Mostrando recursos 1 - 20 de 92

  1. Interpolation process between standard diffusion and fractional diffusion

    Bernardin, Cédric; Gonçalves, Patrícia; Jara, Milton; Simon, Marielle
    We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume (Nonlinearity 25 (4) (2012) 1099–1133; Arch. Ration. Mech. Anal. 220 (2) (2016) 505–542). We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper...

  2. Liouville quantum gravity on the unit disk

    Huang, Yichao; Rhodes, Rémi; Vargas, Vincent
    Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk.

  3. Characterization of a class of weak transport-entropy inequalities on the line

    Gozlan, Nathael; Roberto, Cyril; Samson, Paul-Marie; Shu, Yan; Tetali, Prasad
    We study an weak transport cost related to the notion of convex order between probability measures. On the real line, we show that this weak transport cost is reached for a coupling that does not depend on the underlying cost function. As an application, we give a necessary and sufficient condition for weak transport-entropy inequalities (related to concentration of convex/concave functions) to hold on the line. In particular, we obtain a weak transport-entropy form of the convex Poincaré inequality in dimension one.

  4. The near-critical Gibbs measure of the branching random walk

    Pain, Michel
    Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $\nu_{n,\beta}$ on the $n$th generation, which is also the polymer measure on a disordered tree with inverse temperature $\beta$. The convergence of the partition function $W_{n,\beta}$, after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi (Ann. Probab. 42 (3) (2014) 959–993) in the critical case $\beta=1$ and by Madaule (J. Theoret. Probab. 30 (1) (2017) 27–63) when $\beta>1$. We study here the near-critical case, where $\beta_{n}\to1$, and prove the convergence of $W_{n,\beta_{n}}$, after rescaling, towards a constant...

  5. Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes

    Belomestny, Denis; Trabs, Mathias
    The estimation of the diffusion matrix $\Sigma$ of a high-dimensional, possibly time-changed Lévy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on $\Sigma$. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of $\Sigma$ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.

  6. Optimal discretization of stochastic integrals driven by general Brownian semimartingale

    Gobet, Emmanuel; Stazhynski, Uladzislau
    We study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a pathwise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales (relaxing in particular the non degeneracy conditions usually requested) and we prove that the asymptotic lower bound is attainable.

  7. The size of the last merger and time reversal in $\Lambda$-coalescents

    Kersting, Götz; Schweinsberg, Jason; Wakolbinger, Anton
    We consider the number of blocks involved in the last merger of a $\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n\to\infty$, the sequence of these random variables (a) is tight, (b) converges in distribution to a finite random variable or (c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of invariant measures for the dynamics of the block-counting process, and in case (b) investigate the time-reversal of the block-counting process back from the time...

  8. Thick points of high-dimensional Gaussian free fields

    Chen, Linan
    This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions. Under this setting, we introduce the definition of thick points which, heuristically speaking, are points where the value of the Gaussian Free Field is unusually large. We then establish a result on the Hausdorff dimension of the sets containing thick points.

  9. Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada–Watanabe principle

    Baños, David R.; Duedahl, Sindre; Meyer-Brandis, Thilo; Proske, Frank
    In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.

  10. Fluctuations of bridges, reciprocal characteristics and concentration of measure

    Conforti, Giovanni
    Conditions on the generator of a Markov process to control the fluctuations of its bridges are found. In particular, continuous time random walks on graphs and gradient diffusions are considered. Under these conditions, a concentration of measure inequality for the marginals of the bridge of a gradient diffusion and refined large deviation expansions for the tails of a random walk on a graph are derived. In contrast with the existing literature about bridges, all the estimates we obtain hold for non asymptotic time scales. New concentration of measure inequalities for pinned Poisson random vectors are also established. The quantities expressing...

  11. Weak convergence of obliquely reflected diffusions

    Sarantsev, Andrey
    Burdzy and Chen (Electron. J. Probab. 3 (1998) 29–33) proved results on weak convergence of multidimensional normally reflected Brownian motions. We generalize their work by considering obliquely reflected diffusion processes. We require weak convergence of domains, which is stronger than convergence in Wijsman topology, but weaker than convergence in Hausdorff topology.

  12. Markov processes on the duals to infinite-dimensional classical Lie groups

    Cuenca, Cesar
    We construct a four parameter $z,z',a,b$ family of Markov dynamics that preserve the $z$-measures on the boundary of the branching graph for classical Lie groups of type $B,C,D$. Our guiding principle is the “method of intertwiners” used previously in [J. Funct. Anal. 263 (2012) 248–303] to construct Markov processes that preserve the $zw$-measures.

  13. Biased random walks on the interlacement set

    Fribergh, Alexander; Popov, Serguei
    We study a biased random walk on the interlacement set of $\mathbb{Z}^{d}$ for $d\geq3$. Although the walk is always transient, we can show, in the case $d=3$, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.

  14. The strong Feller property for singular stochastic PDEs

    Hairer, M.; Mattingly, J.
    We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^{4}_{3}$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.

  15. Scaling limit and ageing for branching random walk in Pareto environment

    Ortgiese, Marcel; Roberts, Matthew I.
    We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We show that the system of particles, rescaled in an appropriate way, converges in distribution to a scaling limit that is interesting in its own right. We describe the limit object as a growing collection of “lilypads” built on a Poisson point process in $\mathbb{R}^{d}$. As an application of our main theorem, we show that the maximizer of the system displays the ageing property.

  16. Asymptotics of random domino tilings of rectangular Aztec diamonds

    Bufetov, Alexey; Knizel, Alisa
    We consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, the explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.

  17. Spectral gap for the stochastic quantization equation on the 2-dimensional torus

    Tsatsoulis, Pavlos; Weber, Hendrik
    We study the long time behavior of the stochastic quantization equation. Extending recent results by Mourrat and Weber (Global well-posedness of the dynamic $\phi^{4}$ in the plane (2015) Preprint) we first establish a strong non-linear dissipative bound that gives control of moments of solutions at all positive times independent of the initial datum. We then establish that solutions give rise to a Markov process whose transition semigroup satisfies the strong Feller property. Following arguments by Chouk and Friz (Support theorem for a singular SPDE: the case of gPAM (2016) Preprint) we also prove a support theorem for the laws of...

  18. The velocity of 1d Mott variable-range hopping with external field

    Faggionato, Alessandra; Gantert, Nina; Salvi, Michele
    Mott variable-range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps. ¶ We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the...

  19. Intertwinings of beta-Dyson Brownian motions of different dimensions

    Ramanan, Kavita; Shkolnikov, Mykhaylo
    We show that for all positive $\beta$ the semigroups of $\beta$-Dyson Brownian motions of different dimensions are intertwined. The proof relates $\beta$-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption $\beta\ge1$ in (Probab. Theory Related Fields 163 (2015) 413–463). The corresponding results for $\beta$-Dyson Ornstein–Uhlenbeck processes are also presented.

  20. Universality in a class of fragmentation-coalescence processes

    Kyprianou, A. E.; Pagett, S. W.; Rogers, T.
    We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the thermodynamic limit. Moreover, we discover that in the limit of small fragmentation rate these processes exhibit a universal cluster size distribution regardless of the details of the rates, following a power law with exponent $3/2$.

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