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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 $$n$$.

##### MSC:
 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
##### Keywords:
locally finite graph; Markov operator
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