Page 676 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 676
PROBABILITY
of β are believed to also be relevant in theory (e.g. relevant for the study of self-adjoint and
Schrödinger operators) and applications (e.g. in logistics). For certain rational values of β, β-
ensembles are related to Jack polynomials, but for general β much less is known. In a seminal
article, Dumitriu and Edelman (2002) constructed tridiagonal random matrix models for gen-
eral β-Hermite and β-Laguerre ensembles and defined open problems for research on general
β-ensembles, including finding a unified formula for the level density in general β-case. In
general, level density is defined as distribution of a random eigenvalue of an ensemble (by the
Wigner semicircular law, the limiting distribution of the eigenvalue is semicircular). We derive
the formula for the level density in the general β-case depending on the multivariate Fuss-
Narayana polynomials (related to generalized Fuss-Narayana numbers) and homogenous poly-
nomials from operator-valued free probability theory. In an alternate way, we derive the general
β-case level density formulas using Malliavin-Stein fourth moment-based asymptotic calculus
and study their perturbation invariability (Wang and Yan, 2005; Kozhan, 2017). Derivations al-
low us large possibilities of additional work, and we present extensions to problems of sampling
general β-ensembles (referring to Li and Menon, 2012; Olver et al., 2013; Srakar and Verbic,
2020) and limiting entropy in β-ensembles related point processes (Mészaros, 2020).
A CLT for degenerate diffusions with periodic coefficients, and
application to homogenization of linear PDEs
Ivana Valentic´, ivana.valentic@math.hr
University of Zagreb, Croatia
Coauthor: Nikola Sandric´
Let Lε, ε > 0, be a second-order elliptic differential operator of the form Lε = a(·/ε) +
ε−1b(·/ε) T∇ + 2−1Tr c(·/ε) ∇∇T with a degenerate (possibly vanishing on a set of pos-
itive Lebesgue measure) diffusion coefficient c(x). We first show that the diffusion process
associated to Lε satisfies a functional CLT with Brownian limit as ε → 0, and then by em-
ploying probabilistic representation (the Feynman-Kac formula) of the solutions to the elliptic
boundary-value and the parabolic initial-value problem we conclude the homogenization result.
In the non-degenerate (uniformly elliptic) case these steps can be carried out by combining
classical PDE results and the fact that the underlying diffusion process does not show a singular
behavior in its motion, that is, it is irreducible. In the case of a degenerate diffusion part, this
deficiency is compensated by the assumption that the underlying diffusion process with positive
probability reaches the part of the state space where the diffusion term is non-degenerate. Also,
in this case it is not clear that we can rely on PDE techniques therefore the proofs are completely
based on stochastic analysis tools.
674
of β are believed to also be relevant in theory (e.g. relevant for the study of self-adjoint and
Schrödinger operators) and applications (e.g. in logistics). For certain rational values of β, β-
ensembles are related to Jack polynomials, but for general β much less is known. In a seminal
article, Dumitriu and Edelman (2002) constructed tridiagonal random matrix models for gen-
eral β-Hermite and β-Laguerre ensembles and defined open problems for research on general
β-ensembles, including finding a unified formula for the level density in general β-case. In
general, level density is defined as distribution of a random eigenvalue of an ensemble (by the
Wigner semicircular law, the limiting distribution of the eigenvalue is semicircular). We derive
the formula for the level density in the general β-case depending on the multivariate Fuss-
Narayana polynomials (related to generalized Fuss-Narayana numbers) and homogenous poly-
nomials from operator-valued free probability theory. In an alternate way, we derive the general
β-case level density formulas using Malliavin-Stein fourth moment-based asymptotic calculus
and study their perturbation invariability (Wang and Yan, 2005; Kozhan, 2017). Derivations al-
low us large possibilities of additional work, and we present extensions to problems of sampling
general β-ensembles (referring to Li and Menon, 2012; Olver et al., 2013; Srakar and Verbic,
2020) and limiting entropy in β-ensembles related point processes (Mészaros, 2020).
A CLT for degenerate diffusions with periodic coefficients, and
application to homogenization of linear PDEs
Ivana Valentic´, ivana.valentic@math.hr
University of Zagreb, Croatia
Coauthor: Nikola Sandric´
Let Lε, ε > 0, be a second-order elliptic differential operator of the form Lε = a(·/ε) +
ε−1b(·/ε) T∇ + 2−1Tr c(·/ε) ∇∇T with a degenerate (possibly vanishing on a set of pos-
itive Lebesgue measure) diffusion coefficient c(x). We first show that the diffusion process
associated to Lε satisfies a functional CLT with Brownian limit as ε → 0, and then by em-
ploying probabilistic representation (the Feynman-Kac formula) of the solutions to the elliptic
boundary-value and the parabolic initial-value problem we conclude the homogenization result.
In the non-degenerate (uniformly elliptic) case these steps can be carried out by combining
classical PDE results and the fact that the underlying diffusion process does not show a singular
behavior in its motion, that is, it is irreducible. In the case of a degenerate diffusion part, this
deficiency is compensated by the assumption that the underlying diffusion process with positive
probability reaches the part of the state space where the diffusion term is non-degenerate. Also,
in this case it is not clear that we can rely on PDE techniques therefore the proofs are completely
based on stochastic analysis tools.
674
