Lectures:

Chen, Da-Yue :	The Metastability of the Biased Majority Vote Process. Abstract: The reduction method provides an algorithm to compute large deviation estimates of (possibly non reversible) Markov processes with exponential transition rates. It replaces the original graph minimisation equations of Freidlin and Wentzell by more tractable path minimisation problems. We apply this technique to the study of a biased majority vote process generalising the one studied in Chen. We show that this non reversible dynamics has a two well potential with a unique metastable state, and give an upper bound for its relaxation time.
Chen, Zhen-Qing :	Heat Kernel Estimate for Stable-like Processes on $d$-Sets Abstract: $d$-sets can be regarded as generalizations of fractals. In this talk, we will study stable-like processes on $d$-sets, which include reflected stable processes in Euclidean domains as a special case. More precisely, we will establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the graphs of stable-like processes will also be presented. This talk is based on a joint work with Takashi Kumagai.
Fang, Shi-Zan :	Tangent Processes and its Applications Abstract: The Camerm-Martin's quasi-invariance result for translations plays a fundmental role in the theory of stochastic calculus of variations on the Wiener space. When we deal with a non linear situation, the available Camerm-Martin direction are needed to be enlarged: This gives rise the notion of tangent processes. In this talk, we shall show by examples the role of tangent processes to establish the quasi-invariance results in non linear situations.
Feng, Shui :	Fleming-Viot Process: Large Deviation and Quasi-Potential Abstract: Fleming-Viot process with neutral mutation is a measure-valued process describing the evolution of genotype frequency in a population under the influence of mutation and resampling. In this talk results will be presented on large deviations and quasi-potential of the Fleming-Viot process. The main difficulty is dealing with the degeneracy of the diffusion coefficient at the boundary. These are joint work with D.A. Dawson, and with Jie Xiong.
Gong, Fu-Zhou :	Exponential Integrability of Functions on Loop Spaces
Han, Dong :	Gelation of a Reversible Markov Process of Polymerization Abstract: A reversible Markov process as a chemical polymerization model which permits the coagulation and fragmentation reactions is considered. We present a necessary and sufficient condition for the occurrence of a gelation in the process. We show that a gelation transition may or may not occur, depending on the value of the fragmentation strength, and, in case the gelation takes place, a critical value for the occurrence of the gelation and the mass of the gel can be determined by close forms. (PDF file)
Hsu, Elton P. :	Brownian Motion and Dirichlet Problem at Infinity Abstract: We show how to solve the Dirichlet problem at infinity on a Cartan-Hadamard manifold satisfying very generous curvature conditions by estimating the angular oscillation of Brownian motion on such a manifold. (PDF file)
Li, Zeng-Hu :	Construction of Measure-valued Diffusions Carried by Stochastic Flows Abstract: Let $\$\backslash \{W(s,y):\; s\backslash ge0,y\backslash in\backslash IR\backslash \}\$$ be a two parameter Brownian motion (a time-space white noise). For a smooth and square-integrable function $h(\cdot)$ on $\IR$ and any $r\ge0$ and $a\in\IR$, given $x(r,a,r) = a$ the equation $$ x(r,a,t) = a + \int_r^t\int_{\IR} h(y-x(r,a,s))W(ds,dy), \quad t\ge r,$$ has a unique solution $\{x(r,a,t): t\ge r\}$, which defines a isotropic stochastic flow. We consider a stochastic equation for measure-valued process carried by the flow. The equation is driven by a Poisson point process on the space of one-dimensional excursions. A pathwise unique solution of the equation is proved, which gives a measure-valued diffusion process. (PDF file)
Lu, Yun-Gang :	Quantum Markovian Approximation Abstract: Markovian approximation are used both in physics and mathematics. The basic idea is to replace a rather complicate evolution of a physical system by a simpler one which is determined by a semigroup. We shall give some useful Markovian approximations and present their intuitive idea, discuss the rigorous mathematical treatment.
Mao, Yong-Hua :	Deviation Kernels for One-Dimensional Diffusion Processes Abstract: It is proven that for the non-explosive and ergodic diffusion on the half line with the ransition probability kernel $p(t,x,y)$, the deviation kernel $d(x,y)=\int_0^\ift (p(t,x,y)-1)dt$ exists and is finite if and only if $\int_0^\ift\E^x H_0 \mu(dx)<\ift$, where $H_0$ is the hitting time of $0$ and $\mu$ is the speed measure. The explicit formulas are also obtained. (PDF file)
Schmitt, Bernard :	Rate of Mixing for Gibbs States of Dynamical Systems Abstract: The g measures are discrete stationary processes continuously depending on their past, they extend in a natural way the Markov processes. If we assume regularity of the weight g than the process is strongly mixing and the rate of mixing exponential. We produce a new approach based upon inequalities of Poincare's type for giving constructive estimates of the mixing rate.
Schmuland, Byron :	A Cocycle Proof that Reversible Fleming-Viot Processes have Uniform Mutation Abstract: Why is the mutation operator associated with a reversible Fleming-Viot process uniform? Our explanation is based on Handa's recent result that reversible distributions must be quasi-invariant under a certain flow, forcing the mutation operator to satisfy a cocycle identity. (PDF file)
Wang, Feng-Yu :	Gradient Estimates of Dirichlet Heat Semigroups and Application to Isopermetric Inequalities Abstract: By using probabilistic approaches, some uniform gradient estimates are obtained for Dirichlet heat semigroups on a Riemannian manifold with boundary. As an application, lower bound estimates of isoperimetric constants are presented in terms of functional inequalities. (PDF file)
Wang, Ying-Zhe :	Algebraic Convergence of Markov Chains Abstract: Algebraic convergence in $L^2$-sense is studied for general time-continuous, reversible Markov chains with countable state space, and especially for birth-death chains. Some criteria for the convergence are presented. The results are effective since the convergence region can be completely covered, as illustrated by two examples.
Wu, Li-Ming :	Essential Spectral Radius for Markov Kernel Abstract: Using two new parameters $\beta_{\tau}(P)$ and $\beta_{wc}(P)$ of non-compactness for a positive kernel $P$ on a Polish space $E$, we obtain a new formula of Nussbaum-Gelfand type for the essential spectral radius $r_{ess}(P)$ on $b{\mathcal B}$. Using that formula we show that different known sufficient conditions for geometric ergodicity such as Doeblin's condition, drift condition by means of Lyapunov function, geometric recurrence etc lead to variational formulas of the essential spectral radius. All those can be easily transported on the weighted space $b_u{\mathcal B}$. Some related results on $L^2(\mu)$ are also obtained, especially in the symmetric case. Moreover we prove that for a strongly Feller and topologically transitive Markov kernel, the large deviation principle of Donsker-Varadhan for occupation measures of the associated Markov process holds if and only if the essential spectral radius is zero. The knowledge of $r_{ess}(P)$ allows us to estimate eigenvalues of $P$ in $L^2$ in the symmetric case, and to estimate the geometric convergence rate by means of that in the metric of Wasserstein. Some examples are presented.
Zhang, Tu-Sheng :	Perturbed Reflected Diffusion Processes Abstract: In this talk we will present some recent results on existence and uniqueness of perturbed reflected diffusion processes, which were studied by M.Yor, Le Gall and others before.
Zhang, Xin-Sheng :	On Stochastic Order for Diffusion Processes Abstract: Let $X(t)$ be a diffusion process in $R^d$, i.e., let $X(t)$ satisfy the following stochastic differential equation: $$dX(t)=b(X(t))+\sigma(X(t))dW(t),$$ where $b(x):R^d:\to R^1$ $\sigma(x)$ is a $d\times d$ matrix, and $W(t)$ is a d-dimensional Browian motion. The usual stochastic order and convex order for two diffusion processes will be discussed. Some sufficient and necessary conditions in terms of $b(x)$ and $\sigma(x)$ for two diffusion processes to have usual stochastic order or convex order will be presented.
Zhang, Yu-Hui :	Dual Variational Formulas for the First Dirichlet Eigenvalue on Half-Line Abstract: The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the coefficients of the operators are presented. Moreover, the corresponding problems in the discrete case and the higher-order eigenvalues in the continuous case are also studied. This talk is based on a joint work with Mu-Fa Chen and Xiao-Liang Zhao.(PDF file)
Zhao, Xue-Lei :	Stochastic Analysis on p-Adics Abstract: Some new results on the stochastic analysis on p-adics will be introduced. (PDF file)
Zhou, Xiao-Wen :	Coalescing Brownian Motion, its Dualities and a Measure-Valued Process Abstract: Coalescing Brownian motion captures the interactions among a system of Brownian motions. It plays a key role in analying certain measure-valued processes. Some aspects of coalescing Brownian motion will be discussed in this talk. We will first introduce a characterization theorem. Then we will present two dual relationships involving coalescing Brownian motion. A sketch of the proof will be given. Such a duality can be applied to construct a measure-valued process.