different methods. The essential ingredient of the proof (being of independent interest) is a uniform comparison between the hitting probability for cubes with respect to Brownian motion on the one hand and the random walk generated by the transition kernels \((x, A)\mapsto \lambda_{B (x,r (x))} (A)\) on the other hand.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60J65 Brownian motion
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
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