Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).
 

For the problem of prediction with expert advice in the adversarial setting with geometric stopping, we compute the exact leading order expansion for the long time behavior of the value function using techniques from stochastic analysis and PDEs. Then, we use this expansion to prove that as conjectured in Gravin, Peres and Sivan the comb strategies are indeed asymptotically optimal for the adversary in the case of 4 experts.
 

We provide a detailed, probabilistic interpretation, based on stochastic calculus, for the variational characterization of conservative diffusion as entropic gradient flux. Jordan, Kinderlehrer, and Otto showed in 1998 that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features. These are valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by taking expectations. As a bonus, we obtain the HWI inequality of Otto and Villani relating relative entropy, Fisher information and Wasserstein distance; and from it the celebrated log-Sobolev, Talagrand and Poincare inequalities of functional analysis. (Joint work with W. Schachermayer and B. Tschiderer, from the University of Vienna.)

 

The deep neural network has achieved impressive results in various applications, and is involved in more and more branches of science. However, there are still few theories supporting its empirical success. In particular, we miss the mathematical tool to explain the advantage of certain structures of the network, and to have quantitive error bounds. In our recent work, we used a regularised relaxed control problem to model the deep neural network.  We managed to characterise its optimal control by the invariant measure of a mean-field Langevin system, which can be approximated by the marginal laws. Through this study we understand the importance of the pooling for the deep nets, and are capable of computing an exponential convergence rate for the (stochastic) gradient descent algorithm.

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle
is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin-Wentzell small noise asymptotics. 
Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. We also study large deviation behavior of  interacting particle systems approximating various types of Feynman-Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes. 

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.