Page 675 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 675
PROBABILITY
which generates a moving orthogonal family of eigenfunctions. Fourier-transforming the Fokker–
Planck equation in terms of these eigenfunctions produces a series of simple ordinary differen-
tial equations for the Fourier coefficients of the solution. For example, in the time-homogeneous
first-passage time case, these equations can be expressed as
dω = C da − Λ ω,
dt dt
where ω is the vector of Fourier coefficients, a is the moving boundary, Λ is the diagonal matrix
of the moving Sturm–Liouville eigenvalues λk, and C is the skew-symmetric matrix defined by
1 ∂λk ∂λm
[C]km = λk − λm ∂a ∂a
for k = m.
First passage times of subdiffusive processes over stochastic boundaries
Pierre Patie, pp396@cornell.edu
Cornell University, United States
Let X be the subdiffusive process defined by
Xt = YDt t ≥ 0,
where Y is a Lévy process and Dt = inf{s > 0; σs > t} with σ a subordinator independent of
Y.
We start by providing a composite Wiener-Hopf factorization to characterize the law of the
pair (Ta(κ), (X − σ)Ta(κ)) where
Ta(κ) = inf{t > 0; Xt > a + κt}
with a ∈ R and κ a (possibly degenerate) subordinator independent of Y and σ. We proceed
by providing a detailed analysis of the cases where either σ is a stable subordinator or X is
spectrally negative. Our proofs hinge on a variety of techniques including excursion theory,
change of measure, asymptotic analysis and establishing a link between subdiffusive processes
and a subclass of semi-regenerative processes.
Level densities for general β-ensembles: An operator-valued free
probability perspective
Andrej Srakar, srakara@ier.si
Institute for Economic Research (IER), Slovenia
Coauthor: Miroslav Verbicˇ
Random point processes corresponding to β-ensembles for arbitrary β > 0, or, equivalently, log
gases at inverse temperature β, are being subject to intense study. The orthogonal, unitary, and
symplectic ensembles (β = 1, 2, or 4, respectively) are now well understood, but other values
673
which generates a moving orthogonal family of eigenfunctions. Fourier-transforming the Fokker–
Planck equation in terms of these eigenfunctions produces a series of simple ordinary differen-
tial equations for the Fourier coefficients of the solution. For example, in the time-homogeneous
first-passage time case, these equations can be expressed as
dω = C da − Λ ω,
dt dt
where ω is the vector of Fourier coefficients, a is the moving boundary, Λ is the diagonal matrix
of the moving Sturm–Liouville eigenvalues λk, and C is the skew-symmetric matrix defined by
1 ∂λk ∂λm
[C]km = λk − λm ∂a ∂a
for k = m.
First passage times of subdiffusive processes over stochastic boundaries
Pierre Patie, pp396@cornell.edu
Cornell University, United States
Let X be the subdiffusive process defined by
Xt = YDt t ≥ 0,
where Y is a Lévy process and Dt = inf{s > 0; σs > t} with σ a subordinator independent of
Y.
We start by providing a composite Wiener-Hopf factorization to characterize the law of the
pair (Ta(κ), (X − σ)Ta(κ)) where
Ta(κ) = inf{t > 0; Xt > a + κt}
with a ∈ R and κ a (possibly degenerate) subordinator independent of Y and σ. We proceed
by providing a detailed analysis of the cases where either σ is a stable subordinator or X is
spectrally negative. Our proofs hinge on a variety of techniques including excursion theory,
change of measure, asymptotic analysis and establishing a link between subdiffusive processes
and a subclass of semi-regenerative processes.
Level densities for general β-ensembles: An operator-valued free
probability perspective
Andrej Srakar, srakara@ier.si
Institute for Economic Research (IER), Slovenia
Coauthor: Miroslav Verbicˇ
Random point processes corresponding to β-ensembles for arbitrary β > 0, or, equivalently, log
gases at inverse temperature β, are being subject to intense study. The orthogonal, unitary, and
symplectic ensembles (β = 1, 2, or 4, respectively) are now well understood, but other values
673
