Yueh-Sheng Hsu

Yueh-Sheng Hsu

Postdoc in probability / stochastic analysis / SPDEs

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Variance renormalisation of two dimensional gPAM with differentiated space white noise

Published:

In this work, we consider the gPAM equation with a differentiated space white noise in two dimensions. A particularity of this equation is that it just falls on the borderline of the subcriticality condition; moreover, even if one could have chosen a slightly more regular noise, the variance of certain non-linear functionals of the noise is expected to explode and the local solution theory would still fail. To tame down this variance blowup, a multiplicative renormalisation is introduced. This multiplicative renormalisation was first carried out by [Hairer ‘24] in the case of KPZ, and a general prediction was made there for a wider class of equations. The current work therefore has the objective to show gPAM falls into this picture. It is worth noting that, while a Da Prato-Debussche trick was used in the KPZ case, no such trick is available for gPAM. One thus has to work on the level of singular SPDE machineries such as regularity structure to obtain the desired result.

Sharp convergence rates for singular SPDEs - the case of KPZ

Published:

For subcritical singular SPDEs, the theory of regularity structures typically yields, as the mollification scale \(\varepsilon\to0\), a convergence rate of order \(\varepsilon^\gamma\) for some small \(\gamma>0\). The exponent \(\gamma\) is limited by the gap between the optimal regularity of the driving noise and the regularity in which the noise is measured; in this sense, one can trade regularity of the solution for convergence rate. For instance, under the subcritical and no variance blow-up conditions, in the KPZ equation driven by space-time white noise one obtains \(\gamma<\frac14\), while for \(\Phi^4_3\) one gets \(\gamma<\frac13\). In this work, we show that the optimal exponent for KPZ is \(\gamma=\frac12\). More precisely, we prove if \(u_\varepsilon\) denotes the KPZ solution driven by mollified white noise at scale \(\varepsilon\), then \(\varepsilon^{-1/2}(u_\varepsilon - u_{\varepsilon/2})\) is bounded in \(L^p(\Omega)\) for all \(p \ge 2\) and converges in law to a non-trivial limit \(w\), which solves another singular SPDE driven by an independent white noise. This CLT-type result identifies the optimal convergence rate for KPZ. We also conjecture that the same mechanism yields the optimal rates \(\gamma=1\) for \(\Phi^4_3\) and \(\gamma=2\) for \(\Phi^4_2\), which will be studied in forthcoming work.

teaching

2020-21, 2023-24 - L2 Probability - Tutorial (TD)

Tutorial (TD), Université Paris-Dauphine PSL, CEREMADE, 2023

For the winter semesters of 2020 and 2023, I was responsible for the Bachelor-level tutorial sessions on Probability in Paris-Dauphine University.

2020-2024 - M1 Discrete-time Processes - Tutorial (TD)

Tutorial (TD), Université Paris-Dauphine PSL, CEREMADE, 2024

For four consecutive years (2020 - 2024), I was responsible for the Master-level tutorial sessions on Discrete-time Processes in Paris-Dauphine University, which introduces students to topics such as conditional expectation, martingale theory and Markov chains.

2026 - AKANA AKWTH Stochastic PDEs

Lecture, Technische Unversität Wien, Institut für Analysis und Scientific Computing, 2026

In the summer semester of 2026, I will give a Master-level course on stochastic partial differential equations. The lecture is designed to give an overall introduction to different aspects in modern SPDE theory.