Subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

Prabhu Janakiraman; Alexander Volberg

Home/Archive/arXiv

arXiv · 1012.0948

Subordination by orthogonal martingales in $L^{p}, 1<p\le 2$

Prabhu Janakiraman·Alexander Volberg

View original source ↗Read source document ↗Export metadata ↓

Abstract

We are given two martingales on the filtration of the two dimensional Brownian motion. One is subordinated to another. We want to give an estimate of $L^p$-norm of a subordinated one via the same norm of a dominating one. In this setting this was done by Burkholder in \cite{Bu1}--\cite{Bu8}. If one of the martingales is orthogonal, the constant should drop. This was demonstrated in \cite{BaJ1}, when the orthogonality is attached to the subordinated martingale and when $2\le p<\infty$. This note contains an (almost obvious) observation that the same idea can be used in the case when the orthogonality is attached to a dominating martingale and $1<p\le 2$. Two other complementary regimes are considered in \cite{BJV_La}. When both martingales are orthogonal, see \cite{BJV_Le}. In these two papers the constants are sharp. We are not sure of the sharpness of the constant in the present note.

Explore related subjects

math.PR

Keep this discovery

BibTeX RIS

Prabhu Janakiraman, Alexander Volberg. 2010-12-04. Subordination by orthogonal martingales in $L^{p}, 1<p\le 2$. https://arxiv.org/abs/1012.0948

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗

arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗