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Sub-Gaussian estimates of heat kernels on infinite graphs. (English) Zbl 1010.35016
Let \((\Gamma, \mu)\) be a weighted graph that is an infinite connected locally finite graph endowed by the weight \(\mu_{xy}\) for \(x,y\in \Gamma\) and a measure \(\mu(x)=\sum \mu_{xy}\) (where the sum is taken over all \(y\) connected by the wedge with \(x\)) and \(d(x,y)\) be the distance from \(x\) to \(y\). Let \(P(x,y)=\frac{\mu_{xy}}{\mu(x)}\) be a natural Markov operator on the weighted graph, \(P_n(x,y)\) be the \(n\)th convolution power of \(P\) and \(p_n(x,y)=\frac{P_n(x,y)}{\mu(y)}.\) Fixing \(\alpha>\beta>1\) and \(\gamma=\alpha-\beta\) the authors prove that any weighted graph \((\Gamma, \mu)\) with \(P(x,y)\geq p_0>0\) has the regular volume growth \(V(x,R)\simeq R^\alpha,\) \( x\in \Gamma, R\geq 1\) and the estimate \[ \sum_{n=0}^{\infty}p_n(x,y)\simeq d(x,y)^{-\gamma} \quad \text{for } x\neq y \] holds if and only if \[ p_n(x,y)\leq Cn^{\frac{\alpha}{\beta}} \exp\left(- \left(\frac{d(x,y)^\beta}{Cn}\right)^{\frac{1}{\beta-1}}\right) \] and \[ p_n(x,y)+p_{n+1}(x,y)\geq cn^{-\frac{\alpha}{\beta}} \exp\left(- \left(\frac{d(x,y)^\beta}{cn} \right)^{\frac{1}{\beta-1}}\right), \quad n\geq d(x,y) \] for arbitrary \(x,y\in \Gamma\) and a positive integer \(n\).

35B45 A priori estimates in context of PDEs
60J35 Transition functions, generators and resolvents
60G50 Sums of independent random variables; random walks
35K05 Heat equation
60J45 Probabilistic potential theory
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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