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Joint CNRS-Imperial Workshop on Stochastic Analysis and Applications

The Stochastic Analysis Group at Imperial College London is organising a joint CNRS-Imperial Workshop under the auspices of the Abraham de Moivre International Research Laboratory. The purpose of the workshop is to provide continuing support for the interaction with our CNRS colleagues. The speakers include:  Mihai Gradinaru, Greg Pavliotis, Jean-Francois Chassagneux, Gabriel Stoltz, Eyal Neumann, Dan Crisan, Nizar Touzi, Greg Pavliotis, Martin Hairer, Fabien Panloup, Xue-Mei Li, Tony Lelievre.


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Preliminary programme

See below programme for talk titles and abstracts.

Thursday 7 July

Chair: Dan Crisan

  • 09.15-09.30 Welcome
  • 09.30-09.40 Richard Craster
  • 09.40-10.20 Mihai Gradinaru
  • 10.20-11.00 Greg Pavliotis
  • 11.00-11.20 Coffee break
  • 11.20-12.00 Jean-Francois Chassagneux
  • 12.00-14.00 Lunch

Chair: Xue-Mei Li

  • 14.00-14.40 Gabriel Stoltz
  • 14.40-15.20 Eyal Neumann
  • 15.20-15.40 Coffee break
  • 15.40-16.20 Dan Crisan 
  • 16.20-17.00 Nizar Touzi (remote talk)
  • 18.15 Dinner

Friday 8 July

Chair: Greg Pavliotis

  • 09.00-09.40 Martin Hairer
  • 09.40-10.20 Fabien Panloup 
  • 10.20-10.40 Coffee break
  • 10.40-11.20 Xue-Mei Li


Mihai Gradinaru

Title: Lévy driven non-linear Langevin type equations

Abstract: We will consider a one-dimensional kinetic stochastic model driven by a stable Lévy process, with a non-linear time-inhomogeneous drift. More precisely, a process $(V,X)$ is considered, where $X$ is the position of the particle and its velocity $V$ is the solution of a stochastic differential equation with a drift of the form $t^{-\beta}F(V)$. The behaviour of the process $(V,X)$ will be described when the noise is small or in large time, with a fixed noise.

Greg Pavliotis

Title: Interacting multiagent models in the social sciences: collective behaviour, phase transitions and inference

Abstract: Several mathematical models in the Social Sciences that have been developed in recent years are based on interacting multiagent systems. Often, such systems can be described using interacting diffusion processes. Examples include models for sychronization (the noisy Kuramoto model), systemic risk (the Desai-Zwanzig model), and opinion formation (the noisy Hagselmann-Krause model). For such models, the emergence of collective behaviour, e.g. synchronization and consensus formation, can be interpreted as a disorder-order phase transition between, e.g. a uniform and a clustered/localized state. Such a phase transition can be studied rigorously in the mean field/thermodynamic limit that is described by the McKean-Vlasov equation. In this talk we will present recent results on the rigorous analysis of phase transitions for such systems and on their impact on their dynamical properties. Furthermore, we present inference methodologies, for learning parameters in the mean field model from observations of sufficiently long single trajectories of the interacting particle system.  The effect of phase transitions on this inference problem is elucidated and the development of diagnostic tools for predicting phase transitions is discussed.

Jean-François Chassagneux 

Title: Backward SDEs reflected in non-convex domains 

Abstract: In this talk, I will present some new existence and uniqueness results for classes of multi-dimensional reflected BSDEs. The study of reflected BSDEs is motivated by applications in stochastic control in particular to switching problems. I will first recall some resent results in this direction. I will then introduce reflected BSDEs in non-convex domains and explain the main difficulty encountered in their study. This talk is based on joint works with C. Bénézet, S. Nadtochiy and A. Richou. 

Gabriel Stoltz

Title: Linear response of nonequilibrium stochastic dynamics

Abstract: Transport coefficients relate an external forcing applied to a system to its response in terms of a current. I will first recall, in a hopefully pedagogical way, how these properties are defined for typical ergodic stochastic dynamics such as Langevin dynamics. I will then present approaches to estimate them using either linear response theory and Green-Kubo formulas. I will provide elements of numerical analysis, both to characterize the bias arising from finite time integration and the use of finite timesteps, and to quantify the statistical error in terms of variance. I will in particular hint it a newly developed method based on Girsanov’s change-of-measure theory in the linear response regime, as introduced in works on sensitivity analysis (joint work with Petr Plechac and Ting Wang).

Eyal Neumann

Title: The Effective Radius of Self Repelling Elastic Manifolds

Abstract: Click on this link to read Eyal Neumann’s abstract

Dan Crisan

Title: Solution properties of an incompressible Stochastic Euler system

Abstract: I will present some results for well-posedness of the 3D and 2D Euler equation for the incompressible flow of an ideal fluid perturbed by an additional stochastic divergence-free, Lie-advecting vector field. In 3D, the equation is locally well-posed in regular spaces. A Beale–Kato–Majda type criterion characterizes the blow-up time. In 2D, the eqaution has a unique global strong solution and the initial smoothness of the solution is preserved. I will also present a rough path version of the model.

This is joint work with Oana Lang, Franco Flandoli, Darryl Holm, James Leahy and Torstein Nilssen and is based on the papers:

[1] O Lang, D Crisan, Well-posedness for a stochastic 2D Euler equation with transport noise, Stochastics and Partial Differential Equations: Analysis and Computations, 1-48, 2022.

[2] D. Crisan, F Flandoli, DD Holm, Solution properties of a 3D stochastic Euler fluid equation, Journal of Nonlinear Science 29 (3), 813-870, 2019.

[3] D Crisan, DD Holm, JM Leahy, T Nilssen, Solution properties of the incompressible Euler system with rough path advection, arXiv preprint arXiv:2104.14933

Nizar Touzi

Title: mean field optimal stopping

Abstract: We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process. Using the dynamic programming approach, we provide a characterization of the value function as the unique viscosity solution of the corresponding dynamic programming equation on the Wasserstein space. Under additional smoothness condition, we provide a verification result which characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. Finally, we the convergence of the the finite population multiple optimal stopping problem to the corresponding mean field optimal stopping limit. These results of propagation of chaos are proved by adapting the Barles-Souganidis monotonic scheme method to the present context.

Martin Hairer

Ergodicity of the projective process for linear SPDEs

Abstract: TBC

Fabien Panloup

Title : On the (non)-stationary density of fractional driven SDEs

Abstract : “I will talk about several properties of stationary solutions of fractional SDEs. I will first recall some seminal results by Hairer (2005) on the construction of stationary solutions and associated ergodic results. Then, I will focus on a recent paper with Xue-Mei Li and Julian Sieber where we prove smoothness and Gaussian bounds for the density of the related invariant distribution (under appropriate assumptions) in the additive case. The proofs are based on a novel representation of the stationary density in terms of a Wiener-Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the non-stationary density, which extend previously known results in the additive setting. Avoiding any use of Malliavin calculus in our arguments, our results are obtained under minimal regularity requirements.

Xue-Mei Li

Differential Equations in Fast Moving Environment with auto-correlation

Abstract: We study stochastic SDEs  in a fast oscillating random vector field and driven by auto-correlated fractional Brownian noise, finding a continuum of effective  motions connecting the effective motions of  stochastic averaging of classical SDEs to that of homogenization, which were traditionally treated separately,  with the same tools. As by products, we obtain also  new results in the Markovian SDE setting. This is joint work with Martin Hairer.

[1] Averaging dynamics driven by fractional Brownian motion.  (2020) Annals of Probability

[2] Generating diffusions with fractional Brownian motion. To appear: Communications in Mathematical Physics.

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