-
作者：Cass, Thomas; Hairer, Martin; Litterer, Christian; Tindel, Samy
作者单位：Imperial College London; University of Warwick; Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Universite de Lorraine
摘要：We consider stochastic differential equations of the form dY(t) = V(Y-t)dX(t) + V-0(Y-t)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V-0 and V = (V-1,...,V-d) satisfy Hormander's bracket condition, we demonstrate that Y-t admits a smooth density for any t is an element of (0, T], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the ...
-
作者：Eldan, Ronen
作者单位：Microsoft
摘要：We consider an analogous version of the diffusion-limited aggregation model defined on the hyperbolic plane. We prove that almost surely the aggregate viewed at time infinity will have a positive density.
-
作者：Baur, Erich; Bolthausen, Erwin
作者单位：University of Zurich
摘要：We study exit laws from large balls in Z(d), d >= 3, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary se...
-
作者：Ding, Jian; Eldan, Ronen; Zhai, Alex
作者单位：University of Chicago; Stanford University
摘要：We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns the phenomenon of multiple peaks. Consider a centered Gaussian field whose sites have variance at most 1, and let rho be the standard deviation of its supremum. A theorem of Chatterjee states that when such a Gaussian field is superconcentrated (i.e., rho << 1), it typically attains values near its maximum on multiple almost-orthogonal sites and is said to exhibit multiple peaks. We improve h...
-
作者：Benjamini, Itai; Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel
作者单位：Weizmann Institute of Science; University of Geneva; Ben-Gurion University of the Negev
摘要：We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on Z(d). We prove that the vector space of harmonic functions growing at most linearly is (d + 1)-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides boun...
-
作者：Cerf, Raphael
作者单位：Universite Paris Saclay
摘要：We consider the standard site percolation model on the d-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to 1/2. We improve slightly this lower bound in any dimension d >= 2. Next, starting only with the hypothesis that theta(p) > 0, without using the slab technology, we derive a quantitative estimate establishing long-rang...
-
作者：Pinelis, Iosif
作者单位：Michigan Technological University
摘要：It is shown that, for any given p >= 5, A > 0 and B > 0, the exact upper bound on is an element of vertical bar Sigma X-i vertical bar(p) over all independent zero-mean random variables (r.v.'s) X-1, ..., X-n such that Sigma is an element of X-i(2) = B and Sigma is an element of vertical bar X-i vertical bar(p) = A equals c(p) is an element of vertical bar Pi(lambda) - lambda vertical bar(p), where (lambda, c) is an element of (0, infinity)(2) is the unique solution to the system of equations ...
-
作者：Roeckner, Michael; Zhu, Rongchan; Zhu, Xiangchan
作者单位：University of Bielefeld; Beijing Institute of Technology; Beijing Jiaotong University
摘要：In this paper, we study the 2D stochastic quasi-geostrophic equation on T-2 for general parameter alpha epsilon (0, 1) and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all alpha epsilon (0, 1). In the subcritical case alpha > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven...
-
作者：Lyons, Russell; Peres, Yuval
作者单位：Indiana University System; Indiana University Bloomington; Microsoft
摘要：We introduce a technique using nonbacktracking random walk for estimating the spectral radius of simple random walk. This technique relates the density of nontrivial cycles in simple random walk to that in nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular r...
-
作者：Lawler, Gregory F.; Rezaei, Mohammad A.
作者单位：University of Chicago
摘要：We prove the existence and nontriviality of the d-dimensional 4 Minkowski content for the Schramm-Loewner evolution (SLE kappa) with kappa < 8 and d = 1 + kappa/8 We show that this is a multiple of the natural parameterization.
