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作者：Bai, Shuyang; Taqqu, Murad S.
作者单位：Boston University
摘要：The generalized Rosenblatt process is obtained by replacing the single critical exponent characterizing the Rosenblatt process by two different exponents living in the interior of a triangular region. What happens to that generalized Rosenblatt process as these critical exponents approach the boundaries of the triangle? We show by two different methods that on each of the two symmetric boundaries, the limit is non-Gaussian. On the third boundary, the limit is Brownian motion. The rates of conv...
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作者：Addario-Berry, Louigi; Penington, Sarah
作者单位：McGill University; University of Oxford
摘要：We augment standard branching Brownian motion by adding a competitive interaction between nearby particles. Informally, when particles are in competition, the local resources are insufficient to cover the energetic cost of motion, so the particles' masses decay. In standard BBM, we may define the front displacement at time t as the greatest distance of a particle from the origin. For the model with masses, it makes sense to instead define the front displacement as the distance at which the loc...
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作者：Cass, Thomas; Ogrodnik, Marcel
作者单位：Imperial College London
摘要：The accumulated local p-variation functional [Ann. Probab. 41 (213) 3026-3050] arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local p-variation functional typically decay much faster than the tails of classical p-variation. This observation has been decisive, for example, for proble...
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作者：Corwin, Ivan; Tsai, Li-Cheng
作者单位：Columbia University; Sorbonne Universite; Stanford University
摘要：We prove that under a particular weak scaling, the 4-parameter interacting particle system introduced by Corwin and Petrov [Comm. Math. Phys. 343 (2016) 651-700] converges to the Kardar-Parisi-Zhang (KPZ) equation. This expands the relatively small number of systems for which weak universality of the KPZ equation has been demonstrated.
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作者：Adler, Robert J.; Samorodnitsky, Gennady
作者单位：Technion Israel Institute of Technology; Cornell University
摘要：How likely is the high level of a continuous Gaussian random field on an Euclidean space to have a hole of a certain dimension and depth? Questions of this type are difficult, but in this paper we make progress on questions shedding new light in existence of holes. How likely is the field to be above a high level on one compact set (e.g., a sphere) and to be below a fraction of that level on some other compact set, for example, at the center of the corresponding ball? How likely is the field t...
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作者：Burdzy, Krzysztof; Chen, Zhen-Qing; Marshall, Donald; Ramanan, Kavita
作者单位：University of Washington; University of Washington Seattle; Brown University
摘要：We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including nonsmooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair...
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作者：Hoffman, Christopher; Johnson, Tobias; Junge, Matthew
作者单位：University of Washington; University of Washington Seattle; University of Southern California
摘要：The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite d-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d = 2 and transient for d >= 5. Simulations suggest strong recurrence for d = 2, weak recurrence for d = 3, and transience for d >= 4. Additionally, we prove a 0-1 law for all d-ary trees, and we exhibit a graph on which a 0-...
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作者：Zhang, Xicheng
作者单位：Wuhan University
摘要：Consider the following nonlocal integro-differential operator: for alpha is an element of (0, 2): L-sigma,b((alpha)) f(x) := p.v. integral(|z| < delta) f(x + sigma(x)z) - f(x)/|z|(d+alpha) dz + b(x) . del f(x) + Lf(x), where sigma :R-d -> R-d circle times R-d and b : R-d -> R-d are smooth functions and have bounded partial derivatives of all orders greater than 1, delta is a small positive number, p.v. stands for the Cauchy principal value and L is a bounded linear operator in Sobolev spaces. ...
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作者：Addario-Berry, Louigi; Albenque, Marie
作者单位：McGill University; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS)
摘要：Let M-n be a simple triangulation of the sphere S-2, drawn uniformly at random from all such triangulations with n vertices. Endow M-n with the uniform probability measure on its vertices. After rescaling graph distance by (3/(4n))(1/4), the resulting random measured metric space converges in distribution, in the Gromov-Hausdorff-Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of M-n. Under this labelling, distances to a d...
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作者：Jung, Paul; Owada, Takashi; Samorodnitsky, Gennady
作者单位：Korea Advanced Institute of Science & Technology (KAIST); Purdue University System; Purdue University; Cornell University; Cornell University
摘要：We prove a functional central limit theorem for partial sums of symmetric stationary long-range dependent heavy tailed infinitely divisible processes. The limiting stable process is particularly interesting due to its long memory which is quantified by a Mittag Leffler process induced by an associated Harris chain, at the discrete-time level. Previous results in Owada and Samorodnitsky [Ann. Probab. 43 (2015) 240-285] dealt with positive dependence in the increment process, whereas this paper ...
