We will talk about some stochastic parabolic and hyperbolic partial differential equations (SPDEs), which arise naturally in the context of Liouville quantum gravity. These dynamics are proposed to preserve the Liouville measure, which has been constructed recently in the series of works by David-Kupiainen-Rhodes-Vargas. We construct global solutions to these equations under some conditions and then show the invariance of the Liouville measure under the resulting dynamics. As a by-product, we also answer an open problem proposed by Sun-Tzvetkov recently.
 

Stochastic processes subject to weak noise often show a metastable
behaviour, meaning that they converge to equilibrium extremely slowly;
typically, the convergence time is exponentially large in the inverse
of the variance of the noise (Arrhenius law).
  
In the case of finite-dimensional Ito stochastic differential
equations, the large-deviation theory developed in the 1970s by
Freidlin and Wentzell allows to prove such Arrhenius laws and compute
their exponent. Sharper asymptotics for relaxation times, including the
prefactor of the exponential term (Eyring–Kramers laws) are known, for
instance, if the stochastic differential equation involves a gradient
drift term and homogeneous noise. One approach that has been very
successful in proving Eyring–Kramers laws, developed by Bovier,
Eckhoff, Gayrard and Klein around 2005, relies on potential theory.
  
I will describe Eyring–Kramers laws for some parabolic stochastic PDEs
such as the Allen–Cahn equation on the torus. In dimension 1, an
Arrhenius law was obtained in the 1980s by Faris and Jona-Lasinio,
using a large-deviation principle. The potential-theoretic approach
allows us to compute the prefactor, which turns out to involve a
Fredholm determinant. In dimensions 2 and 3, the equation needs to be
renormalized, which turns the Fredholm determinant into a
Carleman–Fredholm determinant.
  
Based on joint work with Barbara Gentz (Bielefeld), and with Ajay
Chandra (Imperial College), Giacomo Di Gesù (Vienna) and Hendrik Weber
(Warwick). 

References: 
https://dx.doi.org/10.1214/EJP.v18-1802
https://dx.doi.org/10.1214/17-EJP60

We model the trading strategy of an investor who spoofs the limit order book (LOB) to increase the revenue obtained from selling a position in a security. The strategy employs, in addition to sell limit orders (LOs) and sell market orders (MOs), a large number of spoof buy LOs to manipulate the volume imbalance of the LOB. Spoofing is illegal, so the strategy trades off the gains that originate from spoofing against the expected financial losses due to a fine imposed by the financial authorities. As the expected value of the fine increases, the investor relies less on spoofing, and if the expected fine is large enough, it is optimal for the investor not too spoof the LOB because the fine outweighs the benefits from spoofing. The arrival rate of buy MOs increases because other traders believe that the spoofed buy-heavy LOB shows the true supply of liquidity and interpret this imbalance as an upward pressure in prices. When the fine is low, our results show that spoofing considerably increases the revenues from liquidating a position. The profit of the spoof strategy is higher than that of a no-spoof strategy for two reasons. First, the investor employs fewer MOs to draw the inventory to zero and benefits from roundtrip trades, which stem from spoof buy LOs that are ‘inadvertently’ filled and subsequently unwound with sell LOs. Second, the midprice trends upward when the book is buy-heavy, therefore, as time evolves, the spoofer sells the asset at better prices (on average).

The Aldous diffusion is a conjectured Markov process on the
space of real trees that is the continuum analogue of discrete Markov
chains on binary trees. We construct this conjectured process via a
consistent system of stationary evolutions of binary trees with k
labelled leaves and edges decorated with diffusions on a space of
interval partitions constructed in previous work by the same authors.
This pathwise construction allows us to study and compute path
properties of the Aldous diffusion including evolutions of projected
masses and distances between branch points. A key part of proving the
consistency of the projective system is Rogers and Pitman’s notion of
intertwining. This is joint work with Noah Forman, Soumik Pal and
Douglas Rizzolo.                            

 I will talk about optimal estimates for stochastic integrals
in the case when both rough paths and martingales play a role.

This is an ongoing joint work with Peter Friz (TU Berlin).

We derive generalized lower Ricci bounds in terms of signed measures. And we prove associated gradient estimates for the heat flow with Neumann boundary conditions on domains of metric measure spaces obtained through „convexification“ of the domains by means of subtle time changes. This improves upon previous results both in the case of non-convex domains and in the case of convex domains.
 

"We consider a spatial Lambda-Fleming-Viot process, a model in mathematical biology, with a randomly chosen (rough) selection field. We study the scaling limit of this process in different regimes. This leads to the analysis of semi-discrete approximations of singular SPDEs, in particular the Parabolic Anderson Model and allows to extend previous results to weakly nonlinear cases. The subject presented is based on joint works with Aleksander Klimek and Nicolas Perkowski."

In this talk I will introduce a continuous wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as for discrete critical wetting models. I will also discuss the corresponding gradient dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.

We discuss the models of random geometry that are derived
from scaling limits of large graphs embedded in the sphere and
chosen uniformly at random in a suitable class. The case of
quadrangulations with a boundary leads to the so-called
Brownian disk, which has been studied in a number of recent works.
We present a new construction of the Brownian
disk from excursion theory for Brownian motion indexed
by the Brownian tree. We also explain how the structure
of connected components of the Brownian disk above a
given height gives rise to a remarkable connection with
growth-fragmentation processes.

We exhibit a new martingale coupling between two probability measures $\mu$ and $\nu$ in convex order on the real line. This coupling is explicit in terms of the integrals of the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. The integral of $|y-x|$ with respect to this coupling is smaller than twice the Wasserstein distance with index one between $\mu$ and $\nu$. When the comonotonous coupling between $\mu$ and $\nu$ is given by a map $T$, it minimizes the integral of $|y-T(x)|$ among all martingales coupling.

(joint work with William Margheriti)