arXiv subjects

L. Dello Schiavo

Publications and source records attributed to L. Dello Schiavo.

3 recordsLinked to original sources

arXiv2020

The Dirichlet-Ferguson Diffusion on the Space of Probability Measures over a Closed Riemannian Manifold

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 spaces to the Modified Massive Arratia Flow over the unit interval described in V. Konarovskyi, M.-K. von Renesse, Comm. Pure Appl. Math., 72, 0764-0800 (2019). Together with two different constructions of the process, we discuss its ergodicity, invariant sets, finite-dimensional approximations, and Varadhan short-time asymptotics.

math.PR↗

arXiv2018

Characteristic Functionals of Dirichlet Measures

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 study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet-Ferguson measure is also obtained.

math.PR↗

arXiv2016

Moments and Cumulants in Infinite Dimensions with Applications to Poisson, Gamma and Dirichlet-Ferguson Random Measures

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, themselves measures on distributional spaces, and provide a combinatorial interpretation of the associated 'extended' Fock spaces. From the theoretical viewpoint, in the case of the Gamma measure, we trace back such complexity to its 'simplicial part', i.e. the Dirichlet-Ferguson measure, hence to the Dirichlet distribution on the finite-dimensional simplex. We thoroughly explore the combinatorial and algebraic properties of the latter distribution, arising in connection with cycle index polynomials of symmetric groups and dynamical symmetry algebras of confluent Lauricella functions.

math.PR↗