
Lorenzo Dello SchiavoUniversity of Rome Tor Vergata | UNIROMA2 · Dipartimento di Matematica
Lorenzo Dello Schiavo
Doctor of Natural Sciences
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40
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130
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Introduction
Additional affiliations
September 2020 - October 2024
November 2015 - August 2020
Education
October 2015 - November 2019
October 2013 - September 2015
October 2010 - October 2013
Publications
Publications (40)
We prove that the Dean-Kawasaki-type stochastic partial differential equation $$\partial \rho= \nabla\cdot (\sqrt{\rho\,}\, \xi) + \nabla\cdot \left(\rho\, H(\rho)\right)$$ with vector-valued space-time white noise $\xi$, does not admit solutions for any initial measure and any vector-valued bounded measurable function $H$ on the space of measures....
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 fl...
For an arbitrary dimension n$n$, we study: the polyharmonic Gaussian field hL$h_L$ on the discrete torus TLn=1LZn/Zn$\mathbb {T}^n_L = \frac{1}{L} \mathbb {Z}^{n} / \mathbb {Z}^{n}$, that is the random field whose law on RTLn$\mathbb {R}^{\mathbb {T}^{n}_{L}}$ given by cne−bn(−ΔL)n/4h2dh,$$\begin{equation*} \hspace*{-4.5pc}c_n\, \text{e}^{-b_n{\lef...
We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on...
For large classes of even‐dimensional Riemannian manifolds , we construct and analyze conformally invariant random fields. These centered Gaussian fields , called co‐polyharmonic Gaussian fields , are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: . They share a fundamental quasi‐invariance property unde...
We study random perturbations of a Riemannian manifold (M,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textsf{M},\textsf{g})$$\end{document} by means of so-calle...
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 t...
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--{\L}ojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms...
Let $\varUpsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $\pi$. 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...
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\|(-\Delta_L)^{n/4}h\|^2} dh, \end{equation*} where $dh$ is the Lebe...
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...
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.
This is the second paper of a series on configuration spaces $\Upsilon$ over singular spaces $X$. Here, we focus on geometric aspects of the extended metric measure space $(\Upsilon, \mathsf{d}_{\Upsilon}, \mu)$ equipped with the $L^2$-transportation distance $\mathsf{d}_{\Upsilon}$, and a mixed Poisson measure $\mu$. Firstly, we establish the esse...
We consider the open symmetric exclusion (SEP) and inclusion (SIP) processes on a bounded Lipschitz domain $\Omega$, with both fast and slow boundary. For the random walks on $\Omega$ dual to SEP/SIP we establish: a functional-CLT-type convergence to the Brownian motion on $\Omega$ with either Neumann (slow boundary), Dirichlet (fast boundary), or...
We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying the quasi curvature-dimension condition recently introduced in Milman (Commun Pure Appl Math, to appear). We provide several applications to properties of the corresponding heat semigroup. In particular, under the additional assumption of infinitesimal Hilbertianity, we...
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.
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...
We study 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 metrizab...
We construct a canonical differential structure on the configuration space $\boldsymbol\Upsilon$ over a singular base space $X$ and with a general invariant measure $\mu$ on $\boldsymbol\Upsilon$. We present an analytic structure on $\boldsymbol\Upsilon$, constructing a strongly local Dirichlet form $\mathcal E$ on $L^2(\boldsymbol\Upsilon, \mu)$ f...
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...
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 distanc...
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 s...
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 $h^\bullet: \omega\mapsto h^\omega$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^\omega\colon\!\!= e^{2h^\omega}\,...
We prove the Rademacher property for a large class of generalized intrinsic extended pseudo-distances on quasi-regular strongly local Dirichlet spaces, including spaces admitting no square field operator. We discuss various applications of this result, and, in combination with the Sobolev-to-Lipschitz property, we prove Varadhan-type heat-kernel sh...
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 sp...
Let P be any Borel probability measure on the L2-Wasserstein space (P2(M),W2) over a closed Riemannian manifold M. We consider the Dirichlet form E induced by P and by the Wasserstein gradient on P2(M). Under natural assumptions on P, we show that W2-Lipschitz functions on P2(M) are contained in the Dirichlet space D(E) and that W2 is dominated by...
We construct a recurrent diffusion process with values in the space of probability measures over an arbitrary closed Riemannian manifold of dimension $d\ge 2$. The process is associated with the Dirichlet form defined by integration of the Wasserstein gradient w.r.t. the Dirichlet-Ferguson measure, and is the counterpart on multi-dimensional base s...
Let $\mathbb P$ be any Borel probability measure on the $L^2$-Wasserstein space $(\mathscr{P}_2(X),W_2)$ over a closed Riemannian manifold $X$. We consider the Dirichlet energy integral $\mathcal E$ induced by $\mathbb P$ and by the Wasserstein gradient on $\mathscr{P}_2(X)$. Under natural assumptions on $\mathbb P$, we show that $W_2$-Lipschitz fu...
We compute characteristic functionals of Dirichlet-Ferguson measures over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type $A$....
We characterize the Dirichlet-Ferguson measure over a locally compact Polish diffuse probability space as the unique random measure satisfying a Mecke-type identity.
We prove a characterization of the Dirichlet-Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.
We show that the chaos representation of some Compound Poisson Type processes displays an underlying intrinsic combinatorial structure, partly independent of the chosen process. From the computational viewpoint, we solve the arising combinatorial complexity by means of the moments/cumulants duality for the laws of the corresponding processes, thems...
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 cyb...
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 cyb...
Given a non-linear Dirichlet problem, we prove a result of continuous dependence, for both the solution and its gradient, on the zero order term in the case of low Marcinkiewicz summability for the latter, relying on the existence results and estimates recently proved in Boccardo (Ann Mat Pura Appl 188(4):591-601, 2009). As a consequence, solutions...
Recent experimental breakthroughs have finally allowed to implement in-vitro reaction kinetics (the so called 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), p...
Biotechnological expertise and related tools (required e.g. for drug
synthesis) increases daily mainly driven by a continuum of tumultuous
experimental breakthroughs. In particular, recently, scientists have been able
to build in-vitro reaction kinetics (among enzymatic proteins and their
ligands) which code for two-inputs logic gates mimicking the...