Title and Abstracts of Talks

Abstract:

The abstract in PDF format: Krzysztof Bogdan.pdf

Abstract: We prove that two independent continuous time random walks on the infinite component of a supercritical bond percolation of Z2 meet each other infinitely many times.

Abstract: The construction of Stein identities is central to Stein's method for probability approximations. In this talk I will begin with a discussion on Stein's method and ways of constructing Stein identities, with illustrations from a number of examples including the binary expansion of a random integer, the anti-voter model, and the Curie-Weiss model. I will then present a Cramer-type moderate deviation result based on a fairly general Stein identity and apply the result to the three examples mentioned above. This talk is based on a joint work with Xiao Fang and Qi-Man Shao.

Abstract: This talk is concerned with 2D Navier-Stokes equation with Lévy noise. The existence and uniqueness of the global strong and weak solutions and the existence of invariant measures is proved in in our previous paper. But in that framework, it seems that it is impossible to get the strong Feller property. In this talk, we show that the on a suitable state space, the solution is strong Feller. For getting the ergodicity, the priori estimations and stopping time technique play the key role in the proofs.

The abstract in PDF format: ZhaoDong.pdf

Abstract: Consider the center of mass of a supercritical branching-Brownian motion, or that of a supercritical super-Brownian motion. We prove that it is a Brownian motion being slowed down such that it tends to a limiting position almost surely, which, in a sense complements a result of Tribe on the final behavior of a critical super-Brownian motion. This is shown to be true also for a model where branching Brownian motion is modified by attraction/repulsion between particles.

We then put this observation together with the description of the interacting system as viewed from its center of mass, and get the following asymptotic behavior: the system asymptotically becomes a branching Ornstein-Uhlenbeck process (inward for attraction and outward for repulsion), but (1) the origin is shifted to a random point which has normal distribution, and (2) the Ornstein-Uhlenbeck particles are not independent but constitute a system with a degree of freedom which is less than their number by precisely one.

Finally, in the attractive case, we prove a scaling limit theorem for the local mass.

Abstract: How fast does a transient Brownian motion escape to infinity is an interesting question. For example, it is well known that Brownian motion in a Euclidean space of dimension 3 and higher escapes to infinity at approximately the rate of the square root of time. Which geometric quantity controls the rate of escapes an interesting question. We will explain that the speed of Brownian motion can be effectively controlled by the volume growth rate of a complete Riemannian manifold. A precise integral criterion is obtained which in many cases is sharp.

Abstract: This talk is an update on our journey to estimates of transition densities for discontinuous Markov processes in open subsets.

In this talk I will present recent results on two-sided sharp estimates of transition densities for two Levy processes, relativistic stable process and independent sum of two stable processes. This talk is based on some recent works with Zhen-Qing Chen and Renming Song.

Abstract: In this talk, we will discuss general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symmetric jump processes. We will also discuss convergence of Markov chains with random conductance to symmetric jump processes.

Abstract: We establish conditions for the $L^p$-independence of spectral bounds of Feynman-Kac semigroup for continuous additive functionals generated by $\alpha$-order Green-tight measures of Kato class.

Abstract:

Abstract: Consider a family of probability measures $\nu_y : y \in\partial D$ on a bounded open domain $D \in R^d$ with smooth boundary. For any starting point $x \in D$, we run a standard d-dimensional Brownian motion $B(t)\in R^d$ until it first exits D at time $\tau$, at which time it jumps to a point in the domain D according to the measure $\nu_{B(\tau )}$ at the exit time, and starts the Brownian motion afresh. The same evolution is repeated independently each time the process reaches the boundary. The resulting diffusion process is called Brownian motion with jump boundary (BMJ). The spectral gap of non-self-adjoin generator of BMJ, which describes the exponential rate of convergence to the invariant measure, is studied. In particular, we prove the so-called $2/3$-conjecture on the largest spectral gap (fastest rate of convergence) among all possible jump measures in one-dimensional setting. This is joint work with Yuk Leung.

Abstract: Correlations in bipartite quantum states are multifaceted and subtle. The total correlations are well quantized by the quantum mutual information, which is a purely formal and theoretical concept independent of measurements. In order to extract correlations, one has to perform measurements. The observable correlations are the maximum extractable correlations, as quantized by the classical mutual information of local measurement outcomes. A basic issue is the relationship between the total correlations and the observable correlations. The Lindblad conjecture states that the observable correlations account for at least half of the total correlations, or equivalently, correlations are more classical than quantum, and is supported by many intuitive observations. We will disprove this conjecture by counterexamples. In the course, by use of the postulate that classicality emerges from measurement, we present an intrinsic reasoning for interpreting the observable correlations as a measure of classical correlations, and introduce a natural measure of quantum correlations. With this measure, we further introduce a hierarchy of entanglement measures, which not only include the entanglement of formation as a particular instance, but also highlights a peculiar feature of the latter. This is a joint work with Qiang Zhang.

Abstract: Starting from the celebrated integration formula by parts of Phillips in 1953 on the perturbation theory for semi-groups, we derive a general inequality-based perturbation bounds in the total variance norm for the transition semi-groups and the stationary distributions in terms of the strong ergodicity convergence rates. The result can be easily applied to various cases. In particular, for the reversible Markov processes on the countable state spaces, by using a renewal formula we estimate the convergence rates by the uniform expectation of the hitting time of an arbitrary state.

Abstract: A G-Brownian motion has been defined as a continuous path which has independent and stationary increments under a sub-linear expectation, called G-expectation. This new type of Brownian motion provides is a powerful tool for treating the probability uncertainty of stochastic processes with continuous paths. The corresponding Ito's calculus --Ito's integral, Ito's formula, SDE, has been developed within this framework. An important open problem is to formulate and to prove the correspond in martingale representation theorem in which both the first and second derivatives must appear, comparing classical representation theorem, where only the first order derivative appears. This naturally leads to the second order fully nonlinear PDE. This is also the starting point of the corresponding BSDE theory under G-expectation.

Abstract: First, I will mention a scaling limit theorem for a large class of Dawson-Watanabe superprocesses whose underlying spatial motion is a symmetric diffusion with smooth coefficients or a symmetric Levy process on Rd whose Levy exponent ª(´) satisfies some conditions, where the convergence is in strong sense.

Second, I will discuss Kesten-Stigum L log L type theorems for super- diffusions and branching Hunt processes. The Kesten-Stigum type theorems are established under the assumptions that the principle eigenvalue of some associated SchrÄodinger operator is positive and the associated Feynman-Kac semigroups are intrinsic ultracontractive.

AMS Subject Classifications (2000): Primary 60J80; Secondary 60G57,60J45

Keywords and Phrases: super-diffusion, strong limit theorem, branching Hunt processes, Kesten-Stigum L log L type theorem

Abstract: We consider stochastic differential equations on Hilbert spaces with time-dependent coefficients for which no existence and uniqueness results are known. We prove, under suitable assumptions existence and uniqueness of a measure valued solution, for the corresponding Fokker-Planck equation. In particular, we verify the Chapman-Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informly given by the stochastic differential equations.

Abstract: We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol $q(x,\xi)$. If $|q(x,\xi)| < c(1+|\xi|^2)$, the corresponding Feller process can be approximated by Markov chains whose steps are increments of L\'evy processes. This approximation can easily be used for a simulation of the sample path of a Feller process. Further we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process. (joint work with B. Boettcher)

Abstract: For $d\geq 1$ and $\alpha \in (0, 2)$, consider the family of pseudo differential operators $\{\Delta+ b \Delta^{\alpha/2}; b\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$. In this talk, I will present a uniform boundary Harnack principle (BHP) with explicit boundary decay rate for nonnegative functions which are harmonic with respect to $\Delta +b \Delta^{\alpha/2}$ (or equivalently, the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with constant multiple $b^{1/\alpha}$) in $C^{1, 1}$ open sets. Here a ``uniform" BHP means that the comparing constant in the BHP is independent of $b\in [0, 1]$. The proof of this BHP employs a combination of probabilistic and analytic techniques.

Abstract: We present a brief introduction to recent progress in optimal transportation on manifolds and metric spaces. We recall the characterization of the heat equation on Riemannian manifolds M as the gradient flow for the relative entropy on the L2-Wasserstein space of probability measures P(M), regarded as an infinite dimensional Riemannian manifold. Of particular interest are recent extensions to the heat flow on Finisher spaces, Heisenberg groups and Wiener spaces. Convexity properties of the relative entropy Ent(.|m) also play an important role in a powerful concept of generalized Ricci curvature bounds for metric measure spaces (M, d,m).

Moreover, we give a survey on recent results for the Wasserstein diffusion, a canonical reversible process (μt)t_0 on the Wasserstein space P(R). This includes: particle approximation, logarithmic Sobolev inequaltiy, quasi-invariance of its invariant measure, the so-called entropic measure, P_ under push forwards μ 7! h_μ by means of smooth diffeomorphisms h of R. We also indicate how to construct the entropic measure on multi-dimensional spaces, formally given as Z exp(−_ • Ent(.|m))P0(dμ).

Abstract: We consider $n$ independent, identically distributed one-dimensional > Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth > density function $f$. The empirical quantiles, or pointwise order > statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence > of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. > This sequence converges in probability in $C[0,\infty)$ to $q(t)$, the > $\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the > convergence in law in $C[0,\infty)$ of the fluctuation processes $F_n = > n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and > we derive an explicit formula for its covariance function. We also show that > $F$ has many of the same local properties as $B^{1/4}$, the fractional > Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic > variation process, it has H\"older continuous paths with any exponent > $\gamma < 1/4$, and (at least locally) it has increments whose correlation > is negative and of the same order of magnitude as those of $B^{1/4}$.

Abstract: Suppose that $S$ is a subordinator with a nonzero drift and $W$ is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion $X$ defined by $X_t=W(S_t)$.

We give sharp bounds for the Green function of the process $X$ killed upon exiting a bounded open interval and prove a boundary Harnack principle.

In the case when $S$ is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of $X$ in $(0,\infty)$, and sharp bounds for the Poisson kernel of $X$ in a bounded open interval.

Abstract: Some recent progress on reflecting diffusion processes on (non-convex) manifolds with boundary are introduced. We shall present equivalent semigroup/heat kernel properties for lower bounds of curvature and the second fundamental form. In particular, the convexity of the manifold is described by semigroup/heat kernel inequalities.

Abstract: In this talk, we will discuss a class of interacting superprocesses and related SPDEs on a bounded domain.

Abstract: In this talk, I will report a joint work with {\bf Feng-Yu Wang} (published in Bulletin des Sciences Mathematiques, 132 (2008)). By using the super Poincare inequality of a Markov generator $L_0$ on $L^2(\mu)$ over a $\si$-finite measure space $(E,\F,\mu)$, the Schr\"odinger semi-group generated by $L_0-V$ for a class of (unbounded below) potentials $V$ is proved to be $L^2(\mu)$-compact provided $\mu(V\le N)<\infty$ for all $N> 0$. This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on e.g. $\R^d$ under the condition that $V(x)\to \infty$ as $|x|\to \infty.$ Concrete examples are then presented to illustrate our main result.

Abstract: I will review the beautiful history on the optimal estimates of spectral gap and present some recent progresses on transportation inequalities on Riemannian manifolds. The Cheeger's isoperimetric inequality will be also studied.

Abstract: This work is concerned with stability of a class of switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a random switching device. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. For such a two-component process $(X(t),Z(t))$ under consideration, a distinct feature is the switching process $Z(t)$ also depends on the $X(t)$ process. This paper focuses on long-time behavior, namely, stability. After recalling the notion of regularity and stability, it is shown that under suitable conditions, the systems are regular or no finite explosion time. To study stability of the trivial solution (or the equilibrium point 0), systems that are linearizable (in the $x$ variable) in a neighborhood of 0 are considered. Then sufficient conditions for stability and instability are obtained. Next, almost sure stability is examined by treating Lyapunov exponent. The stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients. To close the gap, a transformation technique is used to obtain a necessary and sufficient condition for stability.

Abstract: The sample paths of Levy processes generate various random fractals and many of their properties have been studied since 1960's [see the survey papers of Fristedt (1974), Taylor (1986) and Xiao (2004)].

In this talk we present some recent results on potential theory of multi-parameter Levy processes and show their applications to Hausdorff dimension and exact capacity computation for ordinary Levy processes.

In particular, we verify a conjecture of Bertoin (1997, 1999) on intersections of independent regenerative sets and derive some intersection properties of operator-stable Levy processes.

Abstract: In this talk I will show that parameterized continuous-time Markowitz's mean--variance efficient strategies could reach any given target with arbitrarily high probabilities. This result indicates that the very popular risk measure VaR (Value at Risk) may not be a proper measure in guiding investment practice. This, in turn, motivates the introduction of certain stopped strategies where stock holdings are liquidated whenever the parameterized Markowitz strategies reach the present value of the mean target. The risk aspect of the revised Markowitz strategies are examined via expected discounted loss from the initial budget. A new portfolio selection model is suggested. This talk is based on a joint work with Profeesor Xunyu Zhou of University of Oxford.

Abstract: We will prove that the local time of a Levy process is of finite p-variation in the space variable in the classical sense, a.s. for any p>2, t , if the Levy measure satisfies , and is a rough path of roughness p a.s. for any p>2. This is a new class of rough path processes. Then for any function g of finite q-variation ( ), we establish the integral as a Young integral when and a Lyons' rough path integral when . We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function f if exists and is of finite q-variation when , for both continuous semi-martingales and a class of Levy processes. This is a joint work with Chunrong Feng.

Abstract: We make the following assumptions. (1) Agents’ preferences depend only on the probability distribution of terminal wealth. (2) Agents prefer more to less. (3) The market is perfect and frictionless. (4) The market is arbitrage-free and could be incomplete. Under these assumptions, we show that in general path-dependent strategies are inefficient and not optimal. In addition, we characterize the ones that are cost-efficient. We obtain an explicit formula for the efficiency cost of a strategy as well as for the payoff of the cost-efficient derivative that should be preferred by all investors. We illustrate the study by exhibiting the specific form of a financial derivative that dominates the lookback option, the geometric Asian option or the barrier option.

Abstract: We consider the almost sure local extinction for a class of measure-valued branching processes $X$ with coalescing Brownian spatial motion starting from Lebesgue measure. Let $g(t), t>0$, be any nonnegative, nondecreasing function. Define the extinction time (with respect to $g$) as \[\tau:=\sup\{t\geq 0: X_t([-g(t),g(t)])\neq 0\}.\] Under some mild conditions on branching mechanism, we show that $P\{\tau=\infty\}$ is equal to either 0 or 1 depending on whether an integral associated with $g$ is finite or infinite.

Abstract:

The abstract in PDF format: MKabstract.pdf

Abstract: We first give some constructions of immigration superprocesses with deterministic or stochastic immigration rates by summing up measure-valued excursions according to Pois- son point process. Based on those we prove the existence and uniqueness of the solution to a stochastic integral equation driven by a superprocess and an excursion-valued Poisson point process, which gives a state-dependent immigration structure. Using this approach we can change the branching mechanism of the superprocess. We shall deal with pro- cesses with cµadlµag paths. The constructions given here are closely related with those of Abraham and Delmas (2009, Annal. l'Institut H. Poincar¶e - Probab. et Statist.) and Fu and Li (2004, Osaka J. Math.).

Abstract: We show the localization property for branching Brownian motions in random environment associated with the Poisson random measure in time-space.

Abstract: In this talk, I will show the conservativeness of a class of symmetric diffusion processes with jumps. This is a joint work with Jun Masamune.

Abstract: Let B be an (Ni; d)-fractional Brownian motion with Hurst index i 2 (0; 1) (i = 1; 2), and let B 1 and B2 be independent. We prove that if N11+ N2 2 d, then the intersection local times of B1 and B2 exist, and are jointly continuous. We also establish sharp Holder conditions for the intersection local times. The motivation of this project is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20 (2007)), where the existence of the intersection local times of two independent (1; d)-fractional Brownian motions with the same Hurst index was studied by using a different method. This talk is based on a joint work with Yimin Xiao.

Abstract: We begin with simulation of Brownian motion, and reflected Brownian motion. Then we present the simulation and numerical analysis methods for BSDEs.

Abstract: Consider a family of probability measures on a bounded open Region with a smooth boundary and a positive parameter set , all indexed by . For any starting point inside , we run a diffusion until it first exits , at which time it stays at the exit point for an independent exponential holding time with rate and then leaves by a jump into according to the distribution . Once the process jumps inside, it starts the diffusion afresh. The same evolution is repeated independently each time the process jumped into the domain and the resulting Markov process is called diffusion with holding and jumping boundary (DHJ). We first survey recent development in the case without holding (jumping immediately) and then present a study of DHJ, which is not reversible due to jumping, on its generator, stationary distribution and the speed of convergence. This is joint work with Wenbo Li.