Mostrando recursos 1 - 20 de 104

  1. Chaining, interpolation and convexity II: The contraction principle

    van Handel, Ramon
    The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multiscale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for...

  2. Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential

    Cannizzaro, Giuseppe; Chouk, Khalil
    We study the existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock–Varadhan martingale problem associated to such equations. The approach we exploit is the one of paracontrolled distributions introduced in (Forum Math. Pi 3 (2015) e6). As a result, we make sense of the three-dimensional polymer measure with white noise potential.

  3. Discretisations of rough stochastic PDEs

    Hairer, M.; Matetski, K.
    We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical $\Phi^{4}_{3}$ model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the $\Phi^{4}_{3}$ measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.

  4. Optimal surviving strategy for drifted Brownian motions with absorption

    Tang, Wenpin; Tsai, Li-Cheng
    We study the “Up the River” problem formulated by Aldous (2002), where a unit drift is distributed among a finite collection of Brownian particles on $\mathbb{R}_{+}$, which are annihilated once they reach the origin. Starting $K$ particles at $x=1$, we prove Aldous’ conjecture [Aldous (2002)] that the “push-the-laggard” strategy of distributing the drift asymptotically (as $K\to\infty$) maximizes the total number of surviving particles, with approximately $\frac{4}{\sqrt{\pi}}\sqrt{K}$ surviving particles. We further establish the hydrodynamic limit of the particle density, in terms of a two-phase partial differential equation (PDE) with a moving boundary, by utilizing certain integral identities and coupling techniques.

  5. Dimensions of random covering sets in Riemann manifolds

    Feng, De-Jun; Järvenpää, Esa; Järvenpää, Maarit; Suomala, Ville
    Let ${\mathbf{M}}$, ${\mathbf{N}}$ and ${\mathbf{K}}$ be $d$-dimensional Riemann manifolds. Assume that ${\mathbf{A}}:=(A_{n})_{n\in{\mathbb{N}}}$ is a sequence of Lebesgue measurable subsets of ${\mathbf{M}}$ satisfying a necessary density condition and ${\mathbf{x}}:=(x_{n})_{n\in{\mathbb{N}}}$ is a sequence of independent random variables, which are distributed on ${\mathbf{K}}$ according to a measure, which is not purely singular with respect to the Riemann volume. We give a formula for the almost sure value of the Hausdorff dimension of random covering sets ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}}):=\limsup_{n\to\infty}A_{n}(x_{n})\subset{\mathbf{N}}$. Here, $A_{n}(x_{n})$ is a diffeomorphic image of $A_{n}$ depending on $x_{n}$. We also verify that the packing dimensions of ${\mathbf{E}}({\mathbf{x}},{\mathbf{A}})$ equal $d$ almost surely.

  6. Strong solutions to stochastic differential equations with rough coefficients

    Champagnat, Nicolas; Jabin, Pierre-Emmanuel
    We study strong existence and pathwise uniqueness for stochastic differential equations in $\mathbb{R}^{d}$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and $L^{p}$ bounds for the solution of the corresponding Fokker–Planck PDE, which can be proved separately. This allows a great flexibility regarding the method employed to obtain these last bounds. Hence we are able to obtain general criteria in various cases, including the uniformly elliptic case in any dimension, the one-dimensional case and the...

  7. A variational approach to dissipative SPDEs with singular drift

    Marinelli, Carlo; Scarpa, Luca
    We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which neither continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation,...

  8. A Gaussian small deviation inequality for convex functions

    Paouris, Grigoris; Valettas, Petros
    Let $Z$ be an $n$-dimensional Gaussian vector and let $f:\mathbb{R}^{n}\to \mathbb{R}$ be a convex function. We prove that ¶ \[\mathbb{P}(f(Z)\leq \mathbb{E}f(Z)-t\sqrt{\operatorname{Var}f(Z)})\leq\exp (-ct^{2}),\] for all $t>1$ where $c>0$ is an absolute constant. As an application we derive variance-sensitive small ball probabilities for Gaussian processes.

  9. Multivariate approximation in total variation, II: Discrete normal approximation

    Barbour, A. D.; Luczak, M. J.; Xia, A.
    The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in $\mathbb{Z}^{d}$. We illustrate the use of the method for sums of independent integer valued random vectors, and for random vectors exhibiting an exchangeable pair. We conclude with an application to random colourings of regular graphs.

  10. Multivariate approximation in total variation, I: Equilibrium distributions of Markov jump processes

    Barbour, A. D.; Luczak, M. J.; Xia, A.
    For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein–Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein’s method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimension are the immigration-death processes with Poisson distributions as...

  11. On global fluctuations for non-colliding processes

    Duits, Maurice
    We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models that share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case...

  12. Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes

    Johansson, Kurt; Lambert, Gaultier
    We study mesoscopic linear statistics for a class of determinantal point processes which interpolate between Poisson and random matrix statistics. These processes are obtained by modifying the spectrum of the correlation kernel of the Gaussian Unitary Ensemble (GUE) eigenvalue process. An example of such a system comes from considering the distribution of noncolliding Brownian motions in a cylindrical geometry, or a grand canonical ensemble of free fermions in a quadratic well at positive temperature. When the scale of the modification of the spectrum of the correlation kernel, related to the size of the cylinder or the temperature, is different from...

  13. Interlacements and the wired uniform spanning forest

    Hutchcroft, Tom
    We extend the Aldous–Broder algorithm to generate the wired uniform spanning forests (WUSFs) of infinite, transient graphs. We do this by replacing the simple random walk in the classical algorithm with Sznitman’s random interlacement process. We then apply this algorithm to study the WUSF, showing that every component of the WUSF is one-ended almost surely in any graph satisfying a certain weak anchored isoperimetric condition, that the number of ‘excessive ends’ in the WUSF is nonrandom in any graph, and also that every component of the WUSF is one-ended almost surely in any transient unimodular random rooted graph. The first...

  14. On the cycle structure of Mallows permutations

    Gladkich, Alexey; Peled, Ron
    We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi\in\mathbb{S}_{n}$ is proportional to $q^{\operatorname{inv}(\pi)}$ where $q>0$ and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$. ¶ We focus on the case that $q<1$ and show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2},n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2}\ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-2}\gg n$ then macroscopic cycles, of...

  15. Exponentially concave functions and a new information geometry

    Pal, Soumik; Wong, Ting-Kam Leonard
    A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge–Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two...

  16. SPDE limit of the global fluctuations in rank-based models

    Kolli, Praveen; Shkolnikov, Mykhaylo
    We consider systems of diffusion processes (“particles”) interacting through their ranks (also referred to as “rank-based models” in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course...

  17. First-passage percolation on Cartesian power graphs

    Martinsson, Anders
    We consider first-passage percolation on the class of “high-dimensional” graphs that can be written as an iterated Cartesian product $G\square G\square\dots\square G$ of some base graph $G$ as the number of factors tends to infinity. We propose a natural asymptotic lower bound on the first-passage time between $(v,v,\dots,v)$ and $(w,w,\dots,w)$ as $n$, the number of factors, tends to infinity, which we call the critical time $t^{*}_{G}(v,w)$. Our main result characterizes when this lower bound is sharp as $n\rightarrow\infty$. As a corollary, we are able to determine the limit of the so-called diagonal time-constant in $\mathbb{Z}^{n}$ as $n\rightarrow\infty$ for a large...

  18. Quasi-symmetries of determinantal point processes

    Bufetov, Alexander I.
    The main result of this paper is that determinantal point processes on $\mathbb{R}$ corresponding to projection operators with integrable kernels are quasi-invariant, in the continuous case, under the group of diffeomorphisms with compact support (Theorem 1.4); in the discrete case, under the group of all finite permutations of the phase space (Theorem 1.6). The Radon–Nikodym derivative is computed explicitly and is given by a regularized multiplicative functional. Theorem 1.4 applies, in particular, to the sine-process, as well as to determinantal point processes with the Bessel and the Airy kernels; Theorem 1.6 to the discrete sine-process and the Gamma kernel process....

  19. A fractional kinetic process describing the intermediate time behaviour of cellular flows

    Hairer, Martin; Iyer, Gautam; Koralov, Leonid; Novikov, Alexei; Pajor-Gyulai, Zsolt
    This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: a Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin–Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. ¶ As a consequence of our main theorem, we obtain a homogenization result for the associated advection diffusion...

  20. Free energy in the mixed $p$-spin models with vector spins

    Panchenko, Dmitry
    Using the synchronization mechanism developed in the previous work on the Potts spin glass model, we obtain the analogue of the Parisi formula for the free energy in the mixed even $p$-spin models with vector spins, which include the Sherrington–Kirkpatrick model with vector spins interacting through their scalar product. As a special case, this also establishes the sharpness of Talagrand’s upper bound for the free energy of multiple mixed $p$-spin systems coupled by constraining their overlaps.

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