Short Talk: Ryoichiro Noda

Consider two independent and identically distributed continuous-time Markov chains $X$ and $Y$ on a graph with vertex set $V$. The collision measure $\Pi$, which describes the times and positions of collisions of $X$ and $Y$, is given as a random measure on $\mathbb{R}_{\geq 0} \times V$ by  $\Pi([0,t] \times E) = \int_{0}^{t} 1_{\{X(s) = Y(s)\}} ds.$. We have shown that if graphs converge to a metric space in an appropriate Gromov-Hausdorff-type topology with respect to the effective resistance metric, and if there is a suitable upper bound on the heat kernels of the Markov chains on the graphs, then the two independent Markov chains on the graph and its collision measure converge to stochastic processes on the limiting space and its collision measure. The collision measure of the limiting stochastic processes is constructed using positive continuous additive functionals (PCAFs) in Dirichlet form theory. This applies to many low-dimensional spaces, such as the Sierpi\'{n}ski gasket and the Brownian random tree.