Geometrically constrained stochastic processes are commonly used as models in image processing and shape analysis, chemistry, robotics, and fluid dynamics. The construction, analysis, and simulation of such models involve the application of techniques and insights in probability and statistics, stochastic analysis, geometry, and numerical methods. The goal of this workshop is to bring together researchers from diverse disciplines who are working with geometrically constrained random variables or processes, to exchange ideas, identify open problems, and explore new probabilistic approaches to modeling and analysis.

Organisers

Abstracts

Marc Arnaudon

Title: Long time estimate of the gradient of the semi-group associated with kinetic Brownian motion in the Euclidean plane

Abstract: The kinetic Brownian motion in the plane is a $C^1$ process with velocity the Brownian motion in the unit circle. It is a very simple process, the study of which leads to a surprising variety of technics. Using Malliavin calculus and the Bismut-Elworthy-Li integration by parts formula, we propose long-term estimates of the gradient of its semigroup.

This is a joint work with Magalie Bénéfice, Michel Bonnefont and Delphine Féral.

0-0-0

Fabrice Baudoin

Title: Brownian motion and stochastic areas on full flag manifolds

Abstract: We show that the Brownian motion on the complex full flag manifold can be represented by a matrix-valued diffusion obtained from the unitary Brownian motion. This representation actually leads to an explicit formula for the characteristic function of the joint distribution of the stochastic areas on the full flag manifold. The limit law for those stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries.  Similar results are then proved on the quaternionic full flag manifold.

0-0-0

Adrien Busnot Laurent

Title: Intrinsic sampling of stochastic dynamics on Riemannian manifolds

Abstract: In stochastic optimization, molecular dynamics, quantum physics, or in the training of neural networks, one is interested in sampling from the law of stochastic processes. These dynamics are often subject to geometric constraints (fixed distance between particles, constraints on weights in PINNs, …). In this context, it is crucial to develop numerical approaches that take into account both the geometric and random features of the dynamics. The literature relies mostly on penalty formulations or extrinsic numerical integrators. These solutions are costly, subject to significant time-step restrictions, and require formulation of the dynamics in much higher-dimensional spaces. In this talk, we present a general new class of methods that rely on intrinsic geometric operations, a new robust convergence analysis, and an algebraic approach with Hopf algebras of planar exotic forests and Butcher series for the calculation of order conditions. A second order integrator that generalises the celebrated Leimkuhler-Matthews method is presented. This is joint work with Eugen Bronasco, Baptiste Huguet, and Sébastien Macé.

0-0-0

Marc Corstanje

Title: Simulation of diffusion bridges on the Poincaré disk

Abstract: Diffusion bridges are an essential tool in statistical inference for continuous-time stochastic processes. When inferring information on dynamics that govern a process based on discrete observation of the process, it is often useful to simulate bridge processes in between these points. The simulation of bridge processes has therefore received much attention and we study a particular approach that makes use of h-transforms. In this talk, we are extending the approach of guided processes to a geometric setting. We will focus on the specific case of a Brownian motion evolving in a Poincaré disk with a vector field imposed on it. The focus is on both the simulation of bridge processes as wel as inference on parameters that determine the vector field.

This research is joint work with F. H. van der Meulen, M. Schauer and S. Sommer.

0-0-0

Ana Bela Cruzeiro 

Title: Forward-backward stochastic differential equations on tensor fields and applications

Abstract: We introduce a class of forward-backward stochastic differential equations on tensor fields and use them, in particular, to derive a stochastic representation of the incompressible Navier-Stokes equation on Riemannian manifolds.

This is joint work with X. Chen, W. Ye and Q. Zhang (to appear in Ann. Appl. Probab.)

0-0-0

Marta Ghirardelli

Title: Stability of Numerical Integrators on Riemannian Manifolds.

Abstract: Neural networks (NNs) may be viewed as discretizations of underlying continuous dynamical systems, both at the level of the network architecture and in the gradient flow used for parameter optimization. In both contexts, stability properties of the discretization methods can be relevant, for instance in enhancing adversarial robustness. To address challenges such as exploding or vanishing gradients, neural network feature and/or parameter spaces are often modeled as Riemannian manifolds. In this work, we propose a general framework for analyzing the stability of numerical integrators on Riemannian manifolds. As a concrete application, we analyze the explicit Euler method in the Riemannian setting—referred to as the Geodesic Explicit Euler (GEE) method—and establish precise stepsize stability bounds when the manifold has constant sectional curvature.

0-0-0

Erlend Grong

Title: Landmarks and most probable paths

Abstract: We want to consider a stochastic flow in a Riemannian manifold, and the most likely way one collection of points can be transported by the flow to another such collection.

The setting for considering such multi-points problem is the manifold of landmark configurations.

But even if the original process has an elliptic generator, the induced flow on the manifold of landmark configurations has a generator who will sub-Riemannian, not Riemannian, in nature.

The goal of this talk is therefore to present a framework for most probable paths that is applicable not only for Brownian motions in Riemannian manifold, but also sub-Riemannian Brownian motions as well as anisotropic versions of these processes.

We achieve this by introducing the concept of development most probable paths which can be applied in all of these setting.

Our final goal with this framework is to be able to present most probable ways of transporting landmarks.

The results presented are parts of joint works with Stefan Sommer and Sylvie Vega-Molino.

0-0-0

Karen Habermann

Title: Brownian motion on spaces of polygons

Abstract: We introduce and study Brownian motion on spaces of discrete regular curves in Euclidean space, such as polygons in the plane. Previously, it has been established when these spaces of discrete regular curves are geodesically complete. I will present joint work with Emmanuel Hartman which relies on a general result by Grigor’yan and shows that all spaces of discrete regular curves that are geodesically complete are also stochastically complete, that is, the associated Brownian motion exists for all times. This provides a rigorous footing for performing data statistics, such as data inference and data imputation, on these spaces. We include simulations for sample paths of Brownian motion on spaces of discrete regular curves.

0-0-0

Yington Hou

Title: BPHZ renormalisation of invariant measures via multi-indices

Abstract: In this talk, we provide a method to obtain the renormalised measures in quantum field theory by renormalising the cumulant expansion of the original measures. After reviewing the main ingredients of renormalisation: cumulant expansion, Wick renormalisation, Feynman diagrams, we introduce our approach based on BPHZ renormalisation via multi-indices which are combinatorial objects originating from describing scalar-valued singular SPDEs. To automate the renormalisation procedure, we propose a multi-index counterpart of the Hopf-algebraic program initiated by Connes and Kreimer for the renormalisation of Feynman diagrams. This is a joint work with Yvain Bruned.

0-0-0

Stephan Huckemann

Title: The Probability of the Cut Locus of a Fréchet Mean

Abstract: We show that the cut locus of a Fréchet mean of a random variable on a connected and
complete Riemanian manifold has zero probability, a result known previously in special
cases (Le and Barden, 2014) and conjectured in general. The proof is based on first order
and second order considerations, where the latter are based on a recent result by
Générau (2020) on “Laplacians in the barrier sense”. This generalizes to Fréchet p-means
for p > 2. The former allow also to rule out stickiness on Riemannian manifolds, and for
generalization to 1 <= p < 2, with a conjecture. We close with discussing and conjecturing
extensions to noncomplete manifolds and more general metric spaces.

This is joint work with Alexander Lytchak

Générau, F. (2020). Laplacian of the distance function on the cut locus on a Riemannian manifold.
Nonlinearity 33(8), 3928.
Le, H. and D. Barden (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London
Mathematical Society 46(4), 698–708.
Lytchak, A. and S. F. Huckemann (2025). Zero mass at the cut locus of a Fréchet mean on a Riemannian
manifold. arXiv preprint arXiv:2508.00747

0-0-0

Bas Janssens

Title: Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

Abstract: In his 1982 thesis, Jens Hoppe discovered what became known as the SU(k) -> SDiff(S^2) limit: for large matrix size k, commutation relations between $k \times k$-matrices converge to the Lie algebra of the group of volume-preserving diffeomorphisms of the 2-sphere. In subsequent work by Bordemann, Schlichenmaier, Meinrenken and others in the late 1980s and early 1990s, this curious observation was explained in terms of geometric quantization, and it has found many uses — from
fuzzy sphere models in noncommutative geometry to Zeitlin’s matrix approach in hydrodynamics. In a parallel but unrelated development, the representation theory for centrally extended loop groups LSU(k) (Kac-Moody algebras) revolutionised the way conformal field theories in 1+1 dimension were handled in string theory and condensed matter theory. In this talk, based on joint work with Zhenghan Wang (Microsoft station Q/UCSB), we investigate what happens if you combine these developments. We show that the Kac-Moody cocycles on LSU(k) have `fuzzy sphere limits’ in LSDiff(S^2), and discuss some conjectures on what this might entail for conformal field theories in 2+1 dimensions.

0-0-0

Jonathan Junné

Title : Concentration of empirical measures on Riemannian manifolds

Abstract : We obtain the Wasserstein concentration of empirical measures of independent random variables on Riemannian manifolds with Ricci curvature lower bound, or more generally on metric measure spaces with measure contraction property, following the seminal work of Bolley–Guillin–Villani [1].

As an application, we derive the concentration of the empirical measure of a mean-field interacting particle system approximating the aggregation-diffusion equation. This achieved by coupling the interacting with a proxy non-interacting particle system.

[1] Bolley, F., Guillin, A. & Villani, C. Quantitative Concentration Inequalities for Empirical Measures on Non-compact Spaces. Probab. Theory Relat. Fields 137, 541–593 (2007).

0-0-0

Richard Kraaij

Title: Well-posedness for Hamilton–Jacobi equations via functional inequalities

Abstract: Hamilton-Jacobi (HJ) equations encode the infinitesimal evolution of value functions in (stochastic) control problems and therefore play a central role in also dynamic large deviations and mean-field games. In infinite-dimensional settings, such as the Wasserstein-2 space well-posedness of the associated HJ equation is delicate and remains only partially understood.

We introduce a formulation based on two-sided Hamiltonian bounds via functional inequalities, tailored to infinitesimal Riemannian geodesic spaces. For dynamics driven by lambda-convex energies in the sense of EVI gradient flows, we prove a comparison/uniqueness result. In the Wasserstein-2 space we further establish existence, yielding well-posedness for Hamilton–Jacobi equations associated with linearly controlled EVI gradient flows.

Joint work with Giovanni Conforti (Padova), Luca Tamanini (Brescia), and Daniela Tonon (Padova)

0-0-0

Annika Lang

Title: Simulation of stochastic partial differential equations on hypersurfaces

Abstract: Looking around us, many surfaces including the Earth are no plain Euclidean domains but special cases of Riemannian manifolds.

Uncertain physical phenomena on these surfaces can for example be described by stochastic partial differential equations (SPDEs). In this talk, I will approximate solutions to SPDEs on spheres by spectral methods and discuss their convergence. As an outlook, I will discuss generalizations to hypersurfaces using surface finite element methods.

0-0-0

Alexander Paul Lewis

Title: Mean-Square Convergence for SDEs on Riemannian Manifolds

Abstract: Weak convergence of the Euler approximation of SDEs on Riemannian manifolds is well understood, and has been studied in both intrinsic coordinates and on embedded submanifolds of Euclidean space. Despite this, the mean-square convergence rate of the Euler scheme is still unknown. Moreover, a rigorous derivation of the scheme has yet to be derived.

In this talk, I will derive the manifold Euler method and show that it has (on non-flat manifolds) global convergence rate of order 1/2 when the manifold has empty cut-locus at every point (e.g. Cartan-Hadamard). I will then go further and derive a Milstein method whose mean-square global convergence rate is of order 1.

This is ongoing work with Karthik Bharath and Michael Tretyakov.

0-0-0

Erwin Luesink

Titel: Computational methods for Brownian motion on homogeneous spaces.

Abstract: The construction of Eells, Elworthy and Malliavin is an incredibly general and powerful tool for the construction of Brownian motion on Riemannian manifolds, but does not translate well into numerical methods. By restricting to homogeneous spaces whose algebraic structure is nice enough to admit compatible Riemannian metrics, we can construct efficient numerical methods for the simulation of Brownian motion. Examples include spheres, hyperbolic spaces, symmetric positive definite matrices, Stiefel manifolds and flag manifolds. This is joint work with Sonja Cox and Roy Schieven.

0-0-0

Klas Modin

Title: Statistical mechanics and matrix hydrodynamics

Abstract: In 1949, Lars Onsager, Nobel Laureate in Chemistry, published a paper where he applied statistical mechanics to a system of point vortices. Remarkably, he concluded that ensembles with high enough energy, corresponding to negative thermodynamical temperature, will tend to form clusters of equal-sign vortices. Onsager’s work thereby gave the first theoretical explanation of a phenomenon in 2-D hydrodynamics observed in both natural and laboratory systems: the formation of vortex condensates. In this talk I wish to promote Onsager’s work and relate it to recent work on the simulation of 2-D hydrodynamics via isospectral matrix flows.

0-0-0

Xavier Pennec

Title: Geometric Structures for Simple Statistics on Lie groups: from Riemannian to affine geometric statistics

Abstract: Geometric statistics aim at designing consistent statistical frameworks to study data living in non-linear spaces. I will present the general Riemannian setting and then investigate how this can be generalized to more general affine connection spaces and in particular Lie groups. On Lie groups, one generally use a left (or right)-invariant metric to define geodesics and distances. However, the invariance is only partial as there is generally no bi-invariant metric. This induces an incompatibility of the group structure with the statistical setting because the (Fréchet) mean of inverse transformations is not the inverse of the mean. I will argue that we should drop the left-invariant Riemannian metric to use the canonical symmetric Cartan-Schouten connection (a non-metric connection). Geodesics are now bi-invariant auto-parallel curves instead of being length-minimizers and can be easily computed with the group expenential. Despite the lack of a distance, the mean value can still be defined in a convex neighbourhood as an exponential barycentre. Many local results, like the central limit theorem continue to hold, giving rise to a powerful statistical setting which is fully compatible with the Lie group structure. In finite dimension, this provides strong theoretical bases for the use of one-parameter subgroups and allows to define bi-invariant means on Lie groups even in the absence of a bi-invariant metric. Lie groups can also be considered as globally affine symmetric space structure thanks to the symmetry $s_p(q) = p q^{-1} p$. The symmetric Cartan-Schouten connection turn also to be the cannonical connection associated with this symmetric structure. This notably simplifies some algorithms like parallel transport: for instance, Pole ladder is exact in one step only in such symmetric spaces. This will be illustrated by statistics on diffeomorphisms parametrized by stationary velocity fields (SVF), a framework that provides simple and efficient models of the longitudinal atrophy of the brain in Alzheimer’s disease.

0-0-0

Yvo Pokern

Title: Estimation on Complex Projective Space for Electron Nuclear Double Resonance pectroscopy

Abstract: Electron-nuclear double resonance (ENDOR) spectroscopy is a sophisticated method to indirectly measure intra-molecular distances and orientations for complex biomolecules. Experimental limitations mean that the complex microwave echo signal, recorded as a real (in-phase) and imaginary (in-quadrature) component for each of the different frequencies used to excite the system, has an unidentifiable global phase and magnitude factor. Quotienting this out leads to the natural parameter space on which to perform inference: complex projective space. This work presents estimation methodology, parametric testing and nonparametric estimation based on ENDOR measurements including numerical and first theoretical results on consistency.

0-0-0

Max von Renesse

Title: Functional inequalities for Brownian motion on manifolds with sticky-reflecting boundary diffusion

Abstract: I plan to present some estimates and structure properties for Brownian motions on Riemannian manifolds with sticky-reflecting boundary diffusion, as discussed in these papers:
https://link.springer.com/article/10.1007/s00440-024-01349-2

https://www.semanticscholar.org/paper/A-gradient-flow-that-is-none%3A-Heat-flow-with-Bormann-Monsaingeon/1064b22248622f900ee641bb091d312cb87e918d

0-0-0

Moritz Schauer

Title: Bridge Processes for Jump–Diffusions with Applications to Birth–Death Stochastic Landmark Dynamics

Abstract: In this talk I consider bridge processes for Markov jump–diffusions whose state space has variable dimension due to birth and death events. Using a Doob h-transform, I construct the process conditioned on a terminal state and derive the corresponding modifications of diffusion drifts and jump intensities. The resulting bridge measure admits a representation as diffusion bridges between random jump times. I discuss applications to registration problems and stochastic landmark models with appearing and disappearing landmarks.

0-0-0

Stefan Sommer

Title: Score learning and inference for diffusion processes on shape spaces

Abstract: Steering diffusion processes towards a data distribution is an integral part of diffusion models in generative AI. For geometric data such as shape data, diffusion processes appear as models for stochastic dynamics of e.g. species change through evolution, or for generating data distributions on non-linear spaces, e.g. when defining constructs such as the diffusion mean that relies on geometric equivalents of the Gaussian distribution. Score learning is here key for conditioning on observed data. Thus, score learning provides a connection between generative models and geometric statistics. The talk will concern this connection, bridge simulation on geometric spaces, and application of score learning in geometric contexts. A specific example of this is conditioning diffusion processes in infinite dimensions allowing shape observations to be used for phylogenetic inference in evolutionary biology.

0-0-0

Oliver Street

Title: Symplectic approaches to Langevin diffusions

Abstract: This talk discusses how Langevin diffusions can be interpreted in the context of stochastic Hamiltonian systems with structure-preserving noise and dissipation on Lie groups. We will introduce some results surrounding stochastic Hamiltonian flows, before introducing dissipative effects to discuss how such structures can be used to sample Gibbs measures.

0-0-0

Anton Thalmaier

Title: Geometric a priori estimates by Stochastic Analysis

Abstract: The effect of curvature in the propagation of heat on a Riemannian manifold is a classical problem. A measurement of this behavior is encoded in many geometric functional inequalities.

We show that Stochastic Analysis provides natural tools to quantify this effect.

More specifically, we discuss the interplay of geometry and probability theory in the context of Calderón-Zygmund inequalities and Riesz transforms on weighted Riemannian manifolds.

0-0-0

Tomasz Tyranowski

Title: Learning deterministic and stochastic forced Hamiltonian systems

Abstract: We present a neural network architecture capable of learning the parameter-dependent flow of a Hamiltonian system subject to external forcing, while preserving the underlying Lagrange-d’Alembert structure. We demonstrate that this architecture can learn the flows of time-dependent systems—both deterministic and stochastic—and more accurately emulate the system’s energy evolution compared to general-purpose, non-structure-preserving neural networks. This results in more stable and higher-quality solutions. We also discuss prospective applications to structure-preserving model reduction of stochastic Hamiltonian systems.

0-0-0

Rik Versendaal

Title: The convex hull of hyperbolic Brownian motion

In Euclidean planar geometry, Cauchy’s formula shows that we can compute the perimeter of a closed, convex set by integrating over all directions the width of the set in that direction. This has been used to study the asymptotic behaviour of the convex hull of Brownian motion and random walks in the Euclidean plane.

It turns out that in the hyperbolic plane, especially in the Beltrami-Klein model, one has a
similar formula for the perimeter of a (hyperbolic) convex set. We will introduce this formula, and
show how it can be used to derive asymptotics for the expected perimeter of the convex hull of
hyperbolic Brownian motion. We find that the expected perimeter is twice the lower bound one
obtains by considering that the convex hull contains the line segment from the origin to the
endpoint of the Brownian motion. This is in contrast to the Euclidean Brownian motion with drift,
where the asymptotic expected perimeter in fact equals this lower bound.
Based on joint work with Chinmoy Bhattacharjee and Andrew Wade.

0-0-0

François-Xavier Vialard

Title: Entropic semi-martingale optimal transport.

Abstract: After a brief introduction to standard tools in optimal transport and to entropic regularization, we present how to use the tool of entropic regularization to design numerical schemes to learn the diffusion and drift coefficients. Our proposal relies on renormalizing the entropy of a discretized process. We formulate a time-discrete minimization problem for which we show a sort of gamma-convergence result to a continuous functional that makes appear a Bregman divergence associated to the entropy on gaussian processes.
We finish with some numerical simulations illustrating our main result.

0-0-0

Jean Claude Zambrini

Title: ” The curious destiny of Schrödinger’s 1931-2 problem and its unlikely     consequences ”

Abstract : We shall give a review of a very old variational problem of classical Statistical physics due to Schrödinger ,who observed at the time that its similarities with Quantum Mechanics were so striking that it is hard to believe  they are accidental.

Some modern consequences  will be  reported as well as contact points with apparently unrelated  problems of mathematics and physics, notably :

-Time reversibility in Statistical Physics.

-Stochastic Optimal Control.

-Stochastic (and geometric) dynamical systems.

-Stochastic (or quantum) notions of Integrability.

0-0-0