2014/10/12 by Wolfhard Hansen, Hansen, Wolfhard, Ivan Netuka +1
Mathematics · #31B15 #31C15 #31D05 #60J25 #60J45 #60J65 #60J75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Point processes and geometric inequalities #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1410.3067
openalex publication_date 2014/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (X,\mathcal W) be a balayage space, 1∈ \mathcal W, or - equivalently - let \mathcal W be the set of excessive functions of a Hunt process on a locally compact space X with countable base such that \mathcal W separates points, every function in \mathcal W is the supremum of its continuous minorants and there exist strictly positive continuous u,v∈ \mathcal W such that u/v→ 0 at infinity. We suppose that there is a Green function G>0 for X, a metric ρ on X and a decreasing function g\colon[0,∞)→ (0,∞] having the doubling property and a mild upper decay near 0 such that G≈ g∘ρ (which is equivalent to a 3G-inequality). Then the corresponding capacity for balls of radius r is bounded by a constant multiple of 1/g(r). Assuming that reverse inequalities hold as well and that jumps of the process, when starting at neighboring points, are related in a suitable way, it is proven that positive harmonic functions satisfy scaling invariant Harnack inequalities. Provided that the Ikeda-Watanabe formula holds, sufficient conditions for this relation are given. This shows that rather general Lévy processes are covered by this approach.