Special Session 172: Stochastic and geometric analysis on manifolds and metric measure spaces

Nonlinear gradient flows in spacetime geometry

Mathias Braun EPFL Switzerland Co-Author(s):

Abstract: Recent work in optimal transport and metric geometry has opened new perspectives on spacetime geometry, uncovering deep connections with nonlinear elliptic PDEs, including Lorentzian variants of the $p$-Laplacian. In this talk, I will first outline these ideas and then present recent progress towards a metric theory of gradient flows in Lorentzian signature.

Stochastic perturbation of geodesics on the space of Riemannian metrics

Ana Bela Cruzeiro Instituto Superior Tecnico Portugal Co-Author(s):    Ali Suri

Abstract: The evolution equation on the manifold of Riemannian metrics for the action functional associated with the L2 stochastic kinetic energy is derived.

Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian manifolds

Yuzuru Inahama Kyushu University Japan Co-Author(s):

Abstract: In this talk we discuss a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

On Morrey`s inequality over RCD(K,N)-spaces

Kazuhiro Kuwae Fukuoka University/Department of Applied Mathematics Japan Co-Author(s):    Jun Kitagawa

Abstract: I will talk about a Morrey inequality in the framework of ${\sf RCD}(K,N)$-spaces for $K\in\mathbb{R}$ and $N\in[1,+\infty[$.

Asymptotics of the Nonlocal Perimeter in Grushin Spaces

Huaiqian Li Tianjin University Peoples Rep of China Co-Author(s):

Abstract: We investigate the fractional $s$-perimeter relative to a bounded domain $\Omega$ in the Grushin space, a typical model of sub-Riemannian manifolds that generally lack a group structure. On the one hand, we establish a limiting formula for the $s$-perimeter as $s\rightarrow0^+$, imposing no regularity assumptions on the boundary of $\Omega$. On the other hand, we provide a converse under a mild boundary condition that allows for domains with highly irregular (e.g., fractal) boundaries. Our results significantly weakens the typical Lipschitz or $C^{1,\alpha}$ regularity required in existing literature. The proofs exploit intrinsic properties of the Grushin heat semigroup, thereby circumventing tools such as the $L^\infty$-Liouville property used in earlier Riemannian settings.

On the $W$-entropy on Riemannian manifolds, metric measure spaces, Clifford algebra and quantum Markov semigroups

Xiang-Dong Li Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: In 2002, G. Perelman proved the monotonicity of the $W$-entropy for the Ricci flow. Since then, the $W$-entropy has been studied for various geometric flows on Riemannian manifolds. In this talk, I`ll present some recent results on the study of the $W$-entropy on Riemannian manifolds, metric measure spaces, Clifford algebra and quantum Markov semigroups.

Non-adapted coupling of sub-Riemannian diffusions

Robert Neel Lehigh University USA Co-Author(s):    Liangbing Luo

Abstract: We describe the classical Markov couplings of Riemannian Brownian motions and how these natural constructions fail in sub-Riemannian geometry, even for the simplest case of the Heisenberg group. After reviewing the situation, we describe an improvement and extension of recent constructions of non-adapted reflection couplings on sub-Riemannian model spaces by Banerjee-Gordina-Mariano and B\`en\`efice. Moreover, this construction is relatively simple and geometrically appealing, being based on global symmetries of the underlying spaces.

Nonlinear heat flow

Shin-ichi Ohta University of Osaka Japan Co-Author(s):

Abstract: This talk will be a review of nonlinear heat flow on Finsler manifolds and metric measure spaces.

Dissipative Hamiltonian structure of the Vlasov-Fokker-Planck equation

Sangmin Park California Institute of Technology USA Co-Author(s):

Abstract: The Vlasov-Fokker-Planck equation (VFP) describes the evolution of the probability density of the position and velocity of particles under the influence of external confinement, interaction, friction, and stochastic force. It is well-known that this equation can be formally seen as a dissipative Hamiltonian system in the Wasserstein space of probability measures. Moreover, the geometric structure has possible connections to the conjectured optimal convergence rates of underdamped Langevin Monte Carlo (ULMC), a sampling algorithm known to empirically outperform the (standard) Langevin Monte Carlo. This talk will focus on a time-discrete variational scheme for VFP which we introduce to more rigorously understand the geometric structure. We will begin by introducing the optimal transport problem and the Wasserstein distance, as well as the techniques of gradient flows which form the basis of our variational scheme. After highlighting the connections to ULMC, we will discuss how the proposed variational scheme is (i) consistent with the dissipative Hamiltonian structure, and (ii) (geodesically)-convex at each iteration.

Stability of weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound

Yohei Sakurai Saitama university Japan Co-Author(s):    Yasuaki Fujitani

Abstract: The aim of this talk is to present the validity of weighted Ricci curvature whose dimensional parameter is equal to 1 in view of extrinsic geometric analysis. It has been observed that the 1-weighted Ricci curvature exhibits singular behavior from the view point of the Cheeger-Gromoll splitting theorem and affine geometry. Recently, it has been also pointed out that its non-negativity is equivalent to the so-called substatic condition in the context of the Lorentzian geometry via conformal change of metric. After I review such developments, I will introduce several geometric consequences concerning stable weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound. This talk is based on the joint work with Yasuaki Fujitani (University of Tokyo).

Disintegrated optimal transport for metric fiber bundles and their applications

Asuka Takatsu The University of Tokyo Japan Co-Author(s):    Jun Kitagawa

Abstract: In this talk, I give a definition of the disintegrated Monge--Kantorovich metrics, which are a two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, and demonstrate their applications.

From Schroedinger to Feynman and back

Jean-Claude Tel ++351.916565277 GFM, FCUL and IST, Lisbon Portugal Co-Author(s):    Qiao Huang

Abstract: We shall present recent progress in the stochastic dynamical theory founded on Schroedinger's variational problem (1932). As suggested by him, it shares a number of qualitative properties with Quantum Mechanics and can also be regarded as a mathematically sound version of Feynman's path integral approach.

Gaussian processes under adapted Wasserstein distance

Ting Kam Leonard Wong University of Toronto Canada Co-Author(s):    Madhusoodan Gunasingam and Ting-Kam Leonard Wong

Abstract: The adapted Wasserstein distance is a distance between stochastic processes. It is defined in terms of an adapted optimal transport problem that takes into account the flow of information in term. We consider (filtered) Gaussian processes for which explicit computations and their geometric properties have recently been developed. In particular, the adapted Wasserstein distance (with $p = 2$) can be expressed explicitly in terms of the Cholesky factors, and can be interpreted in terms of a constrained Procrustes problem. In this Gaussian setting, we discuss various properties of the adapted Wasserstein distance and also study the multi-causal barycenter.

Functional inequalities on the Riemannian path space

Bo Wu Fudan University Peoples Rep of China Co-Author(s):    Tianyu Wang

Abstract: In this talk, we first review recent progress in stochastic analysis on Rieamnian path space. We prove an integration by parts formula on path space over a general non-compact Riemannian manifold with a skew symmetric torsion. Using this formula, we derive quasi regular O-U/$L^2$ Dirichelt forms as well as corresponding functional inequalities. This is joint work with Tianyu Wang.

Lipschitz regularity of harmonic map heat flows into CAT(0) spaces

Hui-Chun Zhang Sun Yat-sen University Peoples Rep of China Co-Author(s):

Abstract: In 1964, Eells and Sampson proved the celebrated long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. In 1992, Gromov and Schoen initiated the study of harmonic maps into CAT(0) metric spaces. It naturally motivates the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on CAT(0) spaces and extended Crandall-Liggett`s theory of gradient flows from Banach spaces to CAT(0) spaces to obtain the weak solutions for the harmonic map heat flow into CAT(0) spaces. The weak solutions enjoy the favorable long-time existence, uniqueness and well-established long-time behaviors. It is a long-standing open question whether the weak solutions possess Lipschitz regularity. Very recently, by using elliptic approximation method, Lin, Segatti, Sire, and Wang proved the weak solutions are Lipschitz in space and (1/2)-H\older continuous in time, for a wide class of CAT(0) spaces. In this talk, we will introduce a complete answer to the question, showing that every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality. This is based on a joint work with Xi-Ping Zhu.