Reflecting Brownian Motion and the Gauss–Bonnet–Chern Theorem

We use reflecting Brownian motion(RBM)to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary.The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.

Stochastic formulation of particle kinetics in wall-bounded two-phase flows

This paper presents a generalized framework of stochastic modeling for particle kinetics in wall-bounded flow.We modified a reflected Brownian motion process and straightforwardly obtained a Kramers equation for particle probability density function(PDF).After the wall effects were accounted for as a drift from zero in the mean displacement and suppression in the diffusivity of a particle,an analytical solution was worked out for PDF.Three distinguishable mechanisms were identified to affect the profile of particle probability distribution:external forces,turbophoresis effect,and wall-drift effect.The proposed formulation covers the Huang et al.(2009)model of a wall that produces electrostatic repulsion force and van der Waals force,as well as Monte-Carlo solutions for the Peter and Barenbrug(2002)model under a variety of relaxation times.Moreover,it successfully reproduces the two patterns of particle concentration profiles observed in experiments of sediment-laden open-channel flows.The strength of the wall-drift effect was found to be connected with the interaction frequency between particle and wall.Further exploration of the relationship among flow turbulence,particle inertia,and particle concentration is worthwhile.

Backward stochastic differential equations with rank-based data

In this paper, we investigate Markovian backward stochastic differential equations(BSDEs) with the generator and the terminal value that depend on the solutions of stochastic differential equations with rankbased drift coefficients. We study regularity properties of the solutions of this kind of BSDEs and establish their connection with semi-linear backward parabolic partial differential equations in simplex with Neumann boundary condition. As an application, we study the European option pricing problem with capital size based stock prices.

Heavy Traffic Limit Theorems for a Queue with Poisson ON/OFF Long-range Dependent Sources and General Service Time Distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes （hence service times） can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out （FIFO） and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions （FBMs） or certain type of long- range dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little＇s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.

Diffusion Approximation of a Multitype Re-entrant Line under Smaller-buffer-first-served Policy

This paper studies a multitype re-entrant line under smaller-buffer-first-served policy, which is an extension of first-buffer-first-served re-entrant line. We prove a heavy traffic limit theorem. The key to the proof is to prove the uniform convergence of the corresponding critical fluid model.

The Hot Spots Conjecture on a Class of Domains in R^n with n≥3

In this paper, we define a class of domains in R^n. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.