Recursos de colección
Bárány, Imre; Vu, Van
Dupuis, Paul; Ishii, Hitoshi
Denisov, D.; Dieker, A. B.; Shneer, V.
For a given one-dimensional random walk {S_{n}} with a subexponential step-size distribution, we present a unifying theory to study the sequences {x_{n}} for which $\mathsf{P}\{S_{n}>x\}\sim n\mathsf{P}\{S_{1}>x\}$ as n→∞ uniformly for x≥x_{n}. We also investigate the stronger “local” analogue, $\mathsf{P}\{S_{n}\in(x,x+T]\}\sim n\mathsf{P}\{S_{1}\in(x,x+T]\}$ . Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory.
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When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.
Kim, Panki; Song, Renming
Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L+μ⋅∇−ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ=(μ^{1}, …, μ^{d}) is such that each component μ^{i}, i=1, …, d, is a signed measure belonging to the Kato class K_{d,1} and ν is a (nonnegative) measure belonging to the Kato class K_{d,2}. We show that...
van den Berg, J.
One of the most well-known classical results for site percolation on the square lattice is the equation p_{c}+p_{c}^{*}=1. In words, this equation means that for all values ≠ p_{c} of the parameter p, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed “star” cluster. This result is closely related to the percolation transition being sharp: below p_{c}, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in...
Kesten, Harry; Sidoravicius, Vladas
We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time t, we have an “aggregate” consisting of ℤ∩[0, R(t)] [with R(t) a positive integer]. We also have N(i, t) particles at i, i>R(t). All these particles perform independent continuous-time symmetric simple random walks until the first time t'>t at which some particle tries to jump from R(t)+1 to R(t). The aggregate is then increased to the integers in [0, R(t')]=[0, R(t)+1] [so that R(t')=R(t)+1] and all particles which were at R(t)+1 at time t' − are removed from the system. The problem is to determine how fast R(t)...
Haas, Bénédicte; Miermont, Grégory; Pitman, Jim; Winkel, Matthias
Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s beta-splitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
Bernyk, Violetta; Dalang, Robert C.; Peskir, Goran
Let X=(X_{t})_{t≥0} be a stable Lévy process of index α∈(1, 2) with no negative jumps and let S_{t}=sup_{0≤s≤t} X_{s} denote its running supremum for t>0. We show that the density function f_{t} of S_{t} can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann–Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for f_{t}. Recalling the familiar relation between S_{t} and the first entry time τ_{x} of X into [x, ∞), this further translates into an...
Nolin, Pierre
We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these interfaces. These results build on the recent results for critical percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the scaling ideas developed by Kesten.
Pete, Gábor
We consider a four-vertex model introduced by Bálint Tóth: a dependent bond percolation model on ℤ^{2} in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability ℙ(diameter of the cycle of the origin >n)≈n^{−γ} and the expectation $\mathbb{E}$ (length of a typical cycle with diameter n)≈n^{δ}, with $\gamma=(5-\sqrt{17})/4=0.219\ldots$ and $\delta=(\sqrt{17}+1)/4=1.28\ldots$ . The value of δ comes from...
Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo
We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on ℤ^{d}. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.
Bezerra, Sérgio; Tindel, Samy; Viens, Frederi
This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field W on ℝ_{+}×ℝ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any α<3/5.
Jain, Naresh; Krylov, Nicolai
Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with L_{2} semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.
Chatterjee, Sourav
We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of “local” itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed...
Flury, Markus
We investigate the free energy of nearest-neighbor random walks on ℤ^{d}, endowed with a drift along the first axis and evolving in a nonnegative random potential given by i.i.d. random variables. Our main result concerns the ballistic regime in dimensions d≥4, at which we show that quenched and annealed Lyapunov exponents are equal as soon as the strength of the potential is small enough.
Karatzas, Ioannis; Zamfirescu, Ingrid-Mona
We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities.
Zhan, Dapeng
We prove that the chordal SLE_{κ} trace is reversible for κ∈(0, 4].
Helton, J. William; Lasserre, Jean B.; Putinar, Mihai
We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure μ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with μ.
Beffara, Vincent
Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ≥0. We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2, 1+κ/8).
Budhiraja, Amarjit; Dupuis, Paul; Maroulas, Vasileios
The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.