Week 13.01.2025 – 19.01.2025

The term "affinity" in the thermodynamic sense dates back to the works of Theophile de Donder in the 1920s, where it referred to the chemical potential difference that drives a reaction forward, and quantifies the entropy production and fluctuation properties of the reaction. However, for coupled chemical reactions no single quantity exists that captures the direction, dissipation, and fluctuations of the reactions.

In mesoscopic experiments it is common to observe a single fluctuating current, while the complete set of currents describing the underlying system of coupled reactions is inaccessible. For such scenarios with partial information, we introduce an "effective affinity" associated with the observed current. Like the thermodynamic affinity of single reactions, the effective affinity quantifies dissipative and fluctuation properties of currents. Notably, the effective affinity multiplied by the current lower bounds the rate of dissipation, and the effective affinity determines the direction, first-passage, and extreme value statistics of fluctuating currents. To derive these results, we also introduce a family of martingales associated with generic currents in Markov processes.

I will give a brief introduction into the theory of integral Hankel operators on the positive half-line. For a natural subclass of positive semi-definite integral Hankel operators, I will explain how to set up a direct and inverse spectral problem and how to solve it. This is work in progress with Sergei Treil (Brown).

Many testing problems are readily amenable to randomised tests such as those employing data splitting. However, despite their usefulness in principle, randomised tests have obvious drawbacks. Firstly, two analyses of the same dataset may lead to different results. Secondly, the test typically loses power because it does not fully utilise the entire sample. As a remedy to these drawbacks, we study how to combine the test statistics or p-values resulting from multiple random realisations such as through random data splits. We develop rank-transformed subsampling as a general method for delivering large sample inference about the combined statistic or p-value under mild assumptions. We apply our methodology to a wide range of problems, including testing unimodality in high-dimensional data, testing goodness-of-fit of parametric quantile regression models, testing no direct effect in a sequentially randomised trial and calibrating cross-fit double machine learning confidence intervals. In contrast to existing p-value aggregation schemes that can be highly conservative, our method enjoys type-I error control that asymptotically approaches the nominal level. Moreover, compared to using the ordinary subsampling, we show that our rank transform can remove the first-order bias in approximating the null under alternatives and greatly improve power.