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Gradient estimate of the heat kernel on modified graphs. (English) Zbl 1196.58012

The author obtains a condition on the modification of graphs (in the sense of a finite combinations of an edge modification and a vertex modification) which guarantees the preservation of the Gaussian upper bound for the gradient of the heat kernel, more precisely we have \[ \nabla k_n(x,y)\leq \frac{C}{\sqrt{n}V(x,\sqrt{n})}e^{-cd(x,y)^2/n} \] for all \(x,y\in V\) and \(n\in \mathbb{N}^{\ast}\) where \(V\) is the modified graph, \(V(x,y)\) is the volume of the ball centered at \(x\) with radius \(r\) for the combinatorial distance \(d\), and \(C,c\) are some positive constant.

MSC:

58J37 Perturbations of PDEs on manifolds; asymptotics
60G50 Sums of independent random variables; random walks
58J65 Diffusion processes and stochastic analysis on manifolds
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References:

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