Slit Holomorphic Stochastic Flows and Gaussian Free Field

It was realized recently that the chordal, radial and dipolar Schramm–Löwner evolution (SLEs) are special cases of a general slit holomorphic stochastic flow. We characterize those slit holomorphic stochastic flows which generate level lines of the Gaussian free field. In particular, we describe the modifications of the Gaussian free field (GFF) corresponding to the chordal and dipolar SLE with drifts. Finally, we develop a version of conformal field theory based on the background charge and Dirichlet boundary condition modifications of GFF and present martingale-observables for these types of SLEs.     

A Pricing Model for American Options with Gaussian Interest Rates

In this paper we introduce a new methodology to price American put options under stochastic interest rates. We derive an analytic approximation that can be evaluated very fast and is fairly accurate. The method uses the so-called forward risk adjusted measure to derive analytic prices. We show that for American puts the correlation between the stock price and the interest rate has different influences on European option values and early exercise premiums.     

Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

We consider the discrete two‐dimensional Gaussian free field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.© 2015 Wiley Periodicals, Inc.     

跳扩散和随机利率模型下的欧式双向期权定价

假定股票价格的跳跃过程为一类特殊的更新跳过程,即事件发生时间间隔为相互独立且同服从Gamma分布的随机变量序列.利用鞅定价方法,用较简单的数学推导得到了在随机利率情形下跳扩散模型的欧式双向期权定价公式.     

Toward an Infinite-Component Field Theory with a Double Symmetry: Free Fields

We begin a study of possibilities of describing hadrons in terms of monolocal fields that transform under proper Lorentz group representations that are infinite direct sums of finite-dimensional irreducible representations. The additional requirement that the free-field Lagrangians be invariant under the secondary symmetry transformations generated by the polar or the axial four-vector representation of the orthochronous Lorentz group provides an effective mechanism for selecting the class of representations considered and eliminating an infinite number of arbitrary parameters allowed by the relativistic invariance of the Lagrangians.     

Evolution in a Gaussian Random Field

We consider an evolution process in a Gaussian random field V(q) with the mean ‹V(q)› = 0 and the correlation function W(|q – q|) ‹V(q)V(q)›, where q ^d and d is the dimension of the Euclidean space ^d . For the value ‹G(q,t;q ₀)›, t > 0, of the Green's function of the evolution equation averaged over all realizations of the random field, we use the Feynman–Kac formula to establish an integral equation that is invariant with respect to a continuous renormalization group. This invariance property allows using the renormalization group method to find an asymptotic expression for ‹G(q,t;q ₀)› as |q – q ₀| and t .     

General β‐Jacobi Corners Process and the Gaussian Free Field

We prove that the two‐dimensional Gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general β‐Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to the Heckman‐Opdam hypergeometric functions (of type A). We also discuss the β → ∞ limit. © 2015 Wiley Periodicals, Inc.     

Characterization of the Newtonian Free Particle System in m\geqslant 2 Dependent Variables

We treat the problem of linearizability of a system of second order ordinary differential equations. The criterion we provide has applications to nonlinear Newtonian mechanics, especially in three-dimensional space. Let or , let , let , let and let

Steep Points of Gaussian Free Fields in Any Dimension

Journal of Theoretical Probability - This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets to a more general class of exceptional sets of a...     

A Degenerate Stefan Problem with Two Free Boundaries

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Free 3-Generated Lattices with Two Semi-Normal Generators

We consider a lattice generated by three elements, two of which are semi-normal. We have proved it is infinite and have also found an additional condition for the lattice to be modular. As a result, it is proved that the sublattice generated in the subgroup lattice by two normal subgroups and any subgroup is modular.     

对具有定常故障率和任意系统修复率的可修复系统的猜测分析

研究了具有定常人为故障率 (human error rates)和通常故障率 (common-cause failure rates) ,修复时间任意分布的可修复系统的数学模型 .首先将此系统转换为 Banach空间下的 Volterra积分方程 ,得到了系统非负解的存在性和唯一性结果 .     

Ideal Free Distribution in Two Patches

In this paper, the formats of Julia sets for a class of nonlinear complex dynamic systems with variable coefficients were studied under certain conditions. For the complex dynamic systems in piecewise cases, we proposed some methods to control the forms of their Julia sets and stable domains analytically. What’s more, we illustrated that our methods worked well by computational simulations. Our work provides a better understanding about how to control the Julia sets of certain complex dynamic systems.     

Painlevé XXXIV Asymptotics of Orthogonal Polynomials for the Gaussian Weight with a Jump at the Edge

We study the uniform asymptotics of the polynomials orthogonal with respect to analytic weights with jump discontinuities on the real axis, and the influence of the discontinuities on the asymptotic behavior of the recurrence coefficients. The Riemann–Hilbert approach, also termed the Deift–Zhou steepest descent method, is used to derive the asymptotic results. We take as an example the perturbed Gaussian weight , where θ(x) takes the value of 1 for x < 0 , and a nonnegative complex constant ω elsewhere, and as . That is, the jump occurs at the edge of the support of the equilibrium measure. The derivation is carried out in the sense of a double scaling limit, namely, and . A crucial local parametrix at the edge point where the jump occurs is constructed out of a special solution of the Painlevé XXXIV equation. As a main result, we prove asymptotic formulas of the recurrence coefficients in terms of a special Painlevé XXXIV transcendent under the double scaling limit. The special thirty‐fourth Painlevé transcendent is shown free of poles on the real axis. A consistency check is made with the reduced case when ω= 1 , namely the Gaussian weight: the polynomials in this case are the classical Hermite polynomials. A comparison is also made of the asymptotic results for the recurrence coefficients between the case when the jump happens at the edge and the case with jump inside the support of the equilibrium measure. The comparison provides a formal asymptotic approximation of the Painlevé XXXIV transcendent at positive infinity.     

Diffusion-Approximations for Navier-Stokes Equation in Space-Time Gaussian Velocity Field

Abstract

By approximation methods, the existence of solutions for the Navier-Stokes equation with rapidly oscillating drift term for dimention 2 or 3 is proved. Also by the martingale method, stochastic Navier-Stokes equation is considered.     

Hamiltonian Dynamics of a Particle Interacting with a Wave Field

We study the Hamiltonian equations of motion of a heavy tracer particle interacting with a dense weakly interacting Bose-Einstein condensate in the classical (mean-field) limit. Solutions describing ballistic subsonic motion of the particle through the condensate are constructed. We establish asymptotic stability of ballistic subsonic motion.     

Bayesian normal analysis with an inverse Gaussian prior

Summary This paper considers the Bayesian analysis of normal distribution when its variance has an inverse Gaussian prior density. The result is also generelized in a theorem that is subsequently presented.