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Gaussian bounds and parabolic Harnack inequality on locally irregular graphs. (English) Zbl 1352.60126

Let \(\Gamma=(V,E)\) be a connected locally finite graph without double edges or loops, \(\mu\) be a weight function on edges such that \(\mu_{xy}> 0\) if \((x,y) \in E\) and \(\mu_{xy}= 0\) if \((x,y) \notin E\). A continuous time constant speed random walk \(X_t\) with generator \[ L_{\mu} f(x)=\mu_x^{-1} \sum_{y \in V}(f(y)-f(x))\mu_{xy} \] is considered on \(V\) where \[ \mu_x =\sum_{y \in V}\mu_{xy}. \]
In the article the equivalence of the following three properties is established in appropriate domains.
(1) Gaussian bounds on the heat kernel \(p_t(x,y)=\mu_y^{-1}\mathbb{P}_x(X_t=y)\) where \(\mathbb{P}_x\) describes a law of \(X_t\) starting at \(x\).
(2) A parabolic Harnack inequality for the solution \(u(t,x)\) of the partial-differential equation \[ \frac{\partial u}{\partial t}= L_{\mu} u. \]
(3) Volume doubling and a family of Poincaré inequalities.
More precisely, (3) implies (1) and (2). (3) is described geometrically and it is more easily verified.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
05C81 Random walks on graphs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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