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International
Conference on
Stochastic Analysis and Its Applications |
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PACIFIC INSTITUTE FOR THE MATHEMATICAL SCIENCES (PIMS) INSTITUTE OF MATHEMATICAL STATISTICS DEPARTMENT OF MATHEMATICS (UW)
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Abstracts of Invited TalksSpeaker: Krzysztof Bogdan (Wroclaw University of Technology, Poland) Title: Levy processes and Fourier multipliers Abstract: This is a joint work with Rodrigo Banuelos. We study Fourier multipliers which result from censoring jumps of Levy processes. Using the theory of martingale transforms we prove that these operators are bounded in $L^p(Rd)$ for $1< p <\infty$ and we obtain the same explicit upper bound for their norms as the one known for the second order Riesz transforms. Speaker: Erhan Cinlar (Princeton University) Title: Jump-diffusion Abstract: For Hunt processes with jumps, we seek a treatment that concentrates on the jumps. Embedded at the jump times, there are Markov chains (discrete-time, continuous space) that decompose the original process into a sequence of diffusions. Then, the original resolvent can be written as the potential operator of a Markov chain acting on the resolvent of a diffusion. Similar decompositions are possible for hitting distributions and the transition semigroup. Speaker: Tzuu-Shuh Chiang (Academia Sinica, Taipei) Title: Central Limit Theorem for Diffusion Processes in Discontinuous Drift with Small Perturbation Abstract: Click here to see it in pdf format. Speaker: Michael Cranston (University of California at Irvine) Title: On Unbalanced Large Deviation Results Abstract: The focus of this talk is on the different behavior of large deviations of random additive functionals above the mean versus large deviations below the mean in a variety of random media models. The principal examples are provided by the parabolic Anderson model and by first passage percolation models where the random variables assigned to the bonds may take on negative values. Typically, these large deviations exhibit a strong asymmetry, large deviations above the mean are radically different from large deviations below the mean. We seek to quantify and explain the differences. This talk is based on joint work with D. Gauthier and T. Mountford. Speaker: Steven N. Evans (University California at Berkeley) Title: Subtree Prune and Re-graft: a Reversible Real Tree Valued Markov process Abstract: We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is Aldous's Brownian continuum random tree. This process is inspired by the subtree prune and re-graft (SPR) Markov chains that appear in phylogenetic analysis in biology. A key technical ingredient in this work is the use of a novel Gromov--Hausdorff type distance on the space whose elements are ``weighted'' compact real trees. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion. Speaker: Patrick J. Fitzsimmons (University of California at San Diego) Title: Drift Transformations of Symmetric Diffusions, and Duality Abstract: Starting with a symmetric Markov diffusion process X (with symmetry measure m and L^2(m) infinitesimal generator A) and a suitable core C for the Dirichlet form of X, we describe a class of derivations defined on C. Associated with each such derivation B is a drift transformation of X, obtained through Girsanov's theorem. The transformed process XB is typically non-symmetric, but we are able to show that if the "divergence'' of B is positive, then m is an excessive measure for XB, and the L2(m) infinitesimal generator of XB is an extension of f --> Af+B(f). The methods used are mainly probabilistic, and involve the notions of even and odd continuous additive functionals, and Nakao's stochastic divergence. Speaker: Masatoshi Fukushima (Osaka University, Japan) Title: On Feller's Boundary Problem and Darning Countable holes for Markov Processes Abstract: Click here to see it in pdf format. Speaker: Ron Getoor (University of California at San Diego) Title: The Moderate Markov Dual Process Abstract: Click here to see it in pdf format. Speaker: Masanori Hino (Kyoto University) Title: Reflecting Ornstein-Uhlenbeck Processes on Path Spaces Abstract: Consider a set of continuous maps from [0,1] to a domain in R^d. Although its topological boundary in the path space is not smooth in general, by using the theory of BV functions on the Wiener space and the theory of Dirichlet forms, we discuss the existence of the surface measure and the Skorohod representation of the reflecting Ornstein-Uhlenbeck process on the set. This talk is based on a joint work with H. Uchida. Speaker: Elton P. Hsu (Northwestern University) Title: Generalized Bismut's Formula for Certain SDE in Vector Bundles Abstract: Following Bismut's original work, James Norris proved in 1993 a generalized Bismut's formula for solutions of certain class of heat equations in vector bundles by the perturbation method introduced by Bismut. One interesting feature of this generalized Bismut's formula is that in principle it can be iterated to derive the classical Bismut's formula for higher derivatives. Over the years, Bismut's formula for higher derivatives have been obtained in various forms and using various methods. The original perturbation method was also replaced by the more elegant method of vector bundle valued martingales. We will show that Norris' result can be derived by the new method, thus smoothing out certain technical difficulties in the original paper. We will also give a few new applications. Speaker: Davar Khoshnevisan (University of Utah) Title: Slices of the Brownian Sheet: New Results, and Open Problems Abstract: We can view the Brownian sheet as a sequence of interacting Brownian motions or {\it slices}. Here we present a number of results about the slices of the sheet. A common feature of our results is that they exhibit phase transition. Time permitting, a number of open problems are presented. Speaker: Panki Kim (University of Illinois at Urbana-Champaign) Title: Symmetric Stable Process and Beyond Abstract: Recently there have been a lot of interests in studying discontinuous Markov processes due to their importance in theory as well as applications. In this talk, we discuss some boundary behavior of harmonic function with respect to various types of discontinuous Markov processes: symmetric stable process, relativistic stable process and truncated stable process. Speaker: Kazuhiro Kuwae (Kumamoto University, Japan) Title: On Stratonovich Type Integrals and Lyons-Zheng's Decompositions over Symmetric Markov Processes Abstract: I will talk about the Fisk-Stratonovich type integrals by Dirichlet processes appeared in Fukushima's decomposition in the framework of symmetric Markov processes. Such integrals were firstly defined by Nakao and also by Lyons-Zhang in a different way based on the time reverse operator if the underlying process is a diffusion with no killing inside. We generalize integrals including jump and killing parts, whose definitions are different from what were discussed by Meyer, Protter even if in the framework of semi-martingales. I give a representation of the Fisk-Stratonovich type integrals by the discontinuous part of such Dirichlet processes as an integrator. As a corollary, under the law for quasi-everywhere starting points, we establish an extension of Fukushima's decomposition for the function locally in the domain of forms, whose jump part can be expressed as a convergence of the sum of jumps. We also show that our Fisk-Stratonovich type integrals allow Lyons-Zheng's type decompositions under the law for quasi-everywhere starting point, which strengthen the relation between two definitions on the Fisk-Stratonovich integrals by Nakao and by Lyons-Zhang for diffusions without killing inside. Speaker: Edwin Perkins (University of British Columbia) Title: Pathwise Uniqueness for Parabolic Stochastic PDE's Abstract: Consider the SPDE: du/dt=u"+g(u)dW/dtdx where dW/dtdx is space-time white noise and g is Holder continuous of index h. It is shown that if 2h^3-h>3/4 then pathwise uniqueness holds. The proof is an infinite dimensional extension of the Yamada-Watanabe Theorem. This work is joint with Leonid Mytnik. Speaker: Michael Rockner (Purdue University) Title: Stochastic Porous Media and Fast Diffusion Equations Abstract: We first present a new existence and uniqueness result for stochastic evolution equations on Hilbert spaces. This is a generalization of a classical result by Krylov and Rozovskii based on the so-called variational approach to stochastic partial differential equations (SPDE). The main motivation are applications to nonlinear SPDE of porous media type which also include cases where the nonlinear functions grow slowly at infinity ("fast diffusion equations"). Generally, the main problem is to find the appropriate Gelfand triple to work on. In our case Orlics spaces turn out to be convenient. We show how one must choose the defining Young function for a given nonlinearity. After presenting these applications, we shall summarize results about the qualitative behaviour of solutions and about their invariant measures. Speaker: Jay Rosen (City University of New York) Title: Frequent Points and Harnack Inequalities for Random Walks in Two Dimensions Abstract: For a random walk in $\Z^2$ which does not necessarily have bounded range we study those points which are visited an unusually large number of times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the asymptotics for the number of visits to the most visited site. We also obtain the asymptotics for the number of points which are visited very frequently by time $n$. The key to these results are good Harnack Inequalities for the interior and exterior of a disc. Joint work with Rich Bass. Speaker: Rene Schilling (Universität Marburg, Germany) Title: An Exact Formula for the Hausdorff Dimension of a Levy Image Set Abstract: Let $X_t$ be a $d$-dimensional L\'evy process and let $E\subset [0,\infty)$ be a Borel measurable set. We call the random set $$ X(E)(\omega):= \{y \::\: X_t(\omega)=y, \text{\ for some\ } t\in E \} $$ a L\'evy image set. The aim of this talk is to give a short and elementary proof of a formula recently discovered by D. Khoshnevisan and Y. Xiao (Ann. Probab. 33, No.3, 841-878 (2005)). Speaker: Byron Schmuland (University of Alberta, Canada) Title: Some Curiosities and Questions about the Renewal Theorem. Abstract: We consider some generalizations of the renewal theorem for Markov chains. We will discuss both analytic and probabilistic approaches to finding the asymptotical behaviour of the solution of a renewal equation. Speaker: Oded Schramm (Microsoft Research) Title: Compositions of random transpositions Abstract: Let $Y=(y_1,y_2,...)$, $y_1\ge y_2\ge...$, be the list of sizes of the cycles in the composition of $c n$ transpositions on the set $\{1,2,...,n\}$. We prove that if $c>1/2$ is constant and $n\to\infty$, the distribution of $f(c)Y/n$ converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function $f$ is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process. Speaker: Renming Song (University of Illinois at Urbana-Champaign) Title: Intrinsic Ultracontractivity of Non-symmetric Diffusion Semigroups in Bounded Domains. Abstract: The notion of intrinsic ultracontractivity, introduced by Davies and Simon for symmetric semigroups, is a very important concept and has been studied extensively. However, it seems that, up to now, no one has introduced the concept of intrinsic ultracontractivity for non-symmetric semigroups. In this talk, we first introduce the notion of intrinsic ultracontractivity for non-symmetric semigroups. We will show that the Dirichlet semigroups of non-symmetric diffusion semigroups are intrinsic ultracontractive on very rough domains. Speaker: Karl-Theodor Sturm (Universität Bonn, Germany) Title: Optimal Transportation, Ricci Curvature and Diffusions on the L^2-Wasserstein Space Abstract: Click here to see it in pdf format. Speaker: Wei Sun (Concordia University, Canada) Title: Applications of Dirichlet Forms to Nonlinear Filtering Abstract: Filtering is concerned with estimating the conditional probability distribution of a signal through a partial and noisy sequence of observations of the signal. In this talk we present some applications of Dirichlet forms to nonlinear filtering. In particular, we introduce some recent results on the absolute continuity of the optimal filters with respect to the reference measures and nonlinear filtering of Markov processes in random environments. Speaker: Jason Swanson (University of Wisconsin at Madison) Title: Metastability and Coupling of Markov Processes Abstract: We consider a particle whose motion is governed by a potential function with a finite number of local minima. In this (determinstic) setting, the particle will travel to the "nearest" minimum point and stay there permanently. However, when a small noise term is added to this system, the resulting diffusion process will spend only a finite amount of time in a neighborhood of this minimum. Eventually, the noise term will be strong enough to push the particle out of the domain of attraction of this minimum and into the domain of attraction of another. There the particle will stay for a random amount of time and the cycle continues. Such diffusion models arise in practice, for example, in analyzing biochemical models. In these cases, the diffusion models are too complex for practical applications and it is an important question how to efficiently replace this model with a simpler one. Using large deviations, Freidlin and Wentzell have shown that these "metastable" transition times are asymptotically exponential, suggesting that this diffusion process can be approximated by a continuous-time Markov chain on the finite state space consisting of the stable points of the potential function. Our approach to this is to construct an explicit coupling between the diffusion and such a Markov chain, and to study the conditional law of the diffusion given observations of the Markov chain. This is joint work with Tom Kurtz. Speaker: Masayoshi Takada (Tohoku University, Japan) Title: Large Deviations for Additive Functionals of Symmetric Stable Processes Abstract: We consider a symmetric $\alpha$-stable process and its additive functionals in the Revus correspondence to {\it Green tight} Kato measures. We establish the large deviation principle for the additive functionals by using the G\"artner-Ellis theorem. Speaker: Toshihiro Uemura (University of Hyogo, Japan Title: On an extension of jump-type Dirichlet forms Abstract: We show that any element from the so-called "$L^2$-maximal domain" of a jump-type symmetric Dirichlet form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces $L^{s,2}(R^n)$ for $0 < s < 1$. Speaker: Pierre Vallois (Universite Henri Poincare Nancy 1) Title: Brownian Penalizations Related to Local Time and Excursion Lengths Abstract: Click here to see it in pdf format. Speaker: Zoran Vondracek (University of Zagreb, Croatia) Title: Parabolic Harnack Inequality for the Mixture of Brownian Motion and Stable Process Abstract: Let X be a mixture of independent Brownian motion and symmetric stable process. In this talk we discus sharp bounds for transition density of X, lower bounds for transition densities of the killed process and prove a parabolic Harnack inequality for nonnegative parabolic functions of X. Speaker: Hao Wang (University of Oregon) Title: A Class of Superprocesses in a Random Medium Abstract: A class of superprocesses in a random medium is considered. Due to the interacting between particles, the basic tool, the log-Laplace functional, for independent superprocesses is not available for this class of superprocesses. In order to study properties which are similar to that of independent superprocesses, a conditional log-Laplace functional and conditional sample path decompositions are investigated. This talk is based on joint work with Zenghu Li and Jie Xiong. |
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This page was last modified on Wednesday, March 8, 2006. | |