Transience in growing subgraphs via evolving sets

We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with uniformly bounded above and below, independently time-varying edge conductances, of (effectively) non-decreasing in time vertex conductances (i.e. reversing measure), thereby affirming part of [ABGK, Conj. 7.1].

[1]  C. Sabot,et al.  A random Schr\"odinger operator associated with the Vertex Reinforced Jump Process and the Edge Reinforced Random Walk , 2015 .

[2]  T. Kumagai,et al.  Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances , 2015, 1506.02193.

[3]  熊谷 隆.  Random walks on disordered media and their scaling limits , 2014 .

[4]  T. Kumagai.  Random walks on disordered media and their scaling limits , 2014 .

[5]  A. Dembo,et al.  Walking within growing domains: recurrence versus transience * , 2013, 1312.4610.

[6]  Quenched invariance principle for simple random walk on clusters in correlated percolation models , 2013, 1310.4764.

[7]  V. Climenhaga.  Markov chains and mixing times , 2013 .

[8]  Omer Angel,et al.  Localization for Linearly Edge Reinforced Random Walks , 2012, 1203.4010.

[9]  P. Tarres,et al.  Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model , 2011, 1111.3991.

[10]  L. Saloff-Coste,et al.  Merging for inhomogeneous finite Markov chains, part II: Nash and log-Sobolev inequalities , 2011, 1104.1560.

[11]  Dmitry Dolgopyat,et al.  Random walk in Markovian environment , 2007, math/0702100.

[12]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[13]  A. Grigor’yan,et al.  The discrete integral maximum principle and its applications , 2005 .

[14]  J. Deuschel,et al.  On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ∇ϕ interface model , 2005 .

[15]  A. Chenciner.  A note by Poincare , 2005 .

[16]  Yuval Peres,et al.  Evolving sets and mixing , 2003, STOC '03.

[17]  Y. Peres,et al.  Evolving sets, mixing and heat kernel bounds , 2003, math/0305349.

[18]  M. Barlow.  Random walks on supercritical percolation clusters , 2003, math/0302004.

[19]  É. Remy,et al.  Isoperimetry and heat kernel decay on percolation clusters , 2003, math/0301213.

[20]  G. Giacomin,et al.  On recurrent and transient sets of inhomogeneous symmetric random walks. , 2001 .

[21]  Thierry Delmotte,et al.  Parabolic Harnack inequality and estimates of Markov chains on graphs , 1999 .

[22]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality , 1996 .

[23]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations , 1995 .

[24]  Laurent Saloff-Coste,et al.  GAUSSIAN ESTIMATES FOR MARKOV CHAINS AND RANDOM WALKS ON GROUPS , 1993 .

[25]  A. Grigor’yan.  THE HEAT EQUATION ON NONCOMPACT RIEMANNIAN MANIFOLDS , 1992 .

[26]  L. Saloff-Coste,et al.  A note on Poincaré, Sobolev, and Harnack inequalities , 1992 .

[27]  P. Diaconis,et al.  Strong Stationary Times Via a New Form of Duality , 1990 .

[28]  D. Aronson,et al.  Bounds for the fundamental solution of a parabolic equation , 1967 .