arXiv subjects

Piotr Graczyk

Publications and source records attributed to Piotr Graczyk.

46 records · Page 3Linked to original sources

arXiv2014

Fisher Information and Exponential Families Parametrized by a Segment of Means

We consider natural and general exponential families $(Q_m)_{m\in M}$ on $\mathbb{R}^d$ parametrized by the means. We study the submodels $(Q_{θm_1+(1-θ)m_2})_{θ\in[0,1]}$ parametrized by a segment in the means domain, mainly from the point of view of the Fisher information. Such a parametrization allows for a parsimonious model and is particularly useful in practical situations when hesitating between two parameters $m_1$ and $m_2$. The most interesting examples are obtained when $\mathbb{R}^d$ is a linear space of matrices, in particular for Gaussian and Wishart models.

math.PR↗

arXiv2013

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through log-subharmonic functions, and use it to prove the equivalence of strong hypercontractivity and the strong log Sobolev inequality for such log-subharmonic functions.

math.FA↗

arXiv2012

On the product formula on non-compact Grassmannians

We study the absolute continuity of the convolution $δ_{e^X}^\natural \star δ_{e^Y}^\natural$ of two orbital measures on the symmetric space $SO_0(p,q)/SO(p)\timesSO(q)$, $q>p$. We prove sharp conditions on $X$, $Y\in\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for $\SO_0(p,q)/\SO(p)\times\SO(q)$ will also serve for the spaces $SU(p,q)/S(U(p)\timesU(q))$ and $Sp(p,q)/Sp(p)\timesSp(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $δ_{e^X}^\natural$.

math.RT↗

arXiv2012

Multidimensional Yamada-Watanabe theorem and its applications to particle systems

A multidimensional version of the Yamada-Watanabe theorem is proved. It implies a spectral matrix Yamada-Watanabe theorem. It is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes: Wishart and Jacobi matrix processes. The beta-versions of these particle systems are also considered.

math.PR↗

arXiv2012

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=Δ^{α/2}+ b\cdot\nabla$ with $α\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.

math.AP↗

arXiv2011

Hitting half-spaces or spheres by the Ornstein-Uhlenbeck type diffusions

The purpose of the paper is to provide a general method for computing hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ+ F\cdot\nabla, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in the paper is based on stochastic calculus and on skew product representation of multidimensional Brownian motion and yields more complete results as those based on Feynmann-Kac technique.

math.PR↗

arXiv2011

Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $Ω$-positive (virtual) quadratic map, where $Ω\subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $Ω$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones.

math.ST↗

arXiv2011

On exit time of stable processes

We study the exit time $τ=τ_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^α$ as the Laplace transform of a positive random variable with explicit density. Consequently, $τ$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $τ$ are obtained as an application.

math.PR↗

arXiv2010

On Wiener-Hopf factors for stable processes

We give a series representation of the logarithm of the bivariate Laplace exponent $κ$ of $α$-stable processes for almost all $α\in (0,2]$.

math.PR↗

arXiv2008

Hypercontractivity for log-subharmonic functions

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\RR^n$ and different classes of measures: Gaussian measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on $\RR$, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for {\em any} symmetric measure on $\RR$. For all measures on $\R$ for which we know the (SHC) holds, we prove that a log--Sobolev inequality holds in the log-subharmonic category with a constant {\em smaller} than the one for Gaussian measure in the classical context. This result is extended to all dimensions for compactly-supported measures.

math.FA↗