Midlands Probability Theory Seminars

The department is a major host for the regional Midlands Probability Theory Seminars.

The next MPTS session will take place on Wednesday, February 15th, 2012. The schedule for the afternoon session is as follows:

Please note that all talks will be held in B3.02.

2.00-3.00 Peter Mörters (Bath) Shifting Brownian motion.

Let $B=(B_t)_{t\in\R}$ be a two-sided standard Brownian motion. A stopping time $T$ is an unbiased shift if $(B_{T+t}-B_T)_{t\in\R}$ is a Brownian motion independent of~$B_T$. We solve the Skorokhod embedding problem for unbiased shifts by constructing, for any probability distribution~$\nu$ on the reals, an unbiased shift such that $B_T$ has distribution~$\nu$. We show that our solution has optimal moment properties. The talk is based on joint work with Gunter Last (Karlsruhe) and Hermann Thorisson (Reykjavik).

3.00-3.30 Tea and Coffee break

3.30-4.30 Philippe Carmona (Nantes) The discrete-time parabolic Anderson model with heavy-tailed potential.

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths of the model. This is a joint work with Francesco Caravenna and Nicolas Petrelis.

4.30-5.30 Naotaka Kajino (Bielefeld) Weyl's Laplacian eigenvalue asymptotics for the measurable Riemannian structure on the Sierpinski gasket.

On the Sierpinski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the ''gradient vector fields" of functions, the ''Riemannian volume measure" and the ''geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bound for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal. 36 (2012), 67--115]. In this talk, Weyl's Laplacian eigenvalue asymptotics is presented for this case. The correct scaling order for the asymptotics of the eigenvalues is given by the Hausdorff dimension d of the gasket with respect to the ''geodesic metric", and in the limit of the eigenvalue asymptotics we obtain a constant multiple of the d-dimensional Hausdorff measure. Moreover, we will also see that this Hausdorff measure is Ahlfors regular with respect to the ''geodesic metric" but that it is singular to the ''Riemannian volume measure".


For more information on the Midlands Probability Theory Seminars please contact Dr Larbi Alili who his currently running the seminars.


All are welcome!