L3

The Fokker-Planck equation is one of the major tools of statistical physics in the description of stochastic processes, with numerous applications in physics, chemistry and biology. In the case of heterogeneous diffusions, the formulation of the equation depends on the choice of the discretization of the stochastic integral in the underlying Langevin-equation due to the multiplicative noise. In the Fokker-Planck equation, the choice of discretization then enters as a parameter in the definition of drift and diffusion terms. I show how both short- and long-time limits are affected by this choice. In the long-time limit, the existence of normalizable probability distribution functions is not always guaranteed which can be remedied by invoking elements of infinite ergodic theory. 

[1] S. Giordano, F. Cleri, R. Blossey, Phys Rev E 107, 044111 (2023)

[2] T. Dupont, S. Giordano, F. Cleri, R. Blossey, arXiv:2401.01765 (2024)

This will be an informal discussion seminar based on https://arxiv.org/abs/2211.07592:

The constrained Hamiltonian analysis of geometric actions is worked out before applying the construction to the extended Bondi-Metzner-Sachs group in four dimensions. For any Hamiltonian associated with an extended BMS4 generator, this action provides a field theory in two plus one spacetime dimensions whose Poisson bracket algebra of Noether charges realizes the extended BMS4 Lie algebra. The Poisson structure of the model includes the classical version of the operator product expansions that have appeared in the context of celestial holography. Furthermore, the model reproduces the evolution equations of non-radiative asymptotically flat spacetimes at null infinity.

Positive logic is a generalisation of full first-order logic, where negation is not built in, but can be added as desired. In joint work with Jan Dobrowolski we succesfully generalised the recent development on Kim-independence in NSOP1 theories to the positive setting. One of the important theorems in this development is the independence theorem, whose statement is very similar to the well-known statement for simple theories, and allows us to amalgamate independent types. In this talk we will have a closer look at the proof of this theorem, and what needs to be changed to make the proof work in positive logic compared to full first-order logic.

Causal optimal transport and the related adapted Wasserstein distance have recently been popularized as a more appropriate alternative to the classical Wasserstein distance in the context of stochastic analysis and mathematical finance. In this talk, we establish some interesting consequences of causality for transports on the space of continuous functions between the laws of stochastic differential equations.
 

We first characterize bicausal transport plans and maps between the laws of stochastic differential equations. As an application, we are able to provide necessary and sufficient conditions for bicausal transport plans to be induced by bi-causal maps. Analogous to the classical case, we show that bicausal Monge transports are dense in the set of bicausal couplings between laws of SDEs with unique strong solutions and regular coefficients.

 This is a joint work with Rama Cont.

Diffusion models, which transform noise into new data instances by reversing a Markov diffusion process, have become a cornerstone in modern generative models. A key component of these models is to learn the score function through score matching. While the practical power of diffusion models has now been widely recognized, the theoretical developments remain far from mature. Notably, it remains unclear whether gradient-based algorithms can learn the score function with a provable accuracy. In this talk, we develop a suite of non-asymptotic theory towards understanding the data generation process of diffusion models and the accuracy of score estimation. Our analysis covers both the optimization and the generalization aspects of the learning procedure, which also builds a novel connection to supervised learning and neural tangent kernels.

This is based on joint work with Yinbin Han and Meisam Razaviyayn (USC).

We introduce a methodology based on Multireference Alignment (MRA) for lead-lag detection in multivariate time series, and demonstrate its applicability in developing trading strategies. Specifically designed for low signal-to-noise ratio (SNR) scenarios, our approach estimates denoised latent signals from a set of time series. We also investigate the impact of clustering the time series on the recovery of latent signals. We demonstrate that our lead-lag detection module outperforms commonly employed cross-correlation-based methods. Furthermore, we devise a cross-sectional trading strategy that capitalizes on the lead-lag relationships uncovered by our approach and attains significant economic benefits. Promising backtesting results on daily equity returns illustrate the potential of our method in quantitative finance and suggest avenues for future research.

Global scale ocean currents are strongly constrained by the Earth’s rotation, while this effect is generally negligible at small scales. In between, motions with scales from 1-10km are marginally affected by the Earth’s rotation. These intermediate scales, collectively termed the ocean submesoscale, have been hidden from view until recent years. Evidence from field measurements, numerical models, and satellite data have shown that submesoscales play a particularly important role in the upper ocean where they help to control the transport of material between the ocean surface and interior. In this talk I will review some recent work on submesoscale dynamics and their influence on biogeochemistry and accumulation of microplastics in the surface waters.