Institut Henri Poincaré

Paris, FRANCE

A two days workshop to explore the connections of rough path theory with other fields of mathematics and theoretical physics including stochastic analysis, ergodicity of random dynamical systems, numerical integration methods, Hopf algebras and renormalization in quantum field theory.

#### Invited speakers

- Thomas Cass
- Aurelien Deya
- Peter Friz
- Loďc Foissy
- Martin Hairer
- Arnulf Jentzen
- József Lörinczi
- Jacques Magnen
- Vincent Rivasseau
- Dominique Manchon
- David Nualart
- Josef Teichmann

#### Informations

For further informations please contact Massimiliano Gubinelli (name.surname@dauphine.fr)

Supported by: ANR (project ECRU), CNRS, Institut Henri Poincaré.

Poster (pdf)

#### Program

**June 10**

9:30 starting coffee

10:00 Massimilano Gubinelli

11:00 Thomas Cass

12:00 Aurelien Deya

13:00 lunch

14:30 Peter Friz

15:30 Josef Teichmann

16:30 coffee pause

17:00 Arnulf Jentzen

18:00 closing

**June 11**

9:30 Martin Hairer

10:30 coffee pause

11:00 Dominique Manchon

12:00 Loic Foissy

13:00 lunch

14:30 Vincent Rivasseau et Jacques Magnen

15:30 Jozsef Lorinczi

16:30 coffee pause

17:00 David Nualart

18:00 closing

(Talks are 45 min long + 15 minutes for questions, discussion and a small pause.)

#### Abstracts

Thomas Cass (Oxford University)

Evolving communities with individual preferences.

We consider a community of interacting individuals, each individual having preferences described by some probability measure on rough paths. For certain types of interaction we consider the problem of existence and uniqueness of some forward evolution, which accounts for the individuals preference, and correctly models the interaction with the aggregate behaviour of the community. The evolution of the population need not be governed by any over-arching PDE, but in the case where it is one can match the standard non-linear parabolic PDEs of McKean-Vlasov type with specific examples of communities. Rough paths continuity statements allows for straight forward analysis of propagation of chaos phenomena and large deviations.

Aurelien Deya (Nancy)

Non-linear rough heat equation

We define and solve a heat equation perturbed by an irregular term which may involve the unknown process in a non-linear way. The equation is interpreted in its mild form, and the noisy input is assumed to be generated by a finite-dimensional valued process that allows the construction of a rough path. The interpretation is based on a combination of the standard rough path approach with classical properties of the heat semigroup.

Peter Friz (Berlin)

Applications of rough path theory to stochastic analysis

We will report on some recent applications of rough path theory to stochastic analysis. In particular, we discuss applications of rough path theory to stochastic partial differential equations in viscosity sense.

Loďc Foissy (Université de Reims)

Algčbres de Hopf et chemins rugueux

Nous allons présenter quelques algčbres de Hopf (algčbres de battage, algčbre d'arbres de Connes et Kreimer...) et expliquer comment ces objets peuvent permettre la construction de chemins rugueux.

Martin Hairer (Warwick University and Courant Institute)

Ergodic theory of non-Markovian stochastic processes

We consider evolution equations driven by a noise that is not white in time, so that the resulting process does not have the Markov property. We show that there is an analogue in this setting to the usual Doob-Khashminski ergodicity criterion, provided that the driving noise satisfies a certain "quasi-Markov" property. This can be verified in many cases, including SDEs driven by fractional Brownian motion and thus having long-range memory.

Arnulf Jenzen (Bielefeld University)

Taylor Expansions and Numerical Approximations for Stochastic Partial Differential Equations

In this talk Taylor expansions for the solution process of a stochastic partial differential equation (SPDE) of evolutionary type are presented. Two numerical schemes for SPDEs with additive and non-additive noise respectively are derived on the basis of these Taylor expansions. The key advantage of the two numerical methods is that they break the computational complexity (number of computational operations and random variables needed to the compute the scheme) in comparison to previously considered algorithms for simulating SPDEs with additive space-time white and non-additive trace class noise respectively. The results in this talk are based on joint works with Peter E. Kloeden and Michael Roeckner.

József Lörinczi (Loughborough University)

Gibbs measures for some rough functionals

I present some examples of random processes obtained by Feynman-Kac-type formulae derived to study spectral and ground state properties of some operators originally formulated in quantum theory. The "right hand side" of these formulae gives rise to a measure on path space which can be viewed as a Gibbs measure. Rough paths will be used to define Gibbs specifications pathwise and obtain limits. I discuss cases involving Brownian rough paths, and cases involving stable processes.

Jacques Magnen and Vincent Rivasseau (LPT, Universite Paris Sud)

Constructive field theory

1) The loop vertex expansion is a recent substitute for cluster expansions
which allows to compute connected functions of statistical mechanics or
quantum field theory models without having to divide space into a
(non-canonical) lattice.

2) We introduce an interacting theory for fractional Brownian motion
with Hurst index α ∈ (1/8,1/4), such that the natural
Lévy
area converges in law to a finite Lévy area over fractional Brownian
motion in the ultraviolet limit. The result is obtained using the tools of
constructive field theory, including
renormalization and a cluster expansion.

Dominique Manchon (Université de Clermont-Ferrand)

Connected Hopf algebras and renormalization

We define the Birkhoff decomposition of characters for any connected (graded or filtered) Hopf algebra, and review some applications in quantum physics (Feynman diagrams) and number theory (renormalization of multiple zeta values). An account of the Connes-Kreimer Beta function will be also given.

David Nualart (Kansas University)

Rough paths above the fractional Brownian motion using Volterra's representation

In this talk we will present the construction of a rough path above a multidimensional fractional Brownian motion with a Hurst parameter H in (0,1/2), by means of its representation as a Volterra Gaussian process. The iterated integrals required in the rough paths analysis are expressed explicitly in terms of iterated Stratonovich stochastic integrals involving the kernel of the Volterra representation. This is a joint work with Samy Tindel.

Josef Teichmann (ETH, Zurich)

Another approach to some rough and stochastic partial differential equations

In this note we introduce an alternative approach to some rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and -- for the sake of simiplicity -- finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory we are able to construct unique solutions of the respective R- and SPDEs. An application from Malliavin calculus is presented, where the existence of absolutely continuous densities for projections of solutions of certain SPDEs is proved.