Cornell logo Stochastic Analysis and Mathematical Physics
April 11-13, 2010



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  • Sergio Albeverio (University of Bonn)

    Title: Stochastic FitzHugh-Nagumo equations, asymptotic expansions and quantum graphs

    Abstract:  Deterministic FitzHugh-Nagumo equations are among the most used PDE's in neurobiology. Stochastic perturbations of them. We shall report on recent work dedicated to the study of their stochastic perturbations and the corresponding asymptotics. The systems of such equations describe the dynamics of neural networks. We briefly discuss the relation between the study of such systems and the area of quantum dynamics on their manifolds and on their graph limits.

  • Fabrice Baudoin (Purdue University)

    Title: Heat kernel analysis and generalized Ricci lower bounds in sub-Riemannian geometry

    Abstract:  We study a class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplacian. We construct on these manifolds an analogue of the Riemannian Ricci tensor and prove Bochner type formulas for the sub-Laplacian. As a consequence, it is possible to formulate on these spaces a sub-Riemannian analogue of the so-called curvature dimension inequality. The heat kernel analysis on sub-Riemannian manifolds for which this inequality is satisfied is shown to share many properties in common with the heat kernel analysis on Riemannian manifolds whose Ricci curvature is bounded from below. This is mainly a joint work with N. Garofalo.
  • Rene Carmona (Princeton University)

    Title: Mathematical Challenges of the Emissions Markets
    Abstract:  The first part of the talk will be devoted to a review of the mathematical models used by economists and financial engineers for the quantitative analysis of cap-and-trade schemes. We will concentrate on models inspired by the European Union Emissions Trading Scheme, and we will discuss design issues related to the existing allocation processes and those touted by policy pundits.
    The second part of the talk will address some of the puzzling questions raised by the prices of options on EU Allowances published by the exchanges: Do they really use Black's formula to compute these prices? How reasonable are they? How can we reconcile these prices with the results of the existing equilibrium theories? How do these models suggest we price these options?

  • Eric Carlen (Rutgers University)

    Title: Quantum Entropy and Non-commutative Brascamp-Lieb Inequalities

    Abstract:  We formulate and prove, in sharp form, non-commutative analogs of the Brascamp-Lieb inequality, which is a generalized form of Young's inequality for convolutions. The proof is based in part on a duality between such inequalities and certain subadditivity inequalities for quantum entropy. In proving these, ideas of Len Gross, first employed in his investigation of fermion hypercontractivity, play an important role. This talk is based on joint works with Dario Cordero-Erausquin and Elliott Lieb.
  • Brian Hall (University of Notre Dame)

    Title: Holomorphic function spaces and heat operators for symmetric spaces

    Abstract:  Motivated by Len Gross’s “Fock space” decomposition for functions on a compact Lie group, the speaker introduced in 1994 a Segal-Bargmann transform for compact Lie groups, which is a unitary map from L^2 of the compact group to an L^2 space of holomorphic functions on the associated complex group. The transform itself consists of applying the heat operator on the compact group and then analytically continuing to the complex group. Since then, there has been quite a bit of work, by various authors, in developing Segal-Bargmann-type transforms for various groups and symmetric spaces, both compact and noncompact. I will survey this work, concentrating on the cases where the results can be stated most explicitly.
  • Todd Kemp (UCSD (visiting MIT))

    Title: Chaos and the Fourth Moment

    Abstract:  In 2006, Nualart and collaborators proved a remarkable central limit theorem in the context of the Wiener chaos.  If $X_k$ is a sequence of $n$th Wiener-Ito integrals (in the $n$th chaos), then necessary and sufficient conditions that $X_k$ converge weakly to a normal law are that its (second and) fourth moments converge -- all other moments are controlled by these.  Their proof is heavily steeped in Malliavin calculus.

    Considering $N\times N$-matrix-valued Brownian motion and letting $N\to\infty$, there is an analogue of the Wiener chaos in free probability theory.  In this talk, I will discuss this free "Wigner" chaos, and the free analogue of Nualart's theorem for the semicircle law (the central limit law for the eigenvalues of random matrices).  The qualitative proof uses some beautiful combinatorics; more quantitative estimates require free versions of tools from the Malliavin calculus, which I will also discuss.

    This is joint work with Roland Speicher.
  • H.-H. Kuo (Louisiana State University)

    Title:  An extension of the Ito stochastic integration

    Abstract:  We introduce the class of instantly independent stochastic processes, which serves as the counterpart of the Ito theory of stochastic integration. This class provides a new approach to anticipating stochastic integration. The crucial idea is to evaluate an instantly independent stochastic process at the right endpoint of a subinterval, while in the Ito theory the evaluation of an adapted stochastic process is at the left endpoint. We present some new results on Ito's formula and stochastic differential equations for the new stochastic integral.
  • Michel Ledoux (Université de Toulouse, France)

    Title: Heat kernel measures, logarithmic Sobolev inequalities an measure concentration

    Abstract: Elementary semigroup calculus provides simple proofs of various functional inequalities for heat kernel measures, including Gross's celebrated logarithmic Sobolev inequality, as well as reversed forms. The approach furthermore reveals dimensional forms of the logarithmic Sobolev inequality connecting on the one end to Shannon's entropy power inequality and on the other to the Li-Yau parabolic inequality. The reversed forms allow besides for a converse to the Herbst argument from logarithmic Sobolev inequalities to concentration inequalities in spaces with non-negative curvature, providing thus a new, functional, proof of E. Milman's recent results.
  • Artem Pulemotov (University of Chicago)

    Title: The heat equation and the Ricci flow.

    Abstract:  The first part of the talk is devoted to gradient estimates for the heat equation on a manifold M with a fixed Riemannian metric. We will consider the case where M is complete with M=Ř as well as the case where M is compact with M≠Ř. Our attention will be centered around Li-Yau-type inequalities. The second part of the talk focuses on the situation where the Riemannian metric on M evolves under the Ricci flow. We will describe the known options for imposing the boundary conditions on the flow when M≠Ř. We will then outline the relevant existence theorems. Our ultimate goal will be to discuss Li-Yau-type inequalities for the heat equation on M with respect to the evolving metric. The talk is partially based on my joint work with Mihai Bailesteanu and Xiaodong Cao.
  • Michael Roeckner (University of Bielefeld, Germany)

    Title:  Fokker Planck equations on Hilbert spaces.

    Abstract:  We consider a stochastic differential equation in Hilbert space with time dependent coefficients for which no general 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 informally given by the stochastic differential equation.
  • Ambar Sengupta (Louisiana State University)

    Title: Free Yang-Mills on the Plane
    Abstract:  Quantum Yang-Mills theory on the plane for the gauge group U(N)  has a meaningful large-N limit. We will describe this limit in terms of notions from free probability theory.

  • Dan Stroock (MIT)

    Title: (Tentative) An interesting but often ignored theorem of Len Gross
    Abstract:  Many years ago, Gross gave a definitive formulation of the fact that a Hilbert space does not determine the Banach space in the abstract Wiener space for which it is the Cameron-Martin space.  In this talk, I will recall Gross's theorem and show how it facilitates the proof of a couple of important results.

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