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作者：Angel, Omer; Kolesnik, Brett; Miermont, Gregory
作者单位：University of British Columbia; Ecole Normale Superieure de Lyon (ENS de LYON); Institut Universitaire de France
摘要：The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure. Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any ...
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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：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：Flint, Ian; Torrisi, Giovanni Luca
作者单位：Nanyang Technological University; Consiglio Nazionale delle Ricerche (CNR); Istituto per le Applicazioni del Calcolo Mauro Picone (IAC-CNR)
摘要：We provide a Clark-Ocone formula for square-integrable functionals of a general temporal point process satisfying only a mild moment condition, generalizing known results on the Poisson space. Some classical applications are given, namely a deviation bound and the construction of a hedging portfolio in a pure-jump market model. As a more modern application, we provide a bound on the total variation distance between two temporal point processes, improving in some sense a recent result in this d...
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作者：Freidlin, M.; Koralov, L.; Wentzell, A.
作者单位：University System of Maryland; University of Maryland College Park; Tulane University
摘要：We consider processes that coincide with a given diffusion process outside a finite collection of domains. In each of the domains, there is, additionally, a large drift directed towards the interior of the domain. We describe the limiting behavior of the processes as the magnitude of the drift tends to infinity, and thus the domains become trapping, with the time to exit the domains being exponentially large. In particular, in exponential time scales, metastable distributions between the trapp...
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作者：Holroyd, Alexander E.; Schramm, Oded; Wilson, David B.
作者单位：Microsoft
摘要：Suppose that the vertices of Z(d) are assigned random colors via a finitary factor of independent identically distributed (i.i.d.) vertex-labels. That is, the color of vertex v is determined by a rule that examines the labels within a finite (but random and perhaps unbounded) distance R of v, and the same rule applies at all vertices. We investigate the tail behavior of R if the coloring is required to be proper (i.e., if adjacent vertices must receive different colors). When d >= 2, the optim...
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作者：Konarovskyi, Vitalii
作者单位：Max Planck Society; Ministry of Education & Science of Ukraine; Yuri Fedkovych Chernivtsi National University
摘要：We construct a modified Arratia flow with mass and energy conservation. We suppose that particles have a mass obeying the conservation law, and their diffusion is inversely proportional to the mass. Our main result asserts that such a system exists under the assumption of the uniform mass distribution on an interval at the starting moment. We introduce a stochastic integral with respect to such a flow and obtain the total local time as the density of the occupation measure for all particles.
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作者：Beiglbock, Mathias; Nutz, Marcel; Touzi, Nizar
作者单位：Technische Universitat Wien; Columbia University; Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique
摘要：We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general princip...
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作者：Epstein, Charles L.; Pop, Camelia A.
作者单位：University of Pennsylvania; University of Minnesota System; University of Minnesota Twin Cities
摘要：We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in population genetics, the so-called generalized Kimura diffusion operators Epstein and Mazzeo [SIAM J. Math. Anal. 42 (2010) 568-608; Degenerate Diffusion Operators Arising in Population Biology (2013) Princeton University Press; Applied Mathematics Research Express (2016)]. Our main results are a stochastic representation of weak solutions to a degenerate parabolic equation with sin...
