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Takashi Kumagai

Publications and source records attributed to Takashi Kumagai.

At least 55 records · Page 3Linked to original sources
arXiv2018

Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases for nearest neighbor conductance models, the idea through parabolic Harnack inequalities does not work, since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

math.PR↗
arXiv2018

Cutoff for lamplighter chains on fractals

We show that the total-variation mixing time of the lamplighter random walk on fractal graphs exhibit sharp cutoff when the underlying graph is transient (namely of spectral dimension greater than two). In contrast, we show that such cutoff can not occur for strongly recurrent underlying graphs (i.e. of spectral dimension less than two).

math.PR↗
arXiv2018

Direct observation of near-field induced resonant electron tunneling in a sub-nanometer plasmonic gap

Localized surface plasmon resonance (LSPR) excitation of nanostructures and charge transfer in plasmonic nanocavities plays a central role in nanoscale optoelectronics and in applications for plasmonic devices. However, the direct observation of near-filed induced charge transfer has remained as a challenging experiment. Here we present LSPR-assisted resonant electron tunneling from an Ag or Au tip to the image potential states of a Ag(111) surface using scanning tunneling microscopy (STM). The LSPR-assisted tunneling process results in an incident photon-energy dependent red-shift of the field emission resonances (FERs) in the gap. Using the precise control of the gap distance in the STM junction we demonstrate tuning of the relative contribution from the LSPR-assisted and the normal STM electron tunneling processes. Furthermore, the FER intensity mapping of local defects on the surface allows assigning unambiguously the respective FER levels with and without laser excitation.

cond-mat.mes-hall↗
arXiv2017

Heat kernel estimates for FIN processes associated with resistance forms

Quenched and annealed heat kernel estimates are established for Fontes-Isopi-Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to fractals such as the Sierpinski gasket and carpet.

math.PR↗
arXiv2017

Stability of heat kernel estimates for symmetric non-local Dirichlet forms

In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for $α$-stable-like processes even with $α\ge 2$ when the underlying spaces have walk dimensions larger than $2$, which has been one of the major open problems in this area.

math.PR↗
arXiv2017

Heat kernel estimates for time fractional equations

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

math.PR↗
arXiv2017

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.

math.PR↗
arXiv2017

Products of random walks on finite groups with moderate growth

In this article, we consider products of random walks on finite groups with moderate growth and discuss their cutoffs in the total variation. Based on several comparison techniques, we are able to identify the total variation cutoff of discrete time lazy random walks with the Hellinger distance cutoff of continuous time random walks. Along with the cutoff criterion for Laplace transforms, we derive a series of equivalent conditions on the existence of cutoffs, including the existence of pre-cutoffs, Peres' product condition and a formula generated by the graph diameters. For illustration, we consider products of Heisenberg groups and randomized products of finite cycles.

math.PR↗
arXiv2017

Cutoffs for product chains

In this article, we consider products of ergodic Markov chains and discuss their cutoffs in the total variation. Through a new inequality relating the total variation and the Hellinger distance, we may identify the total variation cutoffs with cutoffs in the Hellinger distance. This provides a new scheme to study the total variation mixing of Markov chains, in particular, product chains. In the theoretical framework, a series of criteria are introduced to examine cutoffs and a comparison of mixing between the product chain and its coordinate chains is made in detail. For illustration, we consider products of two-state chains, cycles and other typical examples.

math.PR↗
arXiv2016

Lamplighter random walks on fractals

We consider on-diagonal heat kernel estimates and the laws of the iterated logarithms for a switch-walk-switch random walk on a lamplighter graph under the condition that the random walk on the underlying graph enjoys sub-Gaussian heat kernel estimates.

math.PR↗
arXiv2016

Laws of the iterated logarithm for symmetric jump processes

Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for $β$-stable-like processes on $α$-sets with $β>0$.

math.PR↗
arXiv2015

Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances

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.

math.PR↗
arXiv2015

Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

math.PR↗
arXiv2015

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.

math.PR↗
arXiv2014

Quenched Invariance Principle for a class of random conductance models with long-range jumps

We study random walks on $\mathbb Z^d$ among random conductances $\{C_{xy}\colon x,y\in\mathbb Z^d\}$ that permit jumps of arbitrary length. Apart from joint ergodicity with respect to spatial shifts, we assume only that the nearest-neighbor conductances are uniformly positive and that $\sum_{x\in\mathbb Z^d} C_{0x}|x|^2$ is integrable. Our focus is on the Quenched Invariance Principle (QIP) which we establish in all $d\ge3$ by a combination of corrector methods and heat-kernel technology. In particular, a QIP thus holds for random walks on long-range percolation graphs with exponents larger than $d+2$ in all $d\ge3$, provided all nearest-neighbor edges are present. We then show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\ge4$ under the conditions close to, albeit not exactly, complementary to those of the recent work of S. Andres, M. Slowik and J.-D. Deuschel.

math.PR↗
arXiv2013

Boundary Harnack inequality for Markov processes with jumps

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, Lévy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schrödinger, drift and jump perturbations of such processes.

math.PR↗
arXiv2013

Fluctuations of recentered maxima of discrete Gaussian Free Fields on a class of recurrent graphs

We provide conditions that ensure that the recentered maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal variance. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in Z but in contrast with the situation in Z^2. We show that our conditions cover a large class of "fractal" graphs.

math.PR↗