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Lorenzo Dello Schiavo

Publications and source records attributed to Lorenzo Dello Schiavo.

At least 19 recordsLinked to original sources
arXiv2026

Polychromatic Partitions, Kingman Theory, and Ewens Sampling Formula

A polychromatic partition is a partition of a set of colored (marked) elements in which each block is recorded only by the number of elements of each color in that block. We introduce a notion of polychromatic partition structure, i.e. a family of random polychromatic partitions consistent under deletion of elements uniformly at random, simultaneously extending (standard) partition structures, multipartition structures, and partition structures of partially exchangeable type. As an example thereof, we define a polychromatic analogue of the celebrated Ewens Sampling Formula, proving polychromatic versions of Kingman's characterization and of Hoppe's urn model. As our main result, we give a complete characterization of polychromatic partition structures, proving a triple affine homeomorphism of Bauer simplices among: polychromatic partition structures, harmonic densities for the polychromatic Hoppe urn, and probability measures on a polychromatic Kingman simplex. Additionally, we prove that this new simplex is homeomorphic to the Martin boundary of the polychromatic Hoppe urn inside its Martin compactification. We discuss various applications including closed non-recursive non-iterative expressions for multivariate moments of several random measures in terms of cycle index polynomials whose monomials are indexed by polychromatic partitions.

math.PR↗
arXiv2026

A 'global' Perspective on the Differential Geometry of Wasserstein Spaces

We develop a global differential calculus on the $L^2$-Wasserstein space over a closed Riemannian manifold, based on derivations of cylinder functions rather than on the standard pointwise approach. Within this framework we define some fundamental geometric tools. In particular, we show that the Levi-Civita connection on the base manifold lifts to the unique torsion-free connection compatible with the 'extended' Otto metric. The corresponding Riemann tensor is exactly the lift of the base Riemann tensor, which shows that - in this framework - the correction terms of the classical gradient formalism are not intrinsic curvature terms, but arise from the projection onto the measure-dependent gradient distribution. This allows us to revisit with a global and purely differential approach the smooth computations by J. Lott, Comm. Math. Phys., 277(2):423-437, 2007, reaching partially different conclusions.

math.DG↗
arXiv2025

Non-bilinear Dirichlet Functionals: Markovianity, locality, invariance

We present in a unified setting the foundations for a theory of non-bilinear Dirichlet functionals on Hilbert spaces. We prove known and new equivalences between non-linear semigroups, non-linear resolvents, non-linear generators, and their energy functionals, including a complete characterization of Markovianity. We introduce and characterize strong notions of invariance for general lower semicontinuous convex functionals, and notions of locality and strong locality for non-bilinear Dirichlet functionals. Contrary to many partial results in the literature, these characterizations are complete and correctly extend the analogous assertions for the bilinear case in full generality.

math.FA↗
arXiv2025

Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation

We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on $L^2$-Wasserstein spaces. For the objects in (1)-(3) we prove existence and uniqueness of solutions, and several characterizations, on an arbitrary locally compact Polish ambient space $M$ with exponentially recurrent Feller driving noise. In the case of the Dean-Kawasaki equation, this amounts to replacing the Laplace operator with some arbitrary diffusive Markov generator $\mathsf{L}$ with ultracontractive semigroup. In addition to a complete discussion of the free case, we consider singular interactions, including, e.g., mean-field repulsive isotropic pairwise interactions of Riesz and logarithmic type under the assumption of local integrability. We further show that each Markov diffusion generator $\mathsf{L}$ on $M$ induces in a natural way a geometry on the space of probability measures over $M$. When $M$ is a manifold and $\mathsf{L}$ is a drifted Laplace-Beltrami operator, this geometry coincides with the geometry of $L^2$-optimal transportation. The corresponding `geometric Brownian motion' coincides with the 'metric measure Brownian motion' in (4).

math.PR↗
arXiv2025

Gradient flows of $(K,N)$-convex functions with negative $N$

We discuss $(K,N)$-convexity and gradient flows for $(K,N)$-convex functionals on metric spaces, in the case of real $K$ and negative $N$. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of $(K,N)$-convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness.

math.FA↗
arXiv2025

The Hellinger-Kantorovich metric measure geometry on spaces of measures

Let $(M,g)$ be a Riemannian manifold with Riemannian distance $\mathsf{d}_g$, and $\mathcal{M}(M)$ be the space of all non-negative Borel measures on $M$, endowed with the Hellinger-Kantorovich distance $\mathsf{H\! K}_{\mathsf{d}_g}$ induced by $\mathsf{d}_g$. Firstly, we prove that $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g}\right)$ is a universally infinitesimally Hilbertian metric space, and that a natural class of cylinder functions is dense in energy in the Sobolev space of every finite Borel measure on $\mathcal{M}(M)$. Secondly, we endow $\mathcal{M}(M)$ with its canonical reference measure, namely A.M. Vershik's multiplicative infinite-dimensional Lebesgue measure $\mathcal{L}_θ$, $θ>0$, and we consider: (a) the geometric structure on $\mathcal{M}(M)$ induced by the natural action on $\mathcal{M}(M)$ of the semi-direct product of diffeomorphisms and densities on $M$, under which $\mathcal{L}_θ$ is the unique invariant measure; and (b) the metric measure structure of $\left(\mathcal{M}(M),\mathsf{H\! K}_{\mathsf{d}_g},\mathcal{L}_θ\right)$, inherited from that of $(M,\mathsf{d}_g,\mathrm{vol}_g)$. We identify the canonical Dirichlet form $\left(\mathcal{E},\mathscr{D}(\mathcal{E})\right)$ of (a) with the Cheeger energy of (b), thus proving that these two structures coincide. We further prove that $\left(\mathcal{E},\mathscr{D}(\mathcal{E})\right)$ is a conservative quasi-regular strongly local Dirichlet form on $\mathcal{M}(M)$, recurrent if and only if $θ\in (0,1]$, and properly associated with the Brownian motion of the Hellinger-Kantorovich geometry on $\mathcal{M}(M)$.

math.FA↗
arXiv2025

Ill-posedness of the pure-noise Dean-Kawasaki equation

We prove that the Dean-Kawasaki-type stochastic partial differential equation $$\partial ρ= \nabla\cdot (\sqrt{ρ\,}\, ξ) + \nabla\cdot \left(ρ\, H(ρ)\right)$$ with vector-valued space-time white noise $ξ$, does not admit solutions for any initial measure and any vector-valued bounded measurable function $H$ on the space of measures. This applies in particular to the pure-noise Dean-Kawasaki equation ($H\equiv 0$). The result is sharp, in the sense that solutions are known to exist for some unbounded $H$.

math.PR↗
arXiv2024

Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension

For large classes of even-dimensional Riemannian manifolds $(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their covariance kernels $k$ which exhibit a precise logarithmic divergence: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field $h$, we define the quantum Liouville measure, a random measure on $M$, heuristically given as $$ dμ_g^{h}(x):= e^{γh(x)-\frac{γ^2}2k(x,x)}\,d \text{vol}_g(x)$$ and rigorously obtained as almost sure weak limit of the right-hand side with $h$ replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. In terms on the quantum Liouville measure, we define the Liouville Brownian motion on $M$ and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorous meaning to the Polyakov-Liouville measure $$ d\boldsymbolν^*_g(h) =\frac1{Z^*_g} \exp\Big(- \int Θ\,Q_g h + m e^{γh} d \text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary compact manifolds.

math.PR↗
arXiv2023

Persistence of Rademacher-type and Sobolev-to-Lipschitz properties

We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces, tensorization, and direct integration. As byproducts we obtain: necessary and sufficient conditions to identify a quasi-regular strongly local Dirichlet form on an extended metric topological $σ$-finite possibly non-Radon measure space with the Cheeger energy of the space; the tensorization of intrinsic distances; the tensorization of the Varadhan short-time asymptotics.

math.MG↗
arXiv2023

Scaling Limits of Random Walks, Harmonic Profiles, and Stationary Non-Equilibrium States in Lipschitz Domains

We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $Ω$, with both fast and slow boundary. For the random walks on $Ω$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $Ω$ with either Neumann (slow boundary), Dirichlet (fast boundary), or Robin (at criticality) boundary conditions; the discrete-to-continuum convergence of the corresponding harmonic profiles. As a consequence, we rigorously derive the hydrodynamic and hydrostatic limits for SEP/SIP on $Ω$, and analyze their stationary non-equilibrium fluctuations. All scaling limit results for SEP/SIP concern finite-dimensional distribution convergence only, as our duality techniques do not require to establish tightness for the fields associated to the particle systems.

math.PR↗
arXiv2023

A Characterization of Maps of Bounded Compression

A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of maps of bounded compression/deformation by means of the measure-algebra functor and corroborates the assertion that maps of bounded deformation are the natural class of morphisms for the category of complete and separable metric measure spaces.

math.MG↗
arXiv2023

Local Conditions for Global Convergence of Gradient Flows and Proximal Point Sequences in Metric Spaces

This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka--Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on $\mathbb{R}^n$. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on $\mathbb{R}^n$.

math.OC↗
arXiv2023

Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Let $\varUpsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $π$. We study the geometry of $\varUpsilon$ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on $\mathscr{P}_{1}(\varUpsilon)$, the space of probability measures over $\varUpsilon$ with finite first moment, and we construct an extended distance $\mathcal{W}$ on $\mathscr{P}_{1}(\varUpsilon)$. The distance $\mathcal{W}$ corresponds, in our setting, to the Benamou--Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with $\mathcal{W}$. We establish non-local infinite-dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below. In particular, we obtain that: (a) the Ornstein--Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by $1$; (c) the distance $\mathcal{W}$ satisfies an HWI inequality.

math.PR↗
arXiv2023

Polyharmonic Fields and Liouville Quantum Gravity Measures on Tori of Arbitrary Dimension: from Discrete to Continuous

For an arbitrary dimension $n$, we study: (a) the Polyharmonic Gaussian Field $h_L$ on the discrete torus $\mathbb{T}^n_L = \frac{1}{L} \mathbb{Z}^{n} / \mathbb{Z}^{n}$, that is the random field whose law on $\mathbb{R}^{\mathbb{T}^{n}_{L}}$ given by \begin{equation*} c_n\, e^{-b_n\|(-Δ_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebesgue measure and $Δ_{L}$ is the discrete Laplacian; (b) the associated discrete Liouville Quantum Gravity measure associated with it, that is the random measure on $\mathbb{T}^{n}_{L}$ \begin{equation*}μ_{L}(dz) = \exp \Big( γh_L(z) - \frac{γ^{2}}{2} \mathbf{E} h_{L}(z) \Big) dz,\end{equation*} where $γ$ is a regularity parameter. As $L\to\infty$, we prove convergence of the fields $h_L$ to the Polyharmonic Gaussian Field $h$ on the continuous torus $\mathbb{T}^n = \mathbb{R}^{n} / \mathbb{Z}^{n}$, as well as convergence of the random measures $μ_L$ to the LQG measure $μ$ on $\mathbb{T}^n$, for all $|γ| < \sqrt{2n}$.

math.PR↗
arXiv2022

Ergodic Decompositions of Dirichlet Forms under Order Isomorphisms

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.

math.FA↗
arXiv2022

A Discovery Tour in Random Riemannian Geometry

We study random perturbations of Riemannian manifolds $(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields $h^\bullet: ω\mapsto h^ω$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^ω\colon\!\!= e^{2h^ω}\,\mathsf{g}$. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise. Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

math.PR↗
arXiv2022

Effective contraction of skinning maps

Using elementary hyperbolic geometry, we give an explicit formula for the contraction constant of the skinning map over moduli spaces of relatively acylindrical hyperbolic manifolds.

math.GT↗