Gaussian bounds and parabolic Harnack inequality on locally irregular graphs (Q343190)

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.