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Mihai N. Pascu

Publications and source records attributed to Mihai N. Pascu.

9 recordsLinked to original sources
arXiv2022

Convex ordering of Pólya random variables and monotonicity of the error estimate of Bernstein-Stancu operators

In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the absolute value of the error of Bernstein-Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proof relies on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of Pólya random variables with respect to the replacement parameter.

math.PR↗
arXiv2018

An improved Popoviciu-type inequality for a new Bernstein-type operator

Recently we introduced a new Bernstein-type operator using Pólya's urn model with negative replacement, and we showed that it satisfies a Popoviciu-type inequality with a constant slightly larger than that of the corresponding inequality for the classical Bernstein operator. In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant in the Popoviciu inequality for the new operator is in fact smaller than the corresponding constant for the Bernstein operator.

math.CA↗
arXiv2017

A new Bernstein-type operator based on Pólya's urn model with negative replacement

Using Pólya's urn model with negative replacement we introduce a new Bernstein-type operator and we show that the new operator improves upon the known estimates for the classical Bernstein operator. We also provide numerical evidence showing that the new operator gives a better approximation when compared to some other classical Bernstein-type operators.

math.CA↗
arXiv2015

Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds

In this paper we introduce three Markovian couplings of Brownian motions on smooth Riemannian manifolds without boundary which sit at the crossroad of two concepts. The first concept is the one of shy coupling put forward in \cite{Burdzy-Benjamini} and the second concept is the lower bound on the Ricci curvature and the connection with couplings made in \cite{ReSt}. The first construction is the shy coupling, the second one is a fixed-distance coupling and the third is a coupling in which the distance between the processes is a deterministic exponential function of time. The result proved here is that an arbitrary Riemannian manifold satisfying some technical conditions supports shy couplings. If in addition, the Ricci curvature is non-negative, there exist fixed-distance couplings. Furthermore, if the Ricci curvature is bounded below by a positive constant, then there exists a coupling of Brownian motions for which the distance between the processes is a decreasing exponential function of time. The constructions use the intrinsic geometry, and relies on an extension of the notion of frames which plays an important role for even dimensional manifolds. In fact, we provide a wider class of couplings in which the distance function is deterministic in Theorem \ref{t:100} and Corollary~\ref{Cor:9}. As an application of the fixed-distance coupling we derive a maximum principle for the gradient of harmonic functions on manifolds with non-negative Ricci curvature. As far as we are aware of, these constructions are new, though the existence of shy couplings on manifolds is suggested by Kendall in \cite{Kendall}.

math.PR↗
arXiv2013

$φ$-strong solutions and uniqueness of 1-dimensional stochastic differential equations

In this paper we consider stochastic differential equations with discontinuous diffusion coefficient of varying sign, for which weak existence and uniqueness holds but strong uniqueness fails. We introduce the notion of $φ$-strong solution, and we show that under certain conditions on the diffusion coefficient a $φ$-strong solution exists and it is unique. We also give a construction of a $φ$-strong solution and a weak solution in terms of the notion of \emph{i.i.d. sign choice} introduced in the paper, and we give an explicit representation of the set of all weak solutions, which in particular explains the reason for the lack of strong existence and uniqueness.

math.PR↗
arXiv2010

Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture

In a series of papers, Burdzy et. al. introduced the \emph{mirror coupling} of reflecting Brownian motions in a smooth bounded domain $D\subset \mathbb{R}^{d}$, and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on $D$. In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domains $D_{1},D_{2}\subset \mathbb{R}^{d}$. As an application of the construction, we derive a unifying proof of the two main results concerning the validity of Chavel's conjecture on the domain monotonicity of the Neumann heat kernel, due to I. Chavel (\cite{Chavel}), respectively W. S. Kendall (\cite{Kendall}).

math.PR↗
arXiv2009

Neighborhoods of univalent functions

The main result shows a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighborhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro-Warschawski-Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.

math.CV↗
arXiv2008

On a conjecture of Laugesen and Morpurgo

A well known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal element of the Neumann heat kernel of the unit ball in $\mathbb{R}^{n}$ ($n\geq1$) is a radially increasing function. In this paper, we use probabilistic arguments to settle this conjecture, and, as an application, we derive a new proof of the Hot Spots conjecture of J. Rauch in the case of the unit disk.

math.PR↗