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Ivan Yaroslavtsev

Publications and source records attributed to Ivan Yaroslavtsev.

10 recordsLinked to original sources
arXiv2026

Characterizations of the UMD property via tail estimates for tangent processes

We characterize the UMD property of a Banach space by tail inequalities for maximal functions of tangent conditionally symmetric processes. More precisely, we prove that a Banach space $V$ is UMD if and only if for some (equivalently, for all) $p\in(0,\infty)$ one has that \[ \mathbb P(\sup_{r\geq 0} \| N_r\|>t)\lesssim_{p,V}\Bigl(\frac{s^p}{t^p}+\mathbb P(\sup_{r\geq 0} \| M_r\|>s)\Bigr), \qquad s,t>0, \] for all tangent conditionally symmetric $V$-valued processes $M$ and $N$. We further show that this estimate is equivalent to suitable Lorentz norm inequalities for the associated maximal functions, and obtain analogous characterizations in the discrete-time, continuous-time, and purely discontinuous settings.

math.FA↗
arXiv2025

Stabilization by transport noise and enhanced dissipation in the Kraichnan model

Stabilization and sufficient conditions for mixing by stochastic transport are shown. More precisely, given a second order linear operator with possibly unstable eigenvalues on a smooth compact Riemannian manifold, it is shown that the inclusion of transport noise can imply global asymptotic stability. Moreover, it is shown that an arbitrary large exponential rate of convergence can be reached, implying enhanced dissipation. The sufficient conditions are shown to be satisfied by the so-called Kraichnan model for stochastic transport of passive scalars in turbulent fluids. In addition, an example is given showing that it can be sufficient to force four modes in order to induce stabilization.

math.PR↗
arXiv2019

The Hilbert transform and orthogonal martingales in Banach spaces

Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E Ψ(N_t) \leq C_{Φ,Ψ,X} \mathbb E Φ(M_t),\;\;\; t\geq 0, $$ where $Φ, Ψ:X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{Φ,Ψ,X}$ coincides with the $Φ,Ψ$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $Φ,Ψ$-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if $Φ$ is symmetric. We also prove that under certain natural assumptions on $Φ$ and $Ψ$, the condition $C_{Φ,Ψ,X}<\infty$ yields the UMD property of the space $X$. As an application, we provide comparison of $L^p$-norms of the periodic Hilbert transform to Wiener and Paley-Walsh decoupling constants. We also study the norms of the periodic, nonperiodic and discrete Hilbert transforms, present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

math.FA↗
arXiv2018

On strongly orthogonal martingales in UMD Banach spaces

In the present paper we introduce the notion of strongly orthogonal martingales. Moreover, we show that for any UMD Banach space $X$ and for any $X$-valued strongly orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$ one has that for any $1<p<\infty$ \[ \mathbb E \|N_t\|^p \leq χ_{p, X}^p \mathbb E \|M_t\|^p,\;\;\; t\geq 0, \] with the sharp constant $χ_{p, X}$ being the norm of a decoupling-type martingale transform and being within the range \[ \max\Bigl\{\sqrt{β_{p, X}}, \sqrt{\hbar_{p,X}}\Bigr\} \leq \max\{β_{p, X}^{γ,+}, β_{p, X}^{γ, -}\} \leq χ_{p, X} \leq \min\{β_{p, X}, \hbar_{p,X}\}, \] where $β_{p, X}$ is the UMD$_p$ constant of $X$, $\hbar_{p, X}$ is the norm of the Hilbert transform on $L^p(\mathbb R; X)$, and $β_{p, X}^{γ,+}$ and $ β_{p, X}^{γ, -}$ are the Gaussian decoupling constants.

math.PR↗
arXiv2018

The UMD property for Musielak--Orlicz spaces

In this paper we show that Musielak--Orlicz spaces are UMD spaces under the so-called $Δ_2$ condition on the generalized Young function and its complemented function. We also prove that if the measure space is divisible, then a Musielak--Orlicz space has the UMD property if and only if it is reflexive. As a consequence we show that reflexive variable Lebesgue spaces $L^{p(\cdot)}$ are UMD spaces.

math.FA↗
arXiv2018

On the martingale decompositions of Gundy, Meyer, and Yoeurp in infinite dimensions

We show that the canonical decomposition (comprising both the Meyer-Yoeurp and the Yoeurp decompositions) of a general $X$-valued local martingale is possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD Banach space if and only if for any $X$-valued local martingale $M$ there exist a continuous local martingale $M^c$, a purely discontinuous quasi-left continuous local martingale $M^q$, and a purely discontinuous local martingale $M^a$ with accessible jumps such that $M = M^c + M^q + M^a$. The corresponding weak $L^1$-estimates are provided. Important tools used in the proof are a new version of Gundy's decomposition of continuous-time martingales and weak $L^1$-bounds for a certain class of vector-valued continuous-time martingale transforms.

math.PR↗
arXiv2018

Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.

math.FA↗
arXiv2016

Cylindrical continuous martingales and stochastic integration in infinite dimensions

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $γ$-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.

math.PR↗