Schedule for: 24w5314 - Stochastics and Geometry

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Informal Meet and Greet at BIRS Lounge (PDC 2nd Floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Langevin dynamics is Newton's law for the motion of N particles subject to friction, thermal fluctuations and potential forces. Aside from its relevance in statistical mechanics, its discretizations are used in Markov chain Monte Carlo to draw samples from its explicit, and moldable, stationary distribution by running the system long enough. Because of its ballistic, as opposed to diffusive, behavior, it is believed to have a better rate of convergence to equilibrium when compared to stochastic gradient dynamics (also known as ``overdamped Langevin"). However, the precise mechanisms leading to geometric ergodicity of the system are more nuanced than stochastic gradient dynamics, especially because the SDE is degenerate elliptic and damping only explicitly acts on the momentum directions. This has led to an abundance of research on the topic. The goal of this talk is to give an overview of methods used to establish convergence to equilibrium Langevin dynamics forced by a wide class of potential functions. Particular attention will be paid to both stochastic and functional analytic methods.

We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincaré inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will report on recent progress in understanding a family of Hastings-Levitov-type planar random growth models, identifying fluid limits and fluctuations in the small-particle limit, in a certain subcritical regime. This is joint work with Amanda Turner and Vittoria Silvestri.

Couplings of sub-Riemannian diffusions, of both reflection and parallel type, have attracted interest in recent years. Unlike in the Riemannian case, Markovian (or even co-adapted) couplings are not optimal, even for model cases, such as the Heisenberg group. After reviewing the situation, we describe an approach to improve and extend recent constructions of non-Markovian reflection couplings by Banerjee-Gordina-Mariano and Bénéfice. Moreover, the construction is relatively simple, being based on global symmetries of the underlying spaces, and is reminiscent of Kuwada’s work on maximal reflection couplings on Riemannian manifolds. This talk is based on joint work with Liangbing Luo.

Onsager-Machlup functional measures how likely a stochastic process stays close to a given path. This functional can be viewed as an infinite dimensional analogue of a probability density function. SLE is a family of measures on simple paths in the plane introduced by O. Schramm obtained from the Loewner transform of a multiple of Brownian motion. It is well-known that the Onsager-Machlup functional for Brownian motion is the Dirichlet energy. We show that the Onsager-Machlup of the SLE_$\kappa$ loop measure, for any \(0 < \kappa <4\), is expressed using the Loewner energy and the central charge c(k) of SLE$_\kappa$. Loewner energy is defined as the Dirichlet energy of the Loewner driving function of the loop, but it also has tight links to many other fields of mathematics. Our proof relies on the conformal restriction covariance of SLE, and on a relation between the renormalized Brownian loop measure and Werner measure. This is based on the joint work (arXiv: 2311.00209) with Yilin Wang (IHES).

The Hamilton-Pontryagin principle in deterministic geometric mechanics is a constrained variational principle that incorporates the Legendre transform. Recently, Street and Takao studied stochastic perturbations of deterministic dynamical systems from a variational approach using a stochastic generalization of the Hamilton-Pontryagin principle. Assuming certain symmetry constraints on the perturbing noise, we develop a reduction theory for the stochastic Hamilton-Pontryagin principle on an arbitrary configuration manifold. This generalizes the corresponding reduction theory for deterministic Lagrangians, known as Lagrange-Poincaré reduction. As an example, we study a stochastic extension of the Kaluza-Klein approach to charged particles.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will start with a gentle introduction to the Yang-Mills model. We will then discuss the stochastic Yang-Mills flow, and our construction for a trivial bundle over 2 and 3 dimensional tori. We will also discuss the meaning of “gauge equivalence” and “orbit space” in the singular setting, and show that the flow has the gauge covariance property, yielding a Markov process on the orbit space. The talk is mostly based on joint work with Ajay Chandra, Ilya Chevyrev and Martin Hairer, but will be more of a survey style.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In a recent breakthrough, Bamler established a precompactness and partial regularity theory for the Ricci flow. Loosely speaking, his work generalizes the theory of noncollapsed Gromov-Hausdorff limits of Einstein metrics to the parabolic setting. In a different direction, a notion of weak solutions for the Ricci flow has been proposed a few years earlier by Naber and myself. In the first part of the talk, I will give a general introduction to these ideas. In the second part of my talk, I will explain how to reconcile these two approaches. Specifically, I will describe recent joint work with Choi, where we prove that ever noncollapsed Ricci limit flow, as provided by Bamler's precompactness theorem, is a weak solution in the sense of Naber and myself.

We present a concise introduction to conformally invariant, log-correlated Gaussian random fields on compact Riemannian manifolds of general even dimension uniquely defined through its covariance kernel given as inverse of the Graham-Jenne-Mason-Sparling (GJMS) operator. The corresponding Gaussian Multiplicative Chaos is a generalization to the $n$-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.

In this talk, we establish the exact second-order moment asymptotics for the parabolic Anderson model in the hyperbolic space with a time-independent, regular, isometry-invariant Gaussian potential. Although the solution is defined under hyperbolic geometry, surprisingly it turns out that the fluctuation exponent is determined by an Euclidean variational problem which is insensitive to the underlying geometry. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics. This is based on joint work with Weijun Xu (Peking University).

We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a sub-Riemannian structure induced by a transitive action by a Lie group. The corresponding sub-Laplacian there is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities and show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we show that the constant is independent of the dimension of the underlying spaces in several examples. Based on joint work with M.Gordina.

Aaron Naber (Northwestern) and Robert Haslhofer (Toronto) characterized solutions of the Einstein equation $Rc = \lambda g$ in terms of both sharp gradient estimates for Brownian motion and a Bochner formula on elliptic path space $P\mathcal{M}$. In this talk, we generalize the classical Bochner formula for the heat flow on evolving manifolds $(M, g_t)_{t\in[0,T ]}$ to an infinite-dimensional Bochner formula for martingales on parabolic path space $P\mathcal{M}$ of space-time $M = M \times [0, T ]$. Our new Bochner formula and the inequalities that follow from it are strong enough to characterize solutions of the Ricci flow in terms of Bochner inequalities on parabolic path space, thus proving the parabolic counterpart of previous results in the elliptic setting. Time-permitting, we shall also discuss gradient and Hessian estimates for martingales on parabolic path space, as well as condensed proofs of the prior characterizations of the Ricci flow.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the context of irreducible Dirichlet metric measure spaces, we demonstrate the discreteness of the Laplacian spectrum and the corresponding diffusion's irreducibility in connected open sets, without assuming regularity of the boundary. This general result can be applied to study various questions, including those related to small deviations of the diffusion and generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces. This is joint work with Marco Carfagnini and Masha Gordina.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A Lie group $G$ is said to be uniformly doubling if there is a uniform upper bound for the volume doubling constants of all left-invariant Riemannian metrics on $G$. When this holds for a particular group $G$, it gives rise to uniform bounds on the constants in many important functional inequalities, such as Poincare inequalities and heat kernel estimates. In earlier work with Maria Gordina and Laurent Saloff-Coste, we showed that the special unitary group $SU(2)$ is uniformly doubling. In this talk, I'll discuss our recent and ongoing joint work to extend these results to the unitary group $U(2)$, including some of the difficulties that arise, and the techniques we use to overcome them. In particular, we will see how, despite this being apparently a problem of purely Riemannian geometry, insight from the study of sub-Riemannian geometry plays an essential role.

I will discuss the evolution of roots of (random) polynomials under flows computed from differential operators. The two main examples will be (1) repeated differentiation, in which we study how the roots of a degree-N polynomial move as the number of derivatives varies from 0 to N; and (2) the heat flow, in which we study how the roots of a high-degree polynomial move with respect to the time variable in the heat equation. In the case of polynomials with all real roots, both problems are well understood—and the answers have a surprising connection to random matrix theory. In the case of polynomials with complex roots, both problems are much more complicated but still very interesting. One idea that has emerged recently applies to both problems, if the initial polynomial has an asymptotic smooth density of roots in the plane. In either case, the roots will then tend to evolve in straight lines with constant velocity, with the velocity of each root being determined by the Cauchy transform of the initial density. I will explain this idea and then present rigorous results for random polynomials with independent coefficients. This is joint work with Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The talk will be self-contained and have lots of pictures and animations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study the boundary trace processes of reflected diffusions on uniform domains. We obtain stable-like heat kernel estimates for such a boundary trace process when the diffusion on the underlying ambient space satisfies sub-Gaussian heat kernel estimates. Our arguments rely on new results of independent interest such as sharp two-sided estimates and the volume doubling property of the harmonic measure, the existence of a continuous extension of the Naïm kernel to the topological boundary, and the Doob--Naïm formula identifying the Dirichlet form of the boundary trace process as the pure-jump Dirichlet form whose jump kernel with respect to the harmonic measure is exactly (the continuous extension of) the Naïm kernel. This is joint work with Naotaka Kajino.

I shall consider models of random $N\times N$ unitary matrices motivated by gauge theory when $N$ goes to infinity. I will focus on two models: the Atiyah-Bott-Goldman measure and the Yang-Mills measure on closed surfaces. Both of them can be viewed as natural random morphisms from a group of loops on a surface to the group of $N\times N$ unitary matrices. We will focus on the behaviour of their composition with the standard trace of matrices. After recalling some recent progress of Michael Magee and Doron Puder on the Atiyah-Bott-Goldman measure, I will present a set of results obtained in recent or ongoing works by T. Lemoine, T. Lévy and myself allowing to prove convergence of these traces, when divided by $N$, towards a deterministic and explicit limit.

In this talk, we introduce Neumann and Dirichlet stable random fields on the Sierpinski gasket in the distribution sense. We first focus on the existence of a density with respect to the Hausdorff measure. When this density field exists, we then study its sample path smoothness. As for Euclidean moving average stable random fields, or these sample paths are unbounded almost surely or the density field admits a modification with Hölder sample paths. Roughly speaking, in the non Gaussian framework, the sample paths can not been smoother than the Riesz fractional kernel. The Hölder regularity follows from an upper bound of the modulus of continuity, that we obtain using a LePage series representation. Finally, the density field also satisfies some scaling and invariance properties. This talk is a joint work with Fabrice Baudoin (Aarhus University) that extend known results on fractional Gaussian random fields.

We investigate central and non-central limit theorems for integral functionals of subordinated Gaussian fields on the Euclidean space, as the integration domain grows. In particular, we consider the case of $p$-domain functionals, where the domain can be written as the Cartesian product of $p$ domains that (possibly) grow at different rates. First, we assume that the covariance function of the Gaussian field is separable and thoroughly investigate under which conditions the study of $p$-domain functionals can be reduced to that of some simpler and classical one-domain functionals. When the considered functionals are in a fixed Wiener chaos, we also provide a quantitative version of the previous result, which improves some bounds in the literature. Second, we extend our study beyond the separable case, by investigating what can be inferred when the covariance function is either in the Gneiting class or is additively separable.

We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will consider the $L^p$ spectrum of the Laplacian on differential forms. In particular, we will show that the resolvent set of the Laplacian on $L^p$ integrable $k$-forms lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. Moreover, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and we provide a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space. The above results are joint work with Zhiqin Lu.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We consider the hypoelliptic sub-Laplacian on a Carnot group G of step 2. This operator is the generator of a Markov semigroup, which is known as the horizontal heat semigroup on the Carnot group. Using some intertwining relationships, we provide a complete description of the spectrum of these operators on . We further consider the Ornstein-Uhlenbeck operator on G and its Levy-type perturbations. We prove that all these perturbations are generators of some ergodic Markov semigroups on G. Moreover, under some mild conditions, we prove isospectrality of these operators. Our methods rely on harmonic analysis on Carnot groups and some intertwining relationships.

The Teichmuller space $T_g$ is a way to parametrize the space of complex structures of a compact surface of a fixed genus $g$. Under the Weil-Petersson metric it is negatively Kahler manifold of dimension $3g - 3$ (for greater than 1) diffeomorphic to a simply connected domain in the complex euclidean space. The general goal is to study the boundary behavior of the Teichmuller space. We show that the corresponding Brownian motion on $T_g$ is stochastically complete, meaning that Brownian motion has infinite life time with probability one. This opens a new approach to study the boundary properties of the Teichmuller space via the Martin compactification by investigating the limiting behavior of Brownian motion as time goes to infinity.