Search ResultsShowing 1-20 of 97
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arXiv:2501.17440 (Published 2025-01-29)
Heat kernel estimates for Schrödinger operators with supercritical killing potentials
Comments: 57 pageIn this paper, we study the Schr\"odinger operator $\Delta-V$, where $V$ is a supercritical non-negative potential belonging to a large class of functions containing functions of the form $b|x|^{-(2+2\beta)}$, $b, \beta>0$. We obtain two-sided estimates on the heat kernel $p(t, x, y)$ of $\Delta-V$, along with estimates for the corresponding Green function. Unlike the case of the fractional Schr\"odinger operator $-(-\Delta)^{\alpha/2}-V$, $\alpha\in (0, 2)$, with supercritical killing potential dealt with in [11], in the present case, the heat kernel $p(t, x, y)$ decays to 0 exponentially as $x$ or $y$ tends to the origin.
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arXiv:2411.14926 (Published 2024-11-22)
Convex combinations of random variables stochastically dominate the parent for a large class of heavy-tailed distributions
Categories: math.PRStochastic dominance of a random variable by a convex combination of its independent copies has recently been shown to hold within the relatively narrow class of distributions with concave odds function. We show that a key property for this stochastic dominance result to hold is the subadditivity of the cumulative distribution function of the reciprocal of the random variable of interest, referred to as the inverted distribution. This enlarges significantly the family of distributions for which the dominance is verified. Moreover, we study the relation between the class of distributions with concave odds function and the class we introduce showing conditions under which the concavity of the odds function implies the subadditivity of inverted distribution-
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arXiv:2411.04248 (Published 2024-11-06)
Maximal $Λ(p)$-subsets of manifolds
We construct maximal $\Lambda(p)$-subsets on a large class of curved manifolds, in an optimal range of Lebesgue exponents $p$. Our arguments combine restriction estimates and decoupling with old and new probabilistic estimates.
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arXiv:2410.00750 (Published 2024-10-01)
Quasi-reversible bullet models
Comments: 24 pages, 12 figuresCategories: math.PRWe consider a large class of bullet models that contains, in particular, the colliding bullet models with creations. For this large class of bullet models, we give sufficient conditions on their parameter to be $\text{rot}(\pi)$-quasi-reversible and to be $\text{rot}(\pi/2)$-quasi-reversible. Moreover, those conditions assure them that one of their invariant measures is described by a Poisson Point Processes. This result applied to the colliding bullet models with creations is a first step to study its property under its non-empty invariant measure.
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arXiv:2409.10037 (Published 2024-09-16)
Singularity of solutions to singular SPDEs
Categories: math.PRBuilding on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $\Phi^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $\Phi^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.
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arXiv:2407.06352 (Published 2024-07-08)
Subadditivity and optimal matching of unbounded samples
We obtain new bounds for the optimal matching cost for empirical measures with unbounded support. For a large class of radially symmetric and rapidly decaying probability laws, we prove for the first time the asymptotic rate of convergence for the whole range of power exponents $p$ and dimensions $d$. Moreover we identify the exact prefactor when $p\le d$. We cover in particular the Gaussian case, going far beyond the currently known bounds. Our proof technique is based on approximate sub- and super-additivity bounds along a geometric decomposition adapted to some features the density, such as its radial symmetry and its decay at infinity.
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arXiv:2404.06383 (Published 2024-04-09)
Maximum Degree in Random Hyperbolic Graphs
Comments: 30 pages, 4 figuresThe random hyperbolic graph, introduced in 2010 by Krioukov, Papadopoulos, Kitsak, Vahdat and Bogu\~{n}\'a, is a graph model suitable for modelling a large class of real-world networks, known as complex networks. Gugelmann, Panagiotou and Peter proved that for curvature parameter $\alpha > 1/2$, the degree sequence of the random hyperbolic graph follows a power-law distribution with controllable exponent up to the maximum degree. To achieve this, they showed, among other results, that with high probability, the maximum degree is $n^{\frac{1}{2\alpha} + o(1)}$, where $n$ is the number of nodes. In this paper, we refine this estimate of the maximum degree, and we extend it to the case $\alpha \leq 1/2$: we first show that, with high probability, the node with the maximum degree is eventually the one that is the closest to the origin of the underlying hyperbolic space. From this, we get the convergence in distribution of the renormalised maximum degree. Except for the critical case $\alpha = 1/2$, the limit distribution belongs to the extreme value distribution family (Weibull distribution in the case $\alpha < 1/2$ and Fr\'echet distribution in the case $\alpha > 1/2$).
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arXiv:2310.10202 (Published 2023-10-16)
Random models on regularity-integrability structures
Comments: 29 pagesWe prove a convergence result for a large class of random models that encompasses the case of the BPHZ models used in the study of singular stochastic PDEs. We introduce for that purpose a useful variation on the notion of regularity structure called a regularity-integrability structure. It allows to deal in a single elementary setting with models on a usual regularity structure and their first order Malliavin derivative.
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arXiv:2304.08225 (Published 2023-04-17)
Percolation threshold for Brownian loop soup on metric graphs
Comments: 8 pages, 1 figureCategories: math.PRIn this short note, we show that the critical threshold for the percolation of the Brownian loop soup on a large class of transient metric graphs (including quasi-transitive graphs such as $\mathbb{Z}^d$, $d\geq 3$) is $1/2$.
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arXiv:2212.04287 (Published 2022-12-08)
Quadratic transportation cost in the conditional central limit theorem for dependent sequences
Categories: math.PRIn this paper, we give estimates of the quadratic transportation cost in the conditional central limit theorem for a large class of dependent sequences. Applications to irreducible Markov chains, dynamical systems generated by intermittent maps and $\tau$-mixing sequences are given.
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arXiv:2210.11225 (Published 2022-10-20)
Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes
Comments: 34 pagesCategories: math.PRLet $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional L\'evy {process} where {$Z^i$'s} are independent 1-dimensional L\'evy {processes} with identical jumping kernel $ \nu^1(r) =r^{-1}\phi(r)^{-1}$. Here $\phi$ is {an} increasing function with weakly scaling condition of order $\underline \alpha, \overline \alpha\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*} \begin{cases} \nu^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$}. \end{cases} \end{align*} Corresponding to the jumping kernel $J$, there exists an anisotropic Markov process $X$, see \cite{KW22}. In this article, we establish sharp two-sided Dirichlet heat kernel estimates for $X$ in $C^{1,1}$ open set, under certain regularity conditions. As an application of the main results, we derive the Green function estimates.
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arXiv:2209.04537 (Published 2022-09-09)
On divergence-free (form-bounded type) drifts
We develop regularity theory for elliptic Kolmogorov operator with divergence-free drift in a large class (or, more generally, drift having singular divergence). A key step in our proofs is "Caccioppoli's iterations", used in addition to the classical De Giorgi's iterations and Moser's method.
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arXiv:2206.05545 (Published 2022-06-11)
Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$
Comments: 23 pagesWe consider a large class of $N\times N$ Gaussian random band matrices with band-width $W$, and prove that for $W \ll N^{1/4}$ they exhibit Anderson localization at all energies. To prove this result, we rely on the fractional moment method, and on the so-called Mermin-Wagner shift (a common tool in statistical mechanics).
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arXiv:2109.03473 (Published 2021-09-08)
Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises
In this paper, we study intermittency properties for various stochastic PDEs with varieties of space time Gaussian noises via matching upper and lower moment bounds of the solution. Due to the absence of the powerful Feynman Kac formula, the lower moment bounds have been missing for many interesting equations except for the stochastic heat equation. This work introduces and explores the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the lower bounds for all moments which match the upper moment bounds. Our upper and lower moments are valid for various interesting equations, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplcians, and stochastic diffusions which are both fractional in time and in space.
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arXiv:2109.00399 (Published 2021-09-01)
Locality for singular stochastic PDEs
Comments: 14 pagesWe develop in this note the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalised equation for a very large class of spacetime dependent renormalization schemes.
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arXiv:2104.05381 (Published 2021-04-12)
Asymptotic of densities of exponential functionals of subordinator
Categories: math.PRIn this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the L\'evy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein-gamma functions. To apply the saddle point method we improved the Stirling type of asymptotic for Bernstein-gamma functions and the latter is of interest beyond this paper as the Bernstein-gamma functions are applicable in different settings especially through their asymptotic behaviour in the complex plane. As an application we have derived the asymptotic of the density and its derivatives for all exponential functionals of non-decreasing, potentially compound Poisson processes which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.
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arXiv:2103.08383 (Published 2021-03-15)
Equivalence-Singularity Dichotomy in Markov Measures
Comments: 14 pagesCategories: math.PRWe establish an equivalence-singularity dichotomy for a large class of one-dimensional Markov measures. Our approach is new in that we deal with one-sided and two-sided chains simultaneously, and in that we do not appeal to any 0-1 law. In fact we deduce a new 0-1 law from the dichotomy.
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arXiv:2101.08298 (Published 2021-01-20)
Dictionary-Sparse Recovery From Heavy-Tailed Measurements
Comments: 23 pagesThe recovery of signals that are sparse not in a basis, but rather sparse with respect to an over-complete dictionary is one of the most flexible settings in the field of compressed sensing with numerous applications. As in the standard compressed sensing setting, it is possible that the signal can be reconstructed efficiently from few, linear measurements, for example by the so-called $\ell_1$-synthesis method. However, it has been less well-understood which measurement matrices provably work for this setting. Whereas in the standard setting, it has been shown that even certain heavy-tailed measurement matrices can be used in the same sample complexity regime as Gaussian matrices, comparable results are only available for the restrictive class of sub-Gaussian measurement vectors as far as the recovery of dictionary-sparse signals via $\ell_1$-synthesis is concerned. In this work, we fill this gap and establish optimal guarantees for the recovery of vectors that are (approximately) sparse with respect to a dictionary via the $\ell_1$-synthesis method from linear, potentially noisy measurements for a large class of random measurement matrices. In particular, we show that random measurements that fulfill only a small-ball assumption and a weak moment assumption, such as random vectors with i.i.d. Student-$t$ entries with a logarithmic number of degrees of freedom, lead to comparable guarantees as (sub-)Gaussian measurements. Our results apply for a large class of both random and deterministic dictionaries. As a corollary of our results, we also obtain a slight improvement on the weakest assumption on a measurement matrix with i.i.d. rows sufficient for uniform recovery in standard compressed sensing, improving on results by Mendelson and Lecu\'e and Dirksen, Lecu\'e and Rauhut.
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arXiv:2009.03609 (Published 2020-09-08)
Random walks on generalized visible lattice points
Comments: 17 pages, comments are welcomeWe consider the proportion of generalized visible lattice points in the plane visited by random walkers. Our work concerns the visible lattice points in random walks in three aspects: (1) generalized visibility along curves; (2) one random walker visible from multiple watchpoints; (3) simultaneous visibility of multiple random walkers. Moreover, we found new phenomenon in the case of multiple random walkers: for visibility along a large class of curves and for any number of random walkers, the proportion of steps at which all random walkers are visible simultaneously is almost surely larger than a positive constant.
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arXiv:2008.07844 (Published 2020-08-18)
Universality of the geodesic tree in last passage percolation
Comments: 39 pagesCategories: math.PRIn this paper we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder of width $o(N^{2/3})$ and length $o(N)$ agrees in the cylinder, with the stationary geodesic sharing the same end point. In the case of the point-to-point model, we consider width $\delta N^{2/3}$ and length up to $\delta^{3/2} N/(\log(\delta^{-1}))^3$ and provide lower and upper bound for the probability that the geodesics agree in that cylinder.