Mostrando recursos 1 - 20 de 536

  1. Nonexistence of Lyapunov exponents for matrix cocycles

    Tian, Xueting
    It follows from Oseledec Multiplicative Ergodic Theorem (or Kingman’s Sub-additive Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a Hölder continuous matrix cocycle $A:X\rightarrow \operatorname{GL}(m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_{\delta}$ set). Here we point out that exponential specification is introduced and plays critical role,...

  2. On the Malliavin differentiability of BSDEs

    Mastrolia, Thibaut; Possamaï, Dylan; Réveillac, Anthony
    In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron–Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin–Sobolev type spaces $\mathbb{D}^{1,p}$.

  3. An irreversible local Markov chain that preserves the six vertex model on a torus

    Borodin, Alexei; Bufetov, Alexey
    We construct an irreversible local Markov dynamics on configurations of up-right paths on a discrete two-dimensional torus, that preserves the Gibbs measures for the six vertex model. An additional feature of the dynamics is a conjecturally nontrivial drift of the height function.

  4. The high-temperature behavior for the directed polymer in dimension $1+2$

    Berger, Quentin; Lacoin, Hubert
    We investigate the high-temperature behavior of the directed polymer model in dimension $1+2$. More precisely we study the difference $\Delta\mathtt{F}(\beta)$ between the quenched and annealed free energies for small values of the inverse temperature $\beta$. This quantity is associated to localization properties of the polymer trajectories, and is related to the overlap fraction of two replicas. Adapting recent techniques developed by the authors in the context of the disordered pinning model (Berger and Lacoin, 2015), we identify the sharp asymptotic high temperature behavior ¶ \[\lim_{\beta\to0}\beta^{2}\log\Delta \mathtt{F}(\beta)=-\pi.\]

  5. Penalized maximum likelihood estimation and effective dimension

    Spokoiny, Vladimir
    This paper extends some prominent statistical results including Fisher Theorem and Wilks phenomenon to the penalized maximum likelihood estimation with a quadratic penalization. It appears that sharp expansions for the penalized MLE $\widetilde{\boldsymbol{\theta}}_{G}$ and for the penalized maximum likelihood can be obtained without involving any asymptotic arguments, the results only rely on smoothness and regularity properties of the of the considered log-likelihood function. The error of estimation is specified in terms of the effective dimension $\mathtt{p}_{G}$ of the parameter set which can be much smaller than the true parameter dimension and even allows an infinite dimensional functional parameter. In the...

  6. On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations

    Chen, Le; Kim, Kunwoo
    In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on $\mathbb{R}$ with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller’s comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at ${\pm}\infty$. These results generalize a recent work by Moreno Flores (Ann. Probab. 42 (2014) 1635–1643), who proves the strict positivity of the solution to the stochastic heat equation with the delta initial...

  7. Scaling limits for the peeling process on random maps

    Curien, Nicolas; Le Gall, Jean-François
    We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or quadrangulation (UIPQ). In particular, our results apply to the metric exploration or peeling by layers algorithm, where the discovered regions are (almost) completed balls, or hulls, centered at the root vertex. The scaling limits of the perimeter and volume of hulls can be expressed in terms of the hull process of the Brownian plane studied in our previous work. Other applications include the metric...

  8. Hammersley’s harness process: Invariant distributions and height fluctuations

    Seppäläinen, Timo; Zhai, Yun
    We study the invariant distributions of Hammersley’s serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards–Wilkinson universality). We prove this for correlated initial data subject to fast enough polynomial decay of strong mixing coefficients, including process-level tightness in the Skorohod space of space–time trajectories.

  9. Strong existence and uniqueness for degenerate SDE with Hölder drift

    Chaudru de Raynal, P. E.
    In this paper, we prove pathwise uniqueness for stochastic degenerate systems with a Hölder drift, for a Hölder exponent larger than the critical value $2/3$. This work extends to the degenerate setting the earlier results obtained by Zvonkin (Mat. Sb. (N.S.) 93(135) (1974) 129–149, 152), Veretennikov (Mat. Sb. (N.S.) 111(153) (1980) 434–452, 480), Krylov and Röckner (Probab. Theory Related Fields 131(2) (2005) 154–196) from non-degenerate to degenerate cases. The existence of a threshold for the Hölder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on...

  10. Some examples of quenched self-averaging in models with Gaussian disorder

    Chen, Wei-Kuo; Panchenko, Dmitry
    In this paper we give an elementary approach to several results of Chatterjee in (Disorder chaos and multiple valleys in spin glasses (2013) arXiv:0907.3381, Comm. Math. Phys. 337 (2015) 93–102), as well as some generalizations. First, we prove quenched disorder chaos for the bond overlap in the Edwards–Anderson type models with Gaussian disorder. The proof extends to systems at different temperatures and covers a number of other models, such as the mixed $p$-spin model, Sherrington–Kirkpatrick model with multi-dimensional spins and diluted $p$-spin model. Next, we adapt the same idea to prove quenched self-averaging of the bond magnetization for one system...

  11. The many-to-few lemma and multiple spines

    Harris, Simon C.; Roberts, Matthew I.
    We develop a simple and intuitive identity for calculating expectations of weighted $k$-fold sums over particles in branching processes, generalising the well-known many-to-one lemma.

  12. Scaling limit of multitype Galton–Watson trees with infinitely many types

    de Raphélis, Loïc
    We introduce a certain class of 2-type Galton–Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton–Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton–Watson trees with finitely many types.

  13. Higher moments of the natural parameterization for SLE curves

    Rezaei, Mohammad A.; Zhan, Dapeng
    In this paper, we will show that the higher moments of the natural parametrization of $\mathit{SLE}$ curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an $\mathit{SLE}$ curve gets near $n$ given points.

  14. A simpler proof for the dimension of the graph of the classical Weierstrass function

    Keller, Gerhard
    Let $W_{\lambda,b}(x)=\sum_{n=0}^{\infty}\lambda^{n}g(b^{n}x)$ where $b\geq2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass function). Building on work by Ledrappier (In Symbolic Dynamics and Its Applications (1992) 285–293), Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) and Tsujii (Nonlinearity 14 (2001) 1011–1027), we provide an elementary proof that the Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda }{\log b}$ for all $\lambda\in(\lambda_{b},1)$ with a suitable $\lambda_{b}<1$. This reproduces results by Barański, Bárány and Romanowska (Adv. Math. 265 (2014) 32–59) without using the dimension theory for hyperbolic measures of Ledrappier and Young (Ann. of Math. (2) 122 (1985) 540–574; Comm. Math. Phys. 117 (1988)...

  15. Height fluctuations in interacting dimers

    Giuliani, Alessandro; Mastropietro, Vieri; Toninelli, Fabio Lucio
    We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of $\mathbb{Z}^{2}$, i.e. subsets of edges such that each vertex is covered exactly once (“close-packing” condition). Dimer configurations are in bijection with discrete height functions, defined on faces $\boldsymbol{\xi}$ of $\mathbb{Z}^{2}$. The non-interacting model is “integrable” and solvable via Kasteleyn theory; it is known that all the moments of the height difference $h_{\boldsymbol{\xi} }-h_{\boldsymbol{\eta} }$ converge to those of the massless Gaussian Free Field (GFF), asymptotically as $|\boldsymbol{\xi} -\boldsymbol{\eta} |\to\infty$. We prove that the same holds for small non-zero interactions, as was conjectured...

  16. Thick points for Gaussian free fields with different cut-offs

    Cipriani, Alessandra; Hazra, Rajat Subhra
    Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.

  17. Random walks on quasi one dimensional lattices: Large deviations and fluctuation theorems

    Faggionato, Alessandra; Silvestri, Vittoria
    Several stochastic processes modeling molecular motors on a linear track are given by random walks (not necessarily Markovian) on quasi 1d lattices and share a common regenerative structure. Analyzing this abstract common structure, we derive information on the large fluctuations of the stochastic process by proving large deviation principles for the first-passage times and for the position. We focus our attention on the Gallavotti–Cohen-type symmetry of the position rate function (fluctuation theorem), showing its equivalence with the independence of suitable random variables. In the special case of Markov random walks, we show that this symmetry is universal only inside a...

  18. Pointwise upper estimates for transition probabilities of continuous time random walks on graphs

    Chen, Xinxing
    Let $X$ be a continuous time random walk on a weighted graph. Given the on-diagonal upper bounds of transition probabilities at two vertices $x_{1}$ and $x_{2}$, we obtain Gaussian upper estimates for the off-diagonal transition probability $\mathbb{P}_{x_{1}}(X_{t}=x_{2})$ in terms of an adapted metric introduced by Davies.

  19. Geodesic PCA in the Wasserstein space by convex PCA

    Bigot, Jérémie; Gouet, Raúl; Klein, Thierry; López, Alfredo
    We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable...

  20. Asymptotics and concentration bounds for bilinear forms of spectral projectors of sample covariance

    Koltchinskii, Vladimir; Lounici, Karim
    Let $X,X_{1},\ldots,X_{n}$ be i.i.d. Gaussian random variables with zero mean and covariance operator $\Sigma=\mathbb{E}(X\otimes X)$ taking values in a separable Hilbert space $\mathbb{H}$. Let ¶ \[\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma\|_{\infty}}\] be the effective rank of $\Sigma$, $\operatorname{tr}(\Sigma)$ being the trace of $\Sigma $ and $\|\Sigma\|_{\infty}$ being its operator norm. Let ¶ \[\hat{\Sigma}_{n}:=n^{-1}\sum_{j=1}^{n}(X_{j}\otimes X_{j})\] be the sample (empirical) covariance operator based on $(X_{1},\ldots,X_{n})$. The paper deals with a problem of estimation of spectral projectors of the covariance operator $\Sigma $ by their empirical counterparts, the spectral projectors of $\hat{\Sigma}_{n}$ (empirical spectral projectors). The focus is on the problems where both the sample size $n$ and the effective rank $\mathbf{r}(\Sigma)$...

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