Abstract
We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.
Highlights
The study of heat kernels of diffusions on manifolds and Markov chains on graphs has a very long and fruitful history
We note that the stability of parabolic Harnack inequalities and estimates of the heat kernel were already established in the framework of time-dependent Dirichlet forms on metric measure spaces by Sturm [St1, St2], and in [DD, GOS] it was proved that for random walks on Zd among uniformly elliptic time-dependent conductances, the two-sided Gaussian heat kernel estimates hold
The purpose of this note is to demonstrate a fundamental difference between discrete time random walks and variable speed random walks even in the framework that the random walks are uniformly elliptic and uniformly lazy. We find that both the upper and lower Gaussian bounds can be violated in these situations, and they are unstable even though (VD)+(PI(2)) are still stable
Summary
The study of heat kernels of diffusions on manifolds and Markov chains on graphs has a very long and fruitful history. We note that the stability of parabolic Harnack inequalities and estimates of the heat kernel were already established in the framework of time-dependent Dirichlet forms on metric measure spaces by Sturm [St1, St2], and in [DD, GOS] it was proved that for random walks on Zd among uniformly elliptic time-dependent conductances, the two-sided Gaussian heat kernel estimates hold. The times {tn} at which the conductances change satisfying tn/n → 1/c, such that the corresponding csrw {Yt}t∈R+ is ballistic and transient almost surely Both walks violate the Gaussian heat kernel on-diagonal lower bound as well as off-diagonal upper bound on Z. The times {tn} at which the conductances change satisfying tn/n → 1/c, such that the corresponding csrw {Yt}t∈R+ is recurrent almost surely Both walks violate the Gaussian heat kernel on-diagonal upper bound on Z2 × Z≥0. By the strong law of large numbers (slln) and (2.3), we have that
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