arXiv subjects

Takashi Kumagai

Publications and source records attributed to Takashi Kumagai.

At least 19 recordsLinked to original sources

arXiv2026

High-order convergence rates of periodic homogenization for symmetric Lévy type operators

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical $α$-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales -- a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.

math.PR↗

arXiv2026

Ultrafast Nano-Imaging and Optical Control of Hyperbolic Phonon Polaritons at hBN/WS$_2$ Heterojunctions

Manipulating nanoscale light-matter interactions on ultrafast time scales is indispensable for future polaritonic devices. Hyperbolic phonon polaritons (HPhPs) in van der Waals materials enable deep subwavelength confinement of electromagnetic fields in the infrared region and long-distance propagation of polaritonic waves. However, achieving ultrafast imaging and optical control of HPhPs remains a major challenge. Here, we demonstrate the direct observation of transient modulation of HPhPs induced by local photocarrier generation in WS$_2$/hBN heterostructures using ultrafast infrared scanning near-field optical microscopy. We implement grating-based spectral filtering of broadband near-field scattering to simultaneously achieve nanoscale and femtosecond spatiotemporal resolution together with fine spectral selectivity. This ultrafast nano-imaging technique reveals that photocarriers in WS$_2$ modulate the polaritonic field amplitudes and wavelengths of HPhPs in hBN. Theoretical simulations corroborate that these changes arise from photoinduced changes in WS$_2$ dielectric properties. This approach offers a versatile platform for exploring ultrafast polaritonic dynamics at the nanoscale.

physics.optics↗

arXiv2025

Quantitative homogenization on time-dependent random conductance models with stable-like jumps

We establish quantitative homogenization results for time-dependent random conductance models with stable-like long range jumps on $\Z^d$, where the transition probability from $x$ to $y$ is given by $w_{t, x,y}|x-y|^{-d-α}$ with $α\in (0,2)$. In particular, time-dependent random coefficients $\{w_{t,x,y}: t\in \R_+, (x,y)\in E\}$ are uniformly bounded from above (but may be degenerate), and satisfy the Kolmogorov continuous condition, where $E=\{(x, y): x \not= y \in \Z^d\}$ is the set of all unordered pairs on $\Z^d$. The proofs are based on $L^2$-estimates and energy estimates for solutions to regionalparabolic equations and multi-scale Poincaré inequalities associated with time-dependent symmetric stable-like random walks with random coefficients.

math.PR↗

arXiv2025

Gradient estimates of the heat kernel for random walks among time-dependent random conductances

In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in a dynamical random environment with symmetric jump rates to nearest neighbours. We assume that these random conductances are stationary and ergodic and, moreover, that they are bounded from below but unbounded from above with finite first moment. We derive sharp on-diagonal estimates for the annealed first and second discrete space derivative of the heat kernel which then yield local limit theorems for the corresponding kernels. Assuming weak algebraic off-diagonal estimates, we then extend these results to the annealed Green function and its first and second derivative. Our proof which extends the result of Delmotte and Deuschel (2005) to unbounded conductances with first moment only, is an adaptation of the recent entropy method of Benjamini et. al. (2015).

math.PR↗

arXiv2025

Scaling limit for Brownian motions on the $l$-level Sierpinski gaskets: The fractal to Euclidean crossover

In two dimensions, the $l$-level Sierpinski gasket $\mathrm{SG}(l)$ is obtained by splitting an equilateral triangle into a collection of $l^2$ equilateral triangles of equal size and with the same total area, retaining only the $l(l+1)/2$ triangles with the same orientation as the original triangle, and then iterating this procedure indefinitely. We show that the canonical diffusions on the spaces $\mathrm{SG}(l)$, $l\geq2$, can be rescaled to yield Brownian motion on the initial triangle. Our argument also applies to the analogous higher-dimensional Sierpinski gaskets. Moreover, we prove a local central limit theorem for the associated transition densities. Key to this is the derivation of a Poincaré inequality, in the proof of which we exploit the Euclidean-type mixing that occurs between the bottlenecks present at each scale of the fractal.

math.PR↗

arXiv2025

Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces

In the context of a metric measure space $(X,d,μ)$, we explore the potential-theoretic implications of having a finite-dimensional Besov space. We prove that if the dimension of the Besov space $B^θ_{p,p}(X)$ is $k>1$, then $X$ can be decomposed into $k$ number of irreducible components (Theorem 1.1). Note that $θ$ may be bigger than $1$, as our framework includes fractals. We also provide sufficient conditions under which the dimension of the Besov space is $1$. We introduce critical exponents $θ_p(X)$ and $θ_p^{\ast}(X)$ for the Besov spaces. As examples illustrating Theorem 1.1, we compute these critical exponents for spaces $X$ formed by glueing copies of $n$-dimensional cubes, the Sierpiński gaskets, and of the Sierpiński carpet.

math.FA↗

arXiv2025

Scattering near-field optical microscopy at 1-nm resolution using ultralow tip oscillation amplitudes

Scattering-type scanning near-field optical microscopy (s-SNOM) allows for the observation of the optical response of material surfaces with a resolution far below the diffraction limit. Based on amplitude-modulation atomic force microscopy (AFM) with typical tapping amplitudes of tens of nanometers, a spatial resolution of 10-100 nm is routinely achieved in s-SNOM. However, optical imaging and spectroscopy of atomic-scale structures remain a substantial challenge. Here, we developed ultralow tip oscillation amplitude s-SNOM (ULA-SNOM), where the ultra-confined field localized at a 1-nm-scale gap between a plasmonic tip and sample is combined with frequency-modulation (non-contact) AFM in a stable cryogenic ultrahigh vacuum environment. Using a silver tip under visible laser illumination with a constant 1-nm amplitude oscillation, we obtain a material-contrast image of silicon islands on a silver surface with 1-nm lateral resolution, which surpasses the conventional limits of s-SNOM. ULA-SNOM paves the way for the acquisition of optical information from atomic-scale structures, such as single photo-active defects and molecules.

physics.optics↗

arXiv2025

Picocavity-Enhanced Raman Spectroscopy of Physisorbed H$_2$ and D$_2$ Molecules

We report on tip-enhanced Raman spectroscopy of H2 and D2 molecules physisorbed within a plasmonic picocavity at 10 K. The intense Raman peaks resulting from the rotational and vibrational transitions are observed at subnanometer gap distances of the junction formed by an Ag tip and an Ag(111) surface, where a picocavity-enhanced field plays a crucial role. A significant redshift of the H-H stretch frequency is observed as the gap distance decreases, while the D-D stretch frequency is unaffected. Density functional theory, path-integral molecular dynamics, and quantum anharmonic vibrational energy calculations suggest that this unexpected isotope effect is explained by a different molecular density between H2 and D2 on the surface.

physics.optics↗

arXiv2024

Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ means the integral in the principal value sense, $p\in(1,\infty)$ and $J(t;x,y)$ is comparable to the kernel of the fractional $p$-Laplacian operator $|x-y|^{-d-sp}$ with $s\in(0,1)$ and uniformly in $(t;x,y)\in\R\times\R^d\times\R^d$. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that $p=2$), in particular with a nonlocal parabolic tail that uses the $L^1$-norm in time for all $p\in (1,\infty)$. The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional $p$-Laplacian type operators with all $p>1$.

math.AP↗

arXiv2024

Quantitative periodic homogenization for symmetric non-local stable-like operators

Homogenization for non-local operators in periodic environments has been studied intensively. So far, these works are mainly devoted to the qualitative results, that is, to determine explicitly the operators in the limit. To the best of authors' knowledge, there is no result concerning the convergence rates of the homogenization for stable-like operators in periodic environments. In this paper, we establish a quantitative homogenization result for symmetric $α$-stable-like operators on $\R^d$ with periodic coefficients. In particular, we show that the convergence rate for the solutions of associated Dirichlet problems on a bounded domain $D$ is of order $$ \varepsilon^{(2-α)/2}\I_{\{α\in (1,2)\}}+\varepsilon^{α/2}\I_{\{α\in (0,1)\}}+\varepsilon^{1/2}|\log \e|^2\I_{\{α=1\}}, $$ while, when the solution to the equation in the limit is in $C^2_c(D)$, the convergence rate becomes $$ \varepsilon^{2-α}\I_{\{α\in (1,2)\}}+\varepsilon^α\I_{\{α\in (0,1)\}}+\varepsilon |\log \e|^2\I_{\{α=1\}}. $$ This indicates that the boundary decay behaviors of the solution to the equation in the limit affects the convergence rate in the homogenization.

math.AP↗

arXiv2024

Quenched local limit theorem for random conductance models with long-range jumps

We establish the quenched local limit theorem for reversible random walk on $\Z^d$ (with $d\ge 2$) among stationary ergodic random conductances that permit jumps of arbitrary length. The proof is based on the weak parabolic Harnack inequalities and on-diagonal heat-kernel estimates for long-range random walks on general ergodic environments. In particular, this partly solves \cite[Open Problem 2.7]{BCKW}, where the quenched invariance principle was obtained. As a byproduct, we prove the maximal inequality with an extra tail term for long-range reversible random walks, which in turn yields the everywhere sublinear property for the associated corrector.

math.PR↗

arXiv2024

Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry-Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

math.PR↗

arXiv2024

On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$

Let $(K,d)$ be a connected compact metric space and $p\in (1, \infty)$. Under the assumption of \cite[Assumption 2.15]{Ki2} and the conductive $p$-homogeneity, we show that $\mathcal{W}^p(K)\subset C(K)$ holds if and only if $p>\operatorname{dim}_{AR}(K,d)$, where $\mathcal{W}^p(K)$ is Kigami's $(1,p)$-Sobolev space and $\operatorname{dim}_{AR}(K,d)$ is the Ahlfors regular dimension.

math.FA↗

arXiv2023

Inelastic Light Scattering in the Vicinity of a Single-Atom Quantum Point Contact in a Plasmonic Picocavity

Here, using low-temperature optical scanning tunneling microscopy (STM), we investigate inelastic light scattering (ILS) in the vicinity of a single-atom quantum point contact (QPC). A vibration mode localized at the single Ag adatom on the Ag(111) surface is resolved in the ILS spectrum, resulting from tip-enhanced Raman scattering (TERS) by the atomically-confined plasmonic field in the STM junction. Furthermore, we trace how TERS from the single adatom evolves as a function of the gap distance. The exceptional stability of the low-temperature STM allows to examine distinctly different electron transport regimes of the picocavity, namely in the tunneling and quantum point contact (QPC) regimes. This measurement shows that the vibration mode localized at the adatom and its TERS intensity exhibits a sharp change upon the QPC formation, indicating that the atomic-level structure has a crucial impact on the plasmonic properties. To gain microscopic insights into picocavity optomechanics, we scrutinize the structure and plasmonic field in the STM junction using time-dependent density functional theory. The simulations reveal that atomic-scale structural relaxation at the single-atom QPC results in a discrete change of the plasmonic field strength, volume, and distribution as well as the vibration mode localized at the single atom. These findings give a qualitative explanation for the experimental observations. Furthermore, we demonstrate that strong ILS is a characteristic feature of QPC by continuously forming, breaking, and reforming the atomic contact, and how the plasmonic resonance evolves throughout the non-tunneling, tunneling, and QPC regimes.

cond-mat.mes-hall↗

arXiv2023

Heat kernel fluctuations for stochastic processes on fractals and random media

It is well-known that stochastic processes on fractal spaces or in certain random media exhibit anomalous heat kernel behaviour. One manifestation of such irregular behaviour is the presence of fluctuations in the short- or long-time asymptotics of the on-diagonal heat kernel. In this note we review some examples for which such fluctuations are known to occur, including Brownian motion on certain deterministic or random fractals, and simple random walks on various examples of random graph trees, such as the incipient infinite cluster of critical percolation on a regular tree and low-dimensional uniform spanning trees. We also announce some new results that add the one-dimensional Bouchaud trap model to this class of examples.

math.PR↗

arXiv2023

Quantitative stochastic homogenization for random conductance models with stable-like jumps

We consider random conductance models with long range jumps on $\Z^d$, where the one-step transition probability from $x$ to $y$ is proportional to $w_{x,y}|x-y|^{-d-α}$ with $α\in (0,2)$. Assume that $\{w_{x,y}\}_{(x,y)\in E}$ are independent, identically distributed and uniformly bounded non-negative random variables with $\Ee w_{x,y}=1$, where $E$ is the set of all unordered pairs on $\Z^d$. We obtain a quantitative version of stochastic homogenization for these random walks, with explicit polynomial rates up to logarithmic corrections.

math.PR↗

arXiv2022

Long range random walks and associated geometries on groups of polynomial growth

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study the large time behavior of its probability of return at time $n$ in terms of the key parameters describing the driving measure and the structure of the underlying group. We obtain assorted estimates including near-diagonal two-sided estimates and the Hölder continuity of the solutions of the associated discrete parabolic difference equation. In each case, these estimates involve the construction of a geometry adapted to the walk.

math.PR↗

arXiv2022

Limit theorems for some long range random walks on torsion free nilpotent groups

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $Γ$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $μ$, we construct a homogeneous nilpotent Lie group $G_\bullet(Γ,μ)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $μ$. In addition, the functional limit theorem is complemented by a local limit theorem.

math.PR↗