Sub-Gaussian short time asymptotics for measure metric Dirichlet spaces (Q867087)

Suppose that \((M, \mu, d)\) is a locally compact separable metric space with \(\mu\) a Radon measure with full support. Let \(({\mathcal E}, {\mathcal F})\) be a regular local Dirichlet form in \(L^2(M, \mu)\) and let \(X\) be the diffusion associated with this Dirichlet form. Assume that \(X\) is conservative and it posseses a transition density \(p_t(x, y)\) with respect to \(\mu\). For any subset \(A, B\) of \(M\), put \[ P_t(A, B)=\int_A\int_Bp_t(x, y)\mu(dx)\mu(dy). \] For any open subset \(G\) of \(M\), \(T_G\) stands for the exit time of \(G\). The main result of this paper can be stated as follows: Suppose that there is \(R_0>0\) such that for all \(r<R_0\) and \(x\in M\), \[ E_xT_{B(x, r)}\simeq r^{\beta} \] for some \(\beta>1\), then for any subsets \(A, B\) of \(M\) with \(0<\mu(A), \mu(B)<\infty\) we have \[ \lim_{t\to 0}t^{\frac1{\beta-1}}\log P_t(A, B)\leq -c(d(A, B))^{\frac{\beta}{\beta-1}} \] for some \(c>0\). Furthermore, if the elliptic Harnack inequality holds, then for any relatively compact open subsets \(A, B\) we have \[ \lim_{t\to 0}t^{\frac1{\beta-1}}\log P_t(A, B)\geq -C(d(A, B))^{\frac{\beta}{\beta-1}} \] for some \(C>0\).