Oxford-Man Institute

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

Abstract: When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Föllmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Abstract: We study measures on half planar maps that satisfy a natural domain Markov property. I will discuss their classification and some of their geometric properties. Joint work with Gourab Ray.

Abstract: We consider the inverse problem of recovering u from a noisy, indirect observation We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0,together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available. We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0). The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A

We consider the fractional Laplacian perturbed by the gradient operator b(x)\nabla for various classes of vector fields b. We construct end estimate the corresponding semigroup.

I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.

I will report on joint work in progress with David Aldous, concerning a curious random metric space on the plane which can be constructed with the help of an improper Poisson line process.