We study minimal thinness in the half-space $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d\>0\}$ for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.

We consider the general Schrödinger operator $L=div\left(A\left(x\right){\nabla }_x\right)-\mu$ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function $G_{\Delta }$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K{ₙ}^{\infty }$ considered by Zhao and Pinchover. As an application we study the cone ${}_L\left(ℝⁿ₊\right)$ of all positive L-solutions continuously vanishing...

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $𝒫$-harmonic space $\Omega$ with countable base of open subsets and satisfying the axiom $D$. When $\Omega$ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $𝒮\left(𝒰\right)$ of nonnegative...

It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence $R$. It is also shown that Ancona’s inequalities extend to $R$, and therefore that the Martin boundary for $R$-potentials coincides with the natural geometric boundary $S^1$, and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, $p^n(x,y)\sim C_{x,y}{R}^{-n}{n}^{-3/2}$.

Soit $L$ un opérateur parabolique sur ${\mathbf{R}}_{n+1}$ écrit sous forme divergence et à coefficients lipschitziens relativement à une métrique adaptée. Nous cherchons à comparer près de la frontière le comportement relatif des $L$-solutions positives sur un domaine “lipschitzien”. Dans un premier temps, nous démontrons un principe de Harnack uniforme pour certaines $L$-solutions positives. Ce principe nous permet alors de démontrer une inégalité de Harnack forte à la frontière pour certains couples de $L$-solutions positives....

In this paper, we study the reduit, the thinness and the non-tangential limit associated to a harmonic structure given by coupled partial differential equations. In particular, we obtain such results for biharmonic equation (i.e. ${▵}^2\varphi =0$) and equations of ${▵}^2\varphi =\varphi$ type.

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of $p$-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions...