# Conference - Stochastic differential geometry and mathematical physics

Marc Arnaudon (Université de Bordeaux)

Title: Stochastic mean curvature flow and intertwined Brownian motion

Abstract: The evolution of a set by deformation of its boundary along mean curvature flow is involved in many physical phenomena. We are interested here in this evolution, to which we add a noise that acts uniformly on its boundary, and a renormalization term.  It is shown that this evolution can be coupled to a Brownian motion which remains within the set, and which at all times is uniformly distributed inside the set. This is the phenomenon of duality and intertwining established by Diaconis and Fill in the context of Markov chains in finite state spaces. Different couplings are proposed,  some of which involving the local time of the Brownian motion, either on the skeleton of the set or on its boundary. These couplings also differ by more or less strong correlation between the Brownian motion inside and the vibrating boundary. When the set is a symmetric real interval, the boundary evolves as a Bessel process of dimension 3, and we recover the 2M-X Pitman theorem as a special case, with one particular coupling. This is a common work with Koléhè Coulibaly (Institut Elie Cartan de Lorraine (IECL), Nancy), and Laurent Miclo (Institut de Mathématiques de Toulouse (IMT), Toulouse).

Davide Barilari (Università degli Studi di Padova)

Title: Surfaces in 3D contact sub-Riemannian: geometry and stochastic processes

Abstract: We discuss differential geometry and stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. By considering the Riemannian approximations to the sub-Riemannian manifold, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices and we illustrate the results with some examples. [Joint work with Ugo Boscain, Daniele Cannarsa and Karen Habermann.]

Fabrice Baudoin (University of Connecticut)

Title: Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions

Abstract: We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion. This is a joint work with Jing Wang (Purdue University).

Robert Baumgarth (Universität Leipzig)

Title: Scattering Theory for the Hodge Laplacian

Abstract: We prove using an integral criterion the existence and completeness of the wave operators $W_{\pm}(\Delta_h^{(p)}, \Delta_g^{(p)}, I_{g,h}^{(p)})$ corresponding to the Hodge-Laplacians $\Delta_\nu^{(p)}$ acting on differential $p$-forms, for $\nu\in\{g,h\}$, induced by two quasi-isometric Riemannian metrics $g$ and $h$ on an complete open smooth manifold $M$. In particular, this result provides a criterion for the absolutely continuous spectra $\sigma_{\mathrm{ac}}(\Delta_g^{(p)}) = \sigma_{\mathrm{ac}}(\Delta_h^{(p)})$ of $\Delta_\nu^{(p)}$ to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion only requires local curvature bounds and some upper local control on the heat kernel acting on functions, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

Ugo Boscain (Ecole Polytechnique)

Title: Geometric confinement of the curvature Laplacian $-\Delta+c K$ on $2D-$almost-Riemannian manifolds

Abstract: Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a $1D$ embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator $\Delta$) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called geometric confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $-\Delta+c K$, for $c>0$ (here $K$ is the Gaussian curvature), which originates in coordinate free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no geometric confinement for these types of operators.

Ilya Chevyrev (University of Edinburgh)

Title: State space for the 3D stochastic quantisation equation of Yang-Mills

Abstract: In this talk I will present a proposed state space of distributions for the stochastic Yang-Mills equations (SYM) in 3D. I will show how the notion of gauge equivalence extends to this space and how one can construct a Markov process on the space of gauge orbits associated with the SYM. This partly extends a recent construction done in the less singular 2D setting. Based on a joint work in progress with Ajay Chandra, Martin Hairer, and Hao Shen.

Title: A stochastic view on the deterministic Navier-Stokes equation

Abstract: We review some recent results on a stochastic approach to the deterministic Navier-Stokes equation, that generalizes Arnold's characterization of the Euler equation in fluid dynamics.

Shizan Fang (Université de Bourgogne)

Title: Ikeda-Watanabe's connection and Navier-Stokes equations.

Abstract: To each velocity, we associate an affine connection introduced by N. Ikeda and S. Watanabe when they rolled a flat Brownian motion on manifolds. We will compute the associated intrinsic Ricci tensor in the sense of B. Driver, which will allow us to express the vorticity form of Navier-Stokes equations: this can be seen as a non linear version of Bochner-Weitzenbock formula.

Franck Gabriel (Ecole Polytechnique Fédérale de Lausanne)

Title: Two-dimensional planar Yang-Mills measure as planar Markovian holomomic fields

Abstract: Yang-Mills theories are gauge theories with a non-abelian symmetry group. On the plane, considering both its gauge symmetry property and the underlying action, it can be seen that the Yang-Mills measure should satisfy a generalization of the defining properties of Lévy processes: it should be described by a planar Markovian holonomy field. Planar Markovian holonomy fields, a class of path-indexed stochastic processes, can be fully described: using both the algebraic and geometric perspectives on the braid group, we describe a correspondence between these fields and Lévy processes on compact groups. In the case where the symmetry group is the permutation group, planar Markovian holomomic fields can also be obtained geometrically from random branched covering models. When the number of sheets of the covering goes to infinity, Wilson’s observables can be easily computed.

Maria Gordina (University of Connecticut)

Title: Ergodicity for Langevin dynamics with singular potentials

Abstract: We discuss Langevin dynamics of $N$ particles on $\mathbb R^d$ interacting via a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted Sobolev norm. The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus. In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on the joint work with F. Baudoin and D. Herzog, as well a recent preprint with E. Camrud, D. Herzog and G. Stoltz.

Erlend Grong (University of Bergen)

Title: Path space on sub-Riemannian manifolds

Abstract: We discuss how we can generalize the concept of Malliavin Calculus to the setting of a sub-Riemannian manifolds. We discuss how concepts such as the Cameron-Martin space and the gradient and damped gradient of functions on path space can be understood in this setting. From there, we discuss how we can obtain functional inequalities related to both an lower and upper bounds for Ricci curvature. These results are from a joint work with Li-Juan Cheng and Anton Thalmaier.

Batu Güneysu (Universität Potsdam)

Title : Feynman-Kac formula for first order perturbations of covariant Laplace-type operators.

Abstract:  It is a classical fact that one can represent the heat semigroup of a Schrödinger operator (that is, a perturbation of the Laplacian by a real-valued potential) as a path integral in terms of Brownian motion: this is the celebrated Feynman-Kac formula. The aim of this talk is to explain a non-selfadjoint variant of this result, where one allows arbitrary first order perturbations of a Bochner-Laplacian that acts on sections of a vector bundle over an arbitrary noncompact Riemannian manifold. In particular, one replaces self-adjoint heat semigroups by holomorphic semigroups. As an application to differential geometry, we obtain an explicit path integral formula for the first degree part of the differential graded Chern character of an even dimensional Riemannian spin manifold, which plays a crucial role in the context of the Duistermaat-Heckman localization formula on loopspace. This is joint work with Sebastian Boldt (Leipzig).

Karen Habermann (University of Warwick)

Title: Fluctuations for Brownian bridge expansions and convergence rates of Lévy area approximations

Abstract: We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian Lévy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with the observation that the Lévy area approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden-Platen-Wright approximation, whilst still only using independent normal random vectors.

Martin Hairer (Imperial College London)

Title: SPDEs with values in manifolds

Abstract: We consider space-time white noise driven SPDEs in one spatial dimension with values in a manifold. Heuristic considerations suggest that the second-order calculus appearing when considering SDEs should then be replaced by a fourth-order calculus. We will see that while this is indeed the case, it turns out that these equations admit a distinguished notion of "solution" which actually has nicer properties than in the case of SDEs!

Makoto Katori (Chuo University)

Title: Zeros of the i.i.d. Gaussian Laurent series on an annulus

Abstract: On an annulus ${\mathbb{A}}_q :=\{z \in {\mathbb{C}}: q < |z| < 1\}$ with a fixed $q \in (0, 1)$,we study a Gaussian analytic function (GAF) defined by the  i.i.d. Gaussian Laurent series. The covariance kernel of the GAF is given by the weighted Szegő kernel of ${\mathbb{A}}_q$ with the weight parameter $r$ studied by Mccullough and Shen. Conditioning the GAF by giving zeros, new GAFs are induced such thatthe covariance kernels are also given by the weighted Szegő kernel of Mccullough and Shen  but the weight parameter $r$ is changed depending on the given zeros. We prove that the zero set of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. If we take the limit $q \to 0$, a simpler but still non-trivial PDPP is obtained on a punctured unit disk ${\mathbb{D}}^{\times} := {\mathbb{D}} \setminus \{0\}$. In the further limit $r \to 0$  the present PDPP is reduced to the  determinantal point process on ${\mathbb{D}}$ studied by Peres and Virág. The present talk is based on a joint work with Tomoyuki Shirai (https://arxiv.org/abs/2008.04177).

Christian Léonard (Université Paris Nanterre)

Title:  Entropic optimal transport, time-reversal and (usual) optimal transport

Abstract: Felix Otto discovered twenty years ago that quadratic optimal transport on a Riemannian manifold $M$ generates the so-called Wasserstein geometry on the space $\mathcal{P}(M)$ of probability measures on $M$. The basic ingredients of this geometry are the McCann displacement interpolations which result from a lift of geodesics of $M$ onto $\mathcal{P}(M)$. Replacing the geodesics of $M$ by Brownian bridges leads to a natural notion of interpolations on $\mathcal{P}(M)$, the so-called entropic interpolations. It is known that entropic interpolations converge to displacement interpolations as the temperature of the Brownian bridges decreases to zero. Not surprisingly, time reversal of some stochastic processes, the Schrödinger bridges, whose marginals are entropic interpolations leads to a precise evaluation of the energetic gap between entropic and displacement interpolations. Some  well-established and heuristic consequences of time-reversing Schrödinger bridges will be presented.

Thierry Lévy (Sorbonne Université)

Title: Matrix-tree theorems and determinantal linear processes

Abstract: The matrix-tree theorem states that the product of the non-zero eigenvalues of the Laplacian on a connected graph is equal to the number of rooted spanning trees of this graph. This theorem can be traced back to the middle of the XIXth century, to papers of Kirchhoff (1847) and Sylvester (1857), both devoted to the resolution of certain systems of linear equations. In the situation where the graph is endowed with a Hermitian fibre bundle and a unitary connection, the matrix-tree theorem was generalised by Forman (for a bundle of rank 1) and Kenyon (for a bundle of rank 2 and a SU(2)-connection), to the effect that the determinant of the covariant Laplacian counts special subgraphs of the graph, namely cycle-rooted spanning forests. On the other hand, from a probabilistic perspective, the uniform spanning tree and uniform cycle-rooted spanning forest fall into the intensely studied class of determinantal point processes. With Adrien Kassel (CNRS, ENS Lyon), we have been trying in the last few years to understand from a combinatorial and probabilistic perspective the covariant Laplacian on a graph endowed with a Hermitian bundle of arbitrary rank and an arbitrary unitary connection. This led us in particular to define a new class of probability measures on Grassmannian manifolds, which generalises determinantal point processes and that we call determinantal linear processes. In this talk, I will describe this class of measures and some of their properties, and how they relate to the covariant Laplacian.

Xue-Mei Li (Imperial College London)

Title: Hessian Estimates.

Abstract: We will discuss Hessian estimates for solutions for Markov generators.

Jacek Malecki (Politechnika Wrocławska)

Title: Archimedes principle for ideal gas

Abstract: We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem. The talk is based on a joint work with Krzysztof Burdzy.

Tai Melcher (University of Virginia)

Title: Regularity properties of some infinite-dimensional hypoelliptic diffusions

Abstract: Smoothness properties are of classical interest in the study of measures in infinite dimensions. These properties have been a particular focus for the end point distributions of diffusions, that is, heat kernel measures. In finite dimensions, hypoellipticity of the generator is a standard hypothesis to assume for (or is in some sense equivalent to) regularity properties of the heat kernel measure. We’ll review progress on generalizations of some hypoelliptic constructions in infinite dimensions, including recent joint work with Fabrice Baudoin and Masha Gordina on infinite-dimensional generalizations of Kolmogorov diffusions.

Gerard Misiolek (Notre-Dame University)

Title: Geometry and Fluids.

Abstract: Hydrodynamics of ideal fluids is an example of an infinite dimensional Riemannian geometry where solutions of the incompressible Euler equations correspond to geodesics in the group of volume preserving diffeomorphisms equipped with a right-invariant metric defined by the fluid’s kinetic energy. This beautiful observation was made by V. Arnold in a pioneering paper published in 1966 in Annales de l'Institut Fourier. It opened the way for introduction of geometric, topological and Lie theoretic methods to the study of fluid dynamics and the field has remained very active ever since. I will explain the basic Riemannian constructions of hydrodynamics and describe some of the results that have been obtained in recent years.

Robert Neel (Lehigh University)

Title: Heat kernels, their derivatives, and the bridge process in small time.

Abstract: Consider a sub-Riemannian manifold with a sub-Laplacian and associated heat kernel and diffusion, and fix two points such that all minimal geodesics between them are strongly normal. We show that the Molchanov method for computing the small-time asymptotics of the heat kernel and its derivatives can be rigorously applied and is capable of giving expansions to all order. Further, we show that there is a family of probability measures on pathspace that, as time goes to 0, converges to a probability measure on the set of minimal geodesics that gives the law of large numbers for the associated bridge processes. This limiting measure can be determined in various cases, essentially, again, via Molchanov's method. This extends earlier work of Hsu and Ballieul-Norris. Moreover, we see that logarithmic derivatives of the heat kernel, to any order, have small-time asymptotic behavior given by the cumulant of geometrically natural random variables with respect to this same limiting measure. The method is fundamentally local and is valid even on incomplete manifolds under appropriate conditions on the distance to infinity. This talk is based on joint work with Ludovic Sacchelli.

Pierre Perruchaud (Notre-Dame University)

Title: The search for an infinite-dimensional Cartan development

Abstract: The description of Brownian motion on manifolds was greatly simplified by Eells, Elworthy and Malliavin as they introduced their now classical construction, using insight from differential geometry. In particular, the so-called Cartan development plays a central role, allowing to transpose not only Brownian motion, but for instance any semimartingale from the Euclidean to the manifold setting. Motivated by applications to stochastic fluids, I will present a possible definition for the Cartan development in a class of infinite dimensional manifolds of diffeomorphisms, and describe how topology, geometry and probability tend to work against each other in this context. This will lead us to discuss the elusive orthonormal frame bundle, which seems to be an important missing piece of the puzzle.

Gregory Schehr (Sorbonne Université)

Title: Non-intersecting Brownian bridges in the flat-to-flat geometry

Abstract: In this talk, I will discuss N non-intersecting Brownian bridges propagating from an initial configuration $\{a_1 < a_2 < \ldots< a_N \}$ at time $t=0$ to a final configuration $\{b_1 < b_2 < \ldots< b_N \}$. I will first show that this problem can be mapped to a non-intersecting Dyson's Brownian bridges with Dyson index $\beta=2$. For the latter I will derive an exact effective Langevin equation that allows to generate very efficiently the non-intersecting bridge configurations. In particular, for the flat-to-flat configuration in the large $N$ limit, where $a_i = b_i = (i-1)/N$, for $i = 1, \cdots, N$, I will use this effective Langevin equation to derive an exact Burgers' equation (in the inviscid limit) for the Green's function and solve this Burgers' equation for arbitrary time $0 \leq t\leq t_f$. Finally, I will discuss connections to some well known problems, such as the Chern-Simons model, the related Stieltjes-Wigert orthogonal polynomials and the Borodin-Muttalib ensemble of determinantal point processes.

Alexander Schmeding (Nord Universitet)

Title: Stochastic PDE from hydrodynamics via infinite-dimensional geometry

Abstract: The motion of a rigid body, can be described as a differential equation on the configuration space of the system which turns out to be a finite-dimensional Lie group. Building on this observation, Arnold postulated that the Euler equation governing an incompressible fluid in a domain can be described as a differential equation on its configuration space, the group of volume preserving diffeomorphisms. Indeed as Ebin and Marsden in 1970 showed, one can reformulate this partial differential equation as an ordinary differential equation, for the price of switching to an infinite dimensional manifold. Using geometric techniques local wellposedness of the Euler equation can then be established. We were recently able to apply this circle of ideas to stochastic versions of the Euler equation (joint with M. Maurelli (Milano) and K. Modin (Chalmers, Gothenburg)). In the talk I will give an introduction to these topics together with an overview on these new developments. Note that the talk will not supposes familiarity with infinite-dimensional manifolds and only mild familiarity with stochastic analysis.

Armen Shirikyan (CY Cergy Paris Université)

Title: Large deviations and entropy production in viscous fluid flows

Abstract: We first review a general framework of the theory of entropic fluctuations in deterministic and stochastic systems. Several questions of interest will be formulated, together with the difficulties encountered when dealing with randomly forced PDEs. We next consider the 2D Navier-Stokes system coupled with a Lagrangian particle. In this context, we discuss a criterion for the validity of the large deviation principle and show that it can be reduced to the verification of some controllability properties of the underlying deterministic problem. We next turn to a study of the large-time asymptotics of the particle and discuss the question of entropy production for it. The talk is based on joint works with V. Jakšić, V. Nersesyan, and C.-A. Pillet.

Stefan Suhr (Ruhr-Universität Bochum)

Title : Recent Developments in Optimal Transport and Lorentzian Geometry

Abstract: The stellar success of optimal transport theory in Riemannian geometry over the last fifteen years invites to consider similar questions in Lorentzian geometry, especially with a view towards general relativity and mathematical physics. In my talk I will introduce the basic ideas of optimal transport for (globally hyperbolic) spacetimes and discuss a few first results. The guiding beacon of my considerations is the characterization of Ricci curvature (and with it the Einstein field equations) via optimal transport theory.

Anton Thalmaier (Université du Luxembourg)

Title: Gradient formulas on manifolds - some new perspectives

Abstract: We recall various first and second order derivative formulas for heat semigroups on manifolds and describe geometric applications related to Calderón-Zygmund type inequalities, as well as new versions of log-Sobolev and transportation inequalities connecting relative entropy, Stein discrepancy and relative Fisher information on Riemannian manifolds.

Emmanuel Trélat (Sorbonne Université)

Title: Spectral analysis of sub-Riemannian Laplacians and Weyl measure

Abstract:  In collaboration with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are selfadjoint hypoelliptic operators satisfying the Hörmander condition. Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl law. When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, Martinet) the Wey law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.

François-Xavier Vialard (Université Gustave Eiffel)

Title: Hdiv generalized minimizing geodesics

Abstract: In this talk we show how to extend Brenier's approach of generalized geodesics for the incompressible Euler equation to the setting of compressible fluid for a generalization of the Camassa-Holm equation. We aim at understanding geodesic on a group of diffeomorphisms endowed with the right invariant metric Hdiv, which is L2 norm of the vector field + L2 norm of its divergence. To introduce this work, we first present unbalanced optimal transport and its link with the generalized Camassa-Holm equation. Then, we propose a simple convex relaxation of the associated minimization problem on the path space on a cone manifold with moment constraints. We show that the relaxation is tight in some cases of interest and conclude with open questions.