Joint CNRS-Imperi al Workshop on Stochastic Analysis and Applications

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

Registration is now closed.< /p>\n

See below prog ramme for talk titles and abstracts.

\n**Chair: Dan Crisan**

- \n<
li class=\"MsoNormal\">
**09.30-09.40**Richard Craster \n**09.40-10.20**Mihai Gradinaru \n**10.20-11. 00**Greg Pavliotis \n**11.00-11.20**Coffee break \n**11.20-12.00**Jean-Fra ncois Chassagneux \n**12.00-14.00**Lun ch \n

**Chair: Xue-Mei Li**

**Chair: Greg Pavliotis**

- \n
**09.00-09.40**Martin Hairer \n**09.40-10.20**Fabien Panloup \
n**10.20-10.40**Coffee break \n**10.40-11.20**Xue-Mei Li \n

**Abstract**: We will cons
ider a one-dimensional kinetic stochastic model driven by a stable Lévy p
rocess\, with a non-linear time-inhomogeneous drift. More precisely\, a pr
ocess $(V\,X)$ is considered\, where $X$ is the position of the particle a
nd 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.

**Abstract:** Sever
al mathematical models in the Social Sciences that have been developed in
recent years are based on interacting multiagent systems. Often\, such sys
tems can be described using interacting diffusion processes. Examples incl
ude models for sychronization (the noisy Kuramoto model)\, systemic risk (
the Desai-Zwanzig model)\, and opinion formation (the noisy Hagselmann-Kra
use model). For such models\, the emergence of collective behaviour\, e.g.
synchronization and consensus formation\, can be interpreted as a disorde
r-order phase transition between\, e.g. a uniform and a clustered/localize
d state. Such a phase transition can be studied rigorously in the mean fie
ld/thermodynamic limit that is described by the McKean-Vlasov equation. In
this talk we will present recent results on the rigorous analysis of phas
e transitions for such systems and on their impact on their dynamical prop
erties. Furthermore\, we present inference methodologies\, for learning pa
rameters in the mean field model from observations of sufficiently long si
ngle trajectories of the interacting particle system. The effect of phas
e transitions on this inference problem is elucidated and the development
of diagnostic tools for predicting phase transitions is discussed.<
/p>\n

**Abstract:** In this talk\, I will pr
esent some new existence and uniqueness results for classes of multi-dimen
sional reflected BSDEs. The study of reflected BSDEs is motivated by appli
cations 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 enco
untered in their study. This talk is based on joint works with C. Bénéze
t\, S. Nadtochiy and A. Richou.

**Abstract: **Transport coefficients relate an ext
ernal 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 dyna
mics. I will then present approaches to estimate them using either linear
response theory and Green-Kubo formulas. I will provide elements of numeri
cal analysis\, both to characterize the bias arising from finite time inte
gration 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 respo
nse regime\, as introduced in works on sensitivity analysis (joint work wi
th Petr Plechac and Ting Wang).

**Abstract: **<
a href=\"https://imperialcollegelondon.box.com/s/t9s9kjnnvvy1e3pgblgqep9ia
0bmmocq\" target=\"_blank\" rel=\"noopener\">Click on this link to read Ey
al Neumann’s abstract

**Abstract:**
I will present some results for well-posedness of the 3D and 2D Euler equ
ation for the incompressible flow of an ideal fluid perturbed by an additi
onal stochastic divergence-free\, Lie-advecting vector field. In 3D\, the
equation is locally well-posed in regular spaces. A Beale–Kato–Majda t
ype criterion characterizes the blow-up time. In 2D\, the eqaution has a u
nique 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 :

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

\n< p class=\"MsoNormal\">[2] D. Crisan\, F Flandoli\, DD Holm\, Solution prop erties of a 3D stochastic Euler fluid equation\, Journal of Nonlinear Scie nce 29 (3)\, 813-870\, 2019.\n[3] D Crisan\, DD Holm\, JM Leahy\, T Nilssen\, Solution properties of the incompressible E uler system with rough path advection\, arXiv preprint arXiv:2104.14933

\n< b>Abstract: We study the optimal stopping problem o f 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 proble m 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 functio n as the unique viscosity solution of the corresponding dynamic programmin g equation on the Wasserstein space. Under additional smoothness condition \, we provide a verification result which characterizes the nature of opti mal stopping policies\, highlighting the crucial need to randomized stoppi ng. Finally\, we the convergence of the the finite population multiple opt imal stopping problem to the corresponding mean field optimal stopping lim it. These results of propagation of chaos are proved by adapting the Barle s-Souganidis monotonic scheme method to the present context.

\n< h3 class=\"MsoPlainText\">Martin Hairer\n**Abstract<
/b>: TBC**

**Abstract **: “I wi
ll talk about several properties of stationary solutions of fractional SDE
s. I will first recall some seminal results by Hairer (2005) on the constr
uction of stationary solutions and associated ergodic results. Then\, I wi
ll focus on a recent paper with Xue-Mei Li and Julian Sieber where we prov
e smoothness and Gaussian bounds for the density of the related invariant
distribution (under appropriate assumptions) in the additive case. The pro
ofs 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-station
ary density\, which extend previously known results in the additive settin
g. Avoiding any use of Malliavin calculus in our arguments\, our results a
re obtained under minimal regularity requirements.

**Abstract**: We study stochastic SDEs in a fast o
scillating random vector field and driven by auto-correlated fractional Br
ownian noise\, finding a continuum of effective motions con
necting the effective motions of stochastic averaging of cl
assical SDEs to that of homogenization\, which were traditionally treated
separately\, with the same tools. As by products\, we obtai
n also new results in the Markovian SDE setting. This is jo
int work with Martin Hairer.

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

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

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