arXiv subjects

Lorenzo Dello Schiavo

Publications and source records attributed to Lorenzo Dello Schiavo.

26 records · Page 2Linked to original sources

arXiv2022

Configuration Spaces over Singular Spaces -- II. Curvature

This is the second paper of a series on configuration spaces $Υ$ over singular spaces $X$. Here, we focus on geometric aspects of the extended metric measure space $(Υ, \mathsf{d}_Υ, μ)$ equipped with the $L^2$-transportation distance $\mathsf{d}_Υ$, and a mixed Poisson measure $μ$. Firstly, we establish the essential self-adjointness and the $L^p$-uniqueness for the Laplacian on $Υ$ lifted from $X$. Secondly, we prove the equivalence of Bakry-Émery curvature bounds on $X$ and on $Υ$, without any metric assumption on $X$. We further prove the Evolution Variation Inequality on $Υ$, and introduce the notion of synthetic Ricci-curvature lower bounds for the extended metric measure space $Υ$. As an application, we prove the Sobolev-to-Lipschitz property on $Υ$ over singular spaces $X$, originally conjectured in the case when $X$ is a manifold by M. Röckner and A. Schield. As a further application, we prove the $L^\infty$-to-$\mathsf{d}_Υ$-Lipschitz regularization of the heat semigroup on $Υ$ and gives a new characterization of the ergodicity of the corresponding particle systems in terms of optimal transport.

math.MG↗

arXiv2021

Sobolev-to-Lipschitz Property on QCD-spaces and Applications

We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature dimension condition recently introduced in E. Milman, The Quasi Curvature-Dimension Condition with applications to sub-Riemannian manifolds, Comm. Pure Appl. Math. (to appear, arXiv:1908.01513v5). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the heat semigroup with respect to the distance, and prove the irreducibility of the heat semigroup. These result apply in particular to large classes of (ideal) sub-Riemannian manifolds.

math.MG↗

arXiv2021

Configuration spaces over singular spaces -- I. Dirichlet-Form and Metric Measure Geometry

We construct a canonical differential structure on the configuration space $Υ$ over a singular base space $X$ and with a general invariant measure $μ$ on $Υ$. We present an analytic structure on $Υ$, constructing a strongly local Dirichlet form $\mathcal E$ on $L^2(Υ, μ)$ for $μ$ in a large class of probability measures. We then investigate the geometric structure of the extended metric measure space $Υ$ endowed with the $L^2$-transportation extended distance $\mathsf{d}_Υ$ and with the measure $μ$. By establishing Rademacher- and Sobolev-to-Lipschitz-type properties for $\mathcal E$, we finally provide a complete identification of the analytic and the geometric structure -- the canonical differential structure induced on $Υ$ by $X$ and $μ$ -- showing that $\mathcal E$ coincides with the Cheeger energy of $(Υ,\mathsf{d}_Υ,μ)$ and that the intrinsic distance of $\mathcal E$ coincides with $\mathsf{d}_Υ$. The class of base spaces to which our results apply includes sub-Riemannian manifolds, RCD spaces, and path/loop spaces over Riemannian manifolds; as for $μ$ our results include quasi-Gibbs measures, in particular: Poisson measures, canonical Gibbs measures, as well as some determinantal/permanental point processes (sine$_β$, Airy$_β$, Bessel$_{α,β}$, Ginibre). A number of applications to interacting particle systems and infinite-dimensional metric measure geometry are also discussed. In particular, we prove the universality of the $L^2$-transportation distance $\mathsf{d}_Υ$ for the Varadhan short-time asymptotics for diffusions on $Υ$, regardless of the choice of $μ$. Many of our results are new even in the case of configuration spaces over Euclidean spaces.

math.PR↗

arXiv2021

Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

We study superpositions and direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin, Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

math.FA↗

arXiv2021

Rademacher-type Theorems and Sobolev-to-Lipschitz Properties for Strongly Local Dirichlet Spaces

We extensively discuss the Rademacher and Sobolev-to-Lipschitz properties for generalized intrinsic distances on strongly local Dirichlet spaces possibly without square field operator. We present many non-smooth and infinite-dimensional examples. As an application, we prove the integral Varadhan short-time asymptotic with respect to a given distance function for a large class of strongly local Dirichlet forms.

math.FA↗

arXiv2020

A Rademacher-type Theorem on $L^2$-Wasserstein Spaces over Closed Riemannian Manifolds

Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(M),W_2)$ over a closed Riemannian manifold $M$. We consider the Dirichlet form $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein gradient on $\mathscr{P}_2(M)$. Under natural assumptions on $\mathbb P$, we show that $W_2$-Lipschitz functions on $\mathscr{P}_2(M)$ are contained in the Dirichlet space $\mathrm{dom}(\mathcal{E})$ and that $W_2$ is dominated by the intrinsic metric induced by $\mathcal E$. We illustrate our results by giving several detailed examples.

math.FA↗

arXiv2016

Complete integrability of information processing by biochemical reactions

Statistical mechanics provides an effective framework to investigate information processing in biochemical reactions. Within such framework far-reaching analogies are established among (anti-) cooperative collective behaviors in chemical kinetics, (anti-)ferromagnetic spin models in statistical mechanics and operational amplifiers/flip-flops in cybernetics. The underlying modeling -- based on spin systems -- has been proved to be accurate for a wide class of systems matching classical (e.g. Michaelis--Menten, Hill, Adair) scenarios in the infinite-size approximation. However, the current research in biochemical information processing has been focusing on systems involving a relatively small number of units, where this approximation is no longer valid. Here we show that the whole statistical mechanical description of reaction kinetics can be re-formulated via a mechanical analogy -- based on completely integrable hydrodynamic-type systems of PDEs -- which provides explicit finite-size solutions, matching recently investigated phenomena (e.g. noise-induced cooperativity, stochastic bi-stability, quorum sensing). The resulting picture, successfully tested against a broad spectrum of data, constitutes a neat rationale for a numerically effective and theoretically consistent description of collective behaviors in biochemical reactions.

cond-mat.stat-mech↗

arXiv2014

Notes on stochastic (bio)-logic gates: the role of allosteric cooperativity

Recent experimental breakthroughs have finally allowed to implement in-vitro reaction kinetics (the so called {\em enzyme based logic}) which code for two-inputs logic gates and mimic the stochastic AND (and NAND) as well as the stochastic OR (and NOR). This accomplishment, together with the already-known single-input gates (performing as YES and NOT), provides a logic base and paves the way to the development of powerful biotechnological devices. The investigation of this field would enormously benefit from a self-consistent, predictive, theoretical framework. Here we formulate a complete statistical mechanical description of the Monod-Wyman-Changeaux allosteric model for both single and double ligand systems, with the purpose of exploring their practical capabilities to express logical operators and/or perform logical operations. Mixing statistical mechanics with logics, and quantitatively our findings with the available biochemical data, we successfully revise the concept of cooperativity (and anti-cooperativity) for allosteric systems, with particular emphasis on its computational capabilities, the related ranges and scaling of the involved parameters and its differences with classical cooperativity (and anti-cooperativity).

cond-mat.dis-nn↗