Summer School on Stochastic Analysis

Abstracts of Talks

Dirk Erhard (Tuesday 10:55)

Weak coupling limit of the Anisotropic KPZ equation

In this talk I will talk about the so called 2D anisotropic KPZ equation, which is a non-linear singular stochastic PDE in the critical regime. As a consequence recent theories like the theory of regularity structures or paracontrolled distributions do not apply. In this talk I will discuss the regime in which the equation is approximated by a smooth sequence of equations with the feature that the non-linearity is multiplied by a constant going to zero. I will argue that even though the non-linearity goes to zero it produces in the limit an additional laplacian and an additional noise term. This talk is based on joint work with Giuseppe Cannizzaro and Fabio Toninelli.

Link to the article: https://arxiv.org/abs/2108.09046

Chiranjib Mukherjee  (Tuesday 14:30)

Directed polymers, SPDEs  and multiplicative chaos 

Weijun Xu (Wednesday 10:55)

Periodic homogenisation problems for some singular SPDEs

Robert Neel (Wednesday 14:30)–Mini Symposium in stochastic differential geometry

Title: Log-derivatives of the Riemannian heat kernel in small time

Abstract: We show that logarithmic derivatives of the heat kernel, for
small time, can be localized under the same conditions as for the heat
kernel itself. In particular, the well-known bounds for compact
Riemannian manifolds, recently extended to the complete case by
Chen-Li-Wu, can be further extended to appropriate subsets of
incomplete Riemannian manifolds. Moreover, for two fixed points, we
show that the leading order of the log-derivatives of the heat kernel
is closely related to the diffusion bridge between the points and the
geometry of the associated minimal geodesics.

This talk is based on joint work with Ludovic Sacchelli.

 Erlend Grong  (Wednesday 3:15)–Mini Symposium in stochastic differential geometry

Title: Sub-Riemannian geometry, most probable paths and transformations.

Abstract: Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack pluss to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds.
Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications.

The results are part of joint work with Stefan Sommer (Copenhagen, Denmark)

Seiichiro Kusuoka (Wednesday 10:55)

Title : The three-dimensional polymer measure with selfinteractions and the stochastic quantization

Abstract : In this talk we consider the polymer measure with selfinteractions, which is called the Edward model. The two-dimensional case had been studied around 2000, and the polymer measure and the associated Dirichlet form were constructed. Here, we consider the three-dimensional case, which requires harder calculation than the two-dimensional case.

It has been known that, to construct the measure we need the renormalization, because of the singularity of the interactions.

Moreover, the renormalization constant coincides with that of the $\Phi ^4$-quantum field measure. In this talk, I will explain the strategies to construct the measure and the Dirichlet form.

This is an ongoing jointwork with Sergio Albeverio, Makoto Nakashima and Song Liang.

 Sarah-Jean Meyer (PhD student Talks) Thursday 3:25

“A forward backward approach to stochastic quantisation”
Abstract:
We propose a forward-backward SDE as a way to stochastically quantise Euclidean quantum field theories. Special emphasis will be put on the synergy with the flow equation method, and how approximate solutions to the Polchinsky flow equation naturally fit into this FBSDE framework. In the case of the sine-Gordon EQFT, our approach can be successfully implemented for a detailed and simple analysis of the first two regions of collapse.
This talk is based on joint work with Massimiliano Gubinelli.

Shigeki Aida (Friday 10:55)

Title : An approach to asymptotic error distributions of rough differential equations

Abstract : We study asymptotic error distributions of rough differential equations driven by the fractional Brownian motion whose Hurst parameter H which satisfies 1/3<H<1/2 for several approximation schemes.

To this end, we introduce an interpolation process between the solution and the approximate solution.

Also we study limit theorems of weighted sum processes of Wiener chaos of order 2.

Our proof is based on the fourth moment theorem for Wiener chaos of finite order, estimates of multidimensional Young integrals and integration by parts formula in the Malliavin calculus.

This is a joint work with Nobuaki Naganuma (Kumamoto University).

Kevin Yang (Friday 14:30)

Title: Boltzmann-Gibbs principle for non-stationary interacting particle systems

Abstract : The goal for this talk is to present the key aspects for proving Boltzmann-Gibbs principles, with an eye towards models without explicit invariant measures. These are the central estimates needed to derive scaling limits for fluctuations in interacting particle systems, yielding SPDEs such as KPZ. To be concrete, we focus on a speed-change exclusion process in one spatial dimension (which can be thought of as a regularized SPDE in its own right). We present further results achieved by the method and pose several open problems at the end. 

Immanuel Zachhuber (Friday 15:15-15:45)

Energy solutions of 3d multiplicative stochastic wave

abstract: I will give a rather simple proof of global well-posedness of the 3d cubic multiplicative stochastic wave equation which is simply a cubic nonlinear wave equation with a (spatial) white noise potential. This result is somewhat surprising given that the wave propagator naively regularises by 1 and the noise has regularity just below -3/2. Based on joint work with M. Gubinelli and B. Ugurcan.

Martin Grothaus  (Monday  August 14, 16:50).

Alexandra Neamtu (Wendesday  August 16th 12-12:30). Finite – time Lyapunov exponents for SPDEs