A Spacial Gradient Estimate for Solutions to the Heat Equation on Graphs

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Heat equation, Gradient estimate, Curvature, Curvature dimension inequality, Continuous time random walk

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The study of positive solutions of the heat equation $\frac{\partial}{\partial \alpha} u = \Delta u$, on both manifolds and graphs, gives an analytic way of extracting geometric information about the object. In the manifold case, one of the most effective ways of studying how solutions to the heat equation evolve is to derive a local “gradient estimate” of heat change, using curvature lower bounds. Recently, notions of curvature for graphs have been developed which enable proving similar estimates for graphs. In this article, we derive a gradient estimate for positive heat solutions that considers only how heat varies in space and the time derivative. This result, due in the manifold case to Hamilton, applies to both finite graphs and infinite graphs of bounded degree. As a corollary, a heat comparison theorem is also developed. This in turn yields results about the mixing of the continuous time random walk on graphs.

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