Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equations are a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers or by positive reals, these corresponding to the discrete or the continuous form of the equations. For dimension d≥3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a method similar to that used to derive the discrete form of the equations in [A. Hammond, F. Rezakhanlou, The kinetic limit of a system of coagulating Brownian particles, Arch. Ration. Mech. Anal. 185 (2007) 1–67]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of the cited work. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.     

A general multidimensional Hermite-Hadamard type inequality

Using a stochastic approach, we establish a multidimensional version of the classical Hermite-Hadamard inequalities which holds for convex functions on general convex bodies. The result is closely related to the Dirichlet problem.     

The Li-Yau-Hamilton estimate and the Yang-Mills heat equation on manifolds with boundary

The paper pursues two connected goals. Firstly, we establish the Li-Yau-Hamilton estimate for the heat equation on a manifold M with nonempty boundary. Results of this kind are typically used to prove monotonicity formulas related to geometric flows. Secondly, we establish bounds for a solution ∇(t) of the Yang-Mills heat equation in a vector bundle over M. The Li-Yau-Hamilton estimate is utilized in the proofs. Our results imply that the curvature of ∇(t) does not blow up if the dimension of M is less than 4 or if the initial energy of ∇(t) is sufficiently small.     

一类随机利率下的变额寿险模型研究

本文对随机利率采用在原点反射的布朗运动以及负二项分布建模，具体以即时给付的综合人寿保险模型为研究对象，对寿险理论中的保费，年金以及责任准备金进行研究，并给出相应的表达式。     

Noise perturbed generalized Mandelbrot sets

Adopting the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, in this paper on the structure characteristics and the discontinuity evolution law of the additive noise perturbed generalized Mandelbrot sets (M-sets) was studied. On the influence of stochastic perturbed parameters of the structure of generalized M-sets was analyzed. The physical meaning of the additive noise perturbed generalized M-sets was expounded.     

Flux and Lateral Conditions for Symmetric Markov Processes

The purpose of this paper is to give an affirmative answer at infinitesimal generator level to the 40 years old Feller’s boundary problem for symmetric Markov processes with general quasi-closed boundaries. For this, we introduce a new notion of flux functional, which can be intrinsically defined via the minimal process X ⁰ in the interior. We then use it to characterize the L ²-infinitesimal generator of a symmetric process that extends X ⁰. Special attention is paid to the case when the boundary consists of countable many points possessing no accumulation points. Research of Masatoshi Fukushima was supported by Grand-in-Aid for Scientific Research of MEXT No.19540125.     

Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) generalizes fBm by letting the local Hölder exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. The aim of this work is twofold: first, we prove that an mBm may be approximated in law by a sequence of “tangent” fBms. Second, using this approximation, we show how to construct stochastic integrals w.r.t. mBm by “transporting” corresponding integrals w.r.t. fBm. We illustrate our method on examples such as the Wick–Itô, Skorohod and pathwise integrals.     

Elements related to the largest complete excursion of a reflected BM stopped at a fixed time. Application to local score

We calculate the density function of $(U^1\left(t\right),{\theta }^1\left(t\right))$, where $U^1\left(t\right)$ is the maximum over $[0,g\left(t\right)]$ of a reflected Brownian motion $U$, where $g\left(t\right)$ stands for the last zero of $U$ before $t$, ${\theta }^1\left(t\right)=f^1\left(t\right)-g^1\left(t\right)$, $f^1\left(t\right)$ is the hitting time of the level $U^1\left(t\right)$, and $g^1\left(t\right)$ is the left-hand point of the interval straddling $f^1\left(t\right)$. We also calculate explicitly the marginal density functions of $U^1\left(t\right)$ and ${\theta }^1\left(t\right)$. Let $U_n^1$ and ${\theta }_n^1$ be the analogs of $U^1\left(t\right)$ and ${\theta }^1\left(t\right)$ respectively where the underlying process $\left(U_n\right)$ is the Lindley process, i.e. the difference between a centered real random walk and its minimum. We prove that $(\frac{U_n^1}{\sqrt{n}},\frac{{\theta }_n^1}{n})$ converges weakly to $(U^1\left(1\right),{\theta }^1\left(1\right))$ as $n\to \infty$.     

Deviation inequalities and the law of iterated logarithm on the path space over a loop group

A law of iterated logarithm (LIL) in small time and an asymptotic estimate of modulus of continuity are proved for Brownian motion on the loop group ?(G) over a compact connected Lie group G. Upper bounds are obtained via infinite-dimensional deviation inequalities for functionals on the path space ?(?(G)) on ?(G), such as the supremum of Brownian motion on ?(G), which are proved from the Clark–Ocone formula on ?(?(G)). The lower bounds rely on analog finite-dimensional results that are proved separately on Riemannian path space.