Past Stochastic Analysis Seminar

13 June 2016
14:15
ANTOINE DAHLQVIST
Abstract

This talk will be about  Lévy processes on compact groups - discrete or continuous - and  two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by  reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments.     There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension.  I shall first sketch this correspondance for  Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior  of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops.  We characterize among them the master field defined by Thierry Lévy by a continuity property.   This is  a joint work with Guillaume Cébron and Franck Gabriel.

  • Stochastic Analysis Seminar
6 June 2016
15:45
Abstract

Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.

The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients

  • Stochastic Analysis Seminar
6 June 2016
14:15
Abstract

We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.

 

  • Stochastic Analysis Seminar
23 May 2016
15:45
KONSTANTIN IZYUROV
Abstract

The planar Ising model is one of the simplest and most studied models in Statistical Mechanics. On one hand, it has a rich and interesting phase transition behaviour. On the other hand, it is "solvable" enough to allow for many rigorous and exact results. This, in particular, makes it one of the prime examples in Conformal Field Theory (CFT). In this talk, I will review my joint work with C. Hongler and D. Chelkak on the scaling limits of correlations in the planar Ising model at criticality. We prove that these limits exist, are conformally covariant and given by explicit formulae consistent with the CFT predictions. This may be viewed as a step towards a rigorous understanding of CFT in the case of the Ising model.TBC

  • Stochastic Analysis Seminar
23 May 2016
14:15
NINA GANTERT
Abstract

We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model:It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle.These invariant measures are often called steady states.

The Einstein relation follows, at least for dimensions three and larger, from an expansion of the steady states as a function of the bias.

The talk is gase on joint work with Jan Nagel and Xiaoqin Guo

 

  • Stochastic Analysis Seminar
16 May 2016
14:15
Abstract

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.

  • Stochastic Analysis Seminar
9 May 2016
14:15
Abstract

The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$  In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.

  • Stochastic Analysis Seminar
7 March 2016
15:45
David Proemel
Abstract

Using Vovk's game-theoretic approach to mathematical finance and probability, it is possible to obtain new results in both areas.We first prove that one can make an arbitrarily large profit by investing in those one-dimensional paths which do not possess a local time of finite p-variation.  Additionally, we provide pathwise Tanaka formulas suitable for our local times and for absolutely continuous functions with sufficient regular derivatives. In the second part we derive a model-independent super-replication theorem in continuous time. Our result covers a broad range of exotic derivatives, including look-back options, discretely monitored Asian options, and options on realized variance.
 This talk is based on joint works with M. Beiglböck, A.M.G. Cox, M. Huesmann and N. Perkowski.


 

  • Stochastic Analysis Seminar
7 March 2016
14:15
Joscha Diehl
Abstract

 

We show how the theories of paracontrolled distributions and regularity structures can be implemented on manifolds, to solve singular SPDEs like the parabolic Anderson model.

This is ongoing work with Bruce Driver (UCSD) and Antoine Dahlqvist (Cambridge)

 

 

  • Stochastic Analysis Seminar

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