-
作者：Champagnat, Nicolas; Jabin, Pierre-Emmanuel
作者单位：Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; University System of Maryland; University of Maryland College Park; University System of Maryland; University of Maryland College Park
摘要：We study strong existence and pathwise uniqueness for stochastic differential equations in 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...
-
作者：Wu, Hao
作者单位：Tsinghua University
摘要：We derive the alternating arm exponents of the critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions (circle minus circle plus), (circle minus free) and (free free), and the alternating interior arm exponents.
-
作者：Pillai, Natesh S.; Smith, Aaron
作者单位：Harvard University; University of Ottawa
摘要：Determining the total variation mixing time of Kac's random walk on the special orthogonal group SO(n) has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not amenable to existing techniques in random matrix theory. To circumvent thi...
-
作者：Kolli, Praveen; Shkolnikov, Mykhaylo
作者单位：Carnegie Mellon University; Princeton University
摘要：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 ...
-
作者：Arapostathis, Ari; Biswas, Anup; Borkar, Vivek S.
作者单位：University of Texas System; University of Texas Austin; Indian Institute of Science Education & Research (IISER) Pune; Indian Institute of Technology System (IIT System); Indian Institute of Technology (IIT) - Bombay
摘要：We consider a dynamical system with finitely many equilibria and perturbed by small noise, in addition to being controlled by an expensive control. The controlled process is optimal for an ergodic criterion with a running cost that consists of the sum of the control effort and a penalty function on the state space. We study the optimal stationary distribution of the controlled process as the variance of the noise becomes vanishingly small. It is shown that depending on the relative magnitudes ...
-
作者：Ding, Jian; Shen, Jianfei
作者单位：University of Chicago
摘要：For a one-dimensional simple random walk (S-t), for each time t we say a site x is a favorite site if it has the maximal local time. In this paper, we show that with probability 1 three favorite sites occurs infinitely often. Our work is inspired by Toth [Ann. Probab. 29 (2001) 484-503], and disproves a conjecture of Erdos and Revesz [In Mathematical Structure-Computational Mathematics-Mathematical Modelling 2 (1984) 152-157] and of Toth [Ann. Probab. 29 (2001) 484-503].
-
作者：Aldous, David
作者单位：University of California System; University of California Berkeley
摘要：Plant differently colored points in the plane; then let random points (Poisson rain) fall, and give each new point the color of the nearest existing point. Previous investigation and simulations strongly suggest that the colored regions converge (in some sense) to a random partition of the plane. We prove a weak version of this, showing that normalized empirical measures converge to Lebesgue measures on a random partition into measurable sets. Topological properties remain an open problem. In ...
-
作者：Binotto, Giulia; Nourdin, Ivan; Nualart, David
作者单位：University of Barcelona; University of Luxembourg; University of Kansas
摘要：The aim of this paper is to establish the weak convergence, in the topology of the Skorohod space, of the nu-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value H = (4l + 2)(-1), where l = l (.) = 1 is the largest natural number satisfying integral(1)(0) alpha(2j) nu(d alpha) = 1/2j+1 for all j = 0,..., l - 1. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic...
-
作者：Zhu, Rongchan; Zhu, Xiangchan
作者单位：Beijing Institute of Technology; Beijing Jiaotong University; University of Bielefeld
摘要：We study the lattice approximations to the dynamical Phi(4)(3) model by paracontrolled distributions proposed in [Forum Math. Pi 3 (2015) e6]. We prove that the solutions to the lattice systems converge to the solution to the dynamical Phi(4)(3) model in probability, locally uniformly in time. Since the dynamical Phi(4)(3) model is not well defined in the classical sense and renormalisation has to be performed in order to define the nonlinear term, a corresponding suitable drift term is added ...
-
作者：Timar, Adam
作者单位：Hungarian Academy of Sciences; HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
摘要：We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves the problem for the FUSF of Caylay graphs completely. These answer questions by Benjamini, Lyons, Peres...
