Workshop in Probability and its Applications

9th December 2013 in Mathematical Institute, University of Oxford.

The idea is to bring together people from around Oxford who are interested in probability and related areas for a one day meeting. We hope to make it a regular event with broad coverage of the topic. The first meeting will be on 9th December 2013. The speakers and will be:

• Jean Bertoin (UZH)
• Jean-Michel Bismut (Paris-Sud)
• Ying Hu (Rennes)
• James Norris (Cambridge)
• Balint Toth (Bristol)

The next workshop will be combined with another meeting and will take place from Monday 17 to Thursday 20 March 2014.

Speakers:

• Jean Bertoin (UZH)
• Jean-Michel Bismut (Paris-Sud)
• Ying Hu (Rennes)
• James Norris (Cambridge)
• Balint Toth (Bristol)

Participants:

• Dmitry Belyaev
• Horatio Boedihardjo
• Zdzislaw Brzezniak
• Philippe Charmoy
• George Deligiannidis
• David Elworthy
• Alison Etheridge
• Chunrong Feng
• Xi Geng
• Christina Goldschm
• Robert Griffiths
• Lajos Gyurko
• Ben Hambly
• Allen Hammond
• Yimin Jiang
• Martin Klimmek
• Sean Ledger
• Jiabin Li
• Arnaud Lionnet
• Terry Lyons
• James Martin
• Michael Monoyios
• Hao Ni
• Harald Oberhauser
• Zhongmin Qian
• Shengxin Ran
• Gesine Reinert
• David Steinsaltz
• Pierre Tarres
• Rongming Wang
• Matthias Winkel
• Jianglun Wu
• Yiming Yang
• Danyu Yang
• Yanhong Zhang
• Zichen Zhang
• Huaizhong Zhao

All talks will be in Mathematical Institute, lecture room L2. The seminar room C4 is reserved for discussions.

 11:00 AM Tea and Coffee 11:30 AM Jean-Michel Bismut The hypoelliptic Laplacian 12:30 PM Buffet Lunch 1:30 PM James Norris Fluid limits in infinite dimensions: an example from gas kinetics 2:30 PM Ying Hu Linear BSDEs driven by space-time white noise and application to optimal control of SPDEs 3:30 PM Tea and Coffee 4:00 PM Balint Toth Random Walks in Divergence-Free Random Drift Field: $H_1$ suffices 5:00 PM Jean Bertoin The cut-tree of large recursive trees 6:30 PM Drink reception, followed by workshop dinner

Jean-Michel Bismut: The hypoelliptic Laplacian.

Abstract: If $X$ is a Riemannian manifold, the hypoelliptic Laplacian $L_b$ is an operator acting on the total space $\chi$ of the tangent bundle of $X$, that is supposed to interpolate between the elliptic Laplacian (when $b\to 0$) and the geodesic flow (when $b\to +\infty$). Up to lower order terms, $L_b$ is a weighted sum of the harmonic oscillator along the fibre $TX$ and of the generator of the geodesic flow. The operator $L_b$ is a geometric version of a Fokker-Planck operator. Its probabilistic counterpart is a Langevin process.

In this deformation, there are conserved quantities. In some cases, the full spectrum of the elliptic Laplacian is preserved by the deformation. In the talk, I will explain the underlying analytic and probabilistic aspects of the construction, and some of the results obtained so far.

James Norris: Fluid limits in infinite dimensions: an example from gas kinetics.

Abstract: Kac's N-particle model for velocity exchange by elastic collisions converges for large N to the spatially homogeneous Boltzmann equation. This was proved by Sznitman in the 70's. I will explain a new argument for this convergence, which gives better quantitative control. The approach is a general one based on fluid limits of Markov chains and finite-dimensional approximation of dynamics.

Hu Ying: Linear BSDEs driven by space-time white noise and application to optimal control of SPDEs.

Abstract: Motivated by the study of optimal control problem of stochastic partial differential equations (SPDEs) driven by space-time white noise, we first study linear (infinite-dimensional) backward stochastic differential equations (BSDEs) driven by space-time white noise. Even though the generator of BSDE is linear, but it is not Lipschitz due to the space-time white noise. We use duality arguments and take advantage of estimates for solutions of linear forward SPDEs to overcome the difficulties. Then we apply our result to establish the stochastic maximum principle for optimal control of SPDEs. This is a joint work with Marco Fuhrman and Gianmario Tessitore.

Jean Bertoin: The cut-tree of large recursive trees.

Abstract: Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size $n$, rescaled by the factor $n^{-1}\ln n$, converges in probability as $n\to \infty$ in the sense of Gromov-Hausdorff-Prokhorov, to the unit interval endowed with the usual distance and Lebesgue measure. This enables us to explain and extend some recent results of Kuba and Panholzer on multiple isolation of nodes in random recursive trees.

Balint Toth: Random Walks in Divergence-Free Random Drift Field: $H_1$ suffices.

Abstract: I will present a central limit theorem under diffusive scaling for the displacement of a random walk on ${\mathbb Z}^d$ in stationary divergence-free random drift field, under the $H_1$-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor be stationary and square integrable. This improves the best existing result in Komorowski, Landim, Olla (2012) where stronger integrability conditions are assumed. T he proof relies on the Relaxed Sector Condition of Horvath, Toth, Veto (2012) and is technically considerably simpler than the proofs in Komorowski, Landim, Olla (2012), or Oelschlager (1988) [where a similar result is proved for diffusion in divergence-free random drift field]. The talk is based on joint work with Illes Horvath and Balint Veto, respectively, with Gady Kozma.