Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

Blow-up of semilinear pde's at the critical dimension. A probabilistic approach

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation in the critical dimension . By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as . In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of -Laplacians with possibly different parameters .

     

Large deviations: An introduction to 2007 Abel prize

2007 Abel prize has been awarded to S R S Varadhan for creating a unified theory of large deviations. We attempt to give a flavour of this branch of probability theory, highlighting the role of Varadhan.     

量子引力的曲率两点真空相关

以平坦的Minkowski时空为背景，得到了任意坐标系和谐和坐标系中，n维GR引力和高导数引力的引力子自由传播子，求得了四种可能的曲率两点真空相关函数的首项.用微扰计算证明了曲率的两点真空相关函数在GR引力中为零，而在高导数引力中不为零.讨论了高导数引力与GR的引力子传播子、曲率相关函数的关系. 关键词： GR 高导数引力 引力子自由传播子 曲率真空相关函数 平移传播子     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.     

Nonautonomous Kato classes of measures and Feynman-Kac propagators

The behavior of the Feynman-Kac propagator corresponding to a time-dependent measure on is studied. We prove the boundedness of the propagator in various function spaces on , and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation.

     

Brownian intersection local times: Exponential moments and law of large masses

Consider independent Brownian motions in , each running up to its first exit time from an open domain , and their intersection local time as a measure on . We give a sharp criterion for the finiteness of exponential moments,

where are nonnegative, bounded functions with compact support in . We also derive a law of large numbers for intersection local time conditioned to have large total mass.

     

Pricing model of interest rate swap with a bilateral default risk

Under the foundation of Duffie & Huang (1996) [7], this paper integrates the reduced form model and the structure model for a default risk measure, giving rise to a new pricing model of interest rate swap with a bilateral default risk. This model avoids the shortcomings of ignoring the dynamic movements of the firm’s assets of the reduced form model but adds only a little complexity and simplifies the pricing formula significantly when compared with Li (1998) [10]. With the help of the Crank-Nicholson difference method, we give the numerical solutions of the new model to study the default risk effects on the swap rate. We find that for a one year interest rate swap with the coupon paid per quarter, the variance of the default fixed rate payer decreases from 0.1 to 0.01 only causing about a 1.35%’s increase in the swap rate. This is consistent with previous results.     

Probabilistic Solutions for a Class of Path-Dependent Hamilton-Jacobi-Bellman Equations

This article shows that the solution of a backward stochastic differential equation under G-expectation provides a probabilistic interpretation for the viscosity solution of a type of path-dependent Hamilton-Jacobi-Bellman equation. Particularly, a G-martingale can be considered as a nonlinear path-dependent partial differential equation (PDE). We also show that certain class of path-dependent PDEs can be transformed into classical multiple state-dependent PDEs. As an application, the path-dependent uncertain volatility model can be described directly by path-dependent Black-Scholes-Barrenblett equations.