IS10: Martingales and their applications in PDEs and Harmonic Analysis

date: 7/14/2025, time: 16:00-17:30, room: ICS 141

Organizer: Adam Osekowski (University of Warsaw, Poland)

Chair: Adam Osekowski (University of Warsaw, Poland)

Gaussian coupling on the Wiener space and stochastic differential equations

Stefan Geiss (Department of Mathematics and Statistics, University of Jyväskylä, Finland)

Using the coupling method on the Wiener space introduced by S. Geiss and Ylinen in \([1]\), we investigate regularity properties of fully path-dependent stochastic differential equations. The used coupling method is robust and allows for results under minimal structural assumptions. Two cases of the coupling method are of special interest: The isotropic coupling to treat the Malliavin Sobolev space \(\mathbb{D}_{1,2}\) and the related Besov spaces obtained by real interpolation, and secondly, a cut-off coupling to treat the \(L_p\)-variation of backward stochastic differential equations where the forward process is the investigated stochastic differential equation. The aim of the talk is two-fold: We re-call the coupling idea for the Gaussian structure on the Wiener space, and then we apply this method to fully path-dependent stochastic differential equations. The talk is based on joint work with Xilin Zhou \([2]\).

Bibliography

\([1]\) S. Geiss and J. Ylinen: Decoupling on the Wiener space, related Besov spaces, and applications to BSDEs. Memoirs AMS 1335, 2021.

\([2]\) S. Geiss and X. Zhou: Regularity of stochastic differential equations on the Wiener space by coupling. arXiv:2412.10836v2, 2025.

Plurisuperharmonic functions and sharp inequalities for analytic BMO functions and martingales

Adam Osękowski (University of Warsaw)

Suppose that \(\mathbb T\) is a unit circle and let \(ABMO(\mathbb T)\) denote the associated class of analytic BMO functions. The purpose of the talk will be to discuss the following sharp versions of John-Nirenberg and Fefferman inequalities for \(ABMO(\mathbb T)\).

Theorem 1. Suppose that \(f\in ABMO(\mathbb{T})\) satisfies \(\|f\|_{BMO}\leq 1\). Then for any \(\lambda <\sqrt{2}\) there is a finite \(C(\lambda)<\infty\) such that \[\int_{\mathbb{T}} |\exp(\lambda f(\zeta))|\mbox{d}\zeta \leq C(\lambda)\left|\exp\left(\int_{\mathbb{T}} f(\zeta)\mbox{d}\zeta\right)\right|.\] Furthermore, if \(\lambda\geq \sqrt{2}\), then the above estimate fails to hold for any finite choice of \(C(\lambda)\).

Theorem 2. Suppose that \(f\) is an element of the Hardy space \(H^1(\mathbb{T})\) and let \(g\) belong to \(ABMO(\mathbb{T})\). If \(\int_{\mathbb{T}} g(\zeta)\mbox{d}\zeta=0\), then we have the sharp bound \[\int_{\mathbb{T}} \overline{f(\zeta)}g(\zeta)\mbox{d}\zeta\leq \sqrt{e^2+1}\|f\|_{H^1(\mathbb{T})}\|g\|_{BMO}.\]

The martingale versions of the above results will also be discussed, as well as the relation of the statements to the existence of certain special plurisuperharmonic functions (cf. \([1]\), \([2]\), \([3]\)).

Bibliography

\([1]\) Hollenbeck, Brian; Verbitsky, Igor E. Best constants for the Riesz projection. J. Funct. Anal. 175 (2000), no. 2, 370–392

\([2]\) Gała̧zka, Tomasz; Osȩkowski, Adam. Sharp analytic version of Fefferman’s inequality. J. Funct. Anal. 288 (2025), no. 2, Paper No. 110707.

\([3]\) Vasyunin, Vasily; Volberg, Alexander. The Bellman function technique in harmonic analysis. Cambridge Studies in Advanced Mathematics, 186. Cambridge University Press, Cambridge, 2020. xvii+445 pp. ISBN: 978-1-108-48689-7

The Dubins Spider

Philip A. Ernst (Imperial College London)

This talk considers the so called Brownian spider, also known as Walsh Brownian motion, as first introduced in the epilogue of Walsh (1978). In an attempt to understand the unboundedness of the variation of the Wiener process, Lester Dubins asked the following question: for a Brownian spider with n rays how does one design a stopping time to maximize the "coverage" of Brownian motion on the spider for a given expected time? In this formulation, the coverage of the spider is measured as the sum of the lengths of its arms. This talk discusses some advances on the Dubins spider problem that have been made by Ernst (2016) and Ernst (2025). We shall also touch on a solution to a more "friendly" variant of the Dubins spider problem given by Bednarz, Ernst, and Osekowski (2024).