## 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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