Convergence rate estimates for semigroups and heat kernels associated with resistance forms

Abstract: In this paper, we derive quantitative convergence rates for stochastic processes associated with resistance forms. While the qualitative convergence of heat kernels and semigroups under the Gromov-Hausdorff-vague convergence of underlying measured resistance metric spaces has been investigated previously, their quantitative convergence rates have remained unexplored. We establish explicit convergence rates for the associated semigroups and heat kernels under the assumptions of measure regularity and lower resistance estimates. Furthermore, we introduce a new metric that induces the Gromov-Hausdorff-vague topology, and is convenient for evaluation. As applications of our main results, we present two illustrative examples. First, we derive first estimate on the convergence rate for the random walk approximation of Brownian motion on the Sierpinski gasket. Second, we apply our results to the one-dimensional Bouchaud trap model, successfully extending the previously known parameter regime to all cases where homogenization occurs and improving the convergence rate estimates in the existing regime by at least a quadratic factor.