Following Rosati and Perkowski’s work on constructing the first version of a rough super Brownian motion, we generalize the rough super Brownian motion to the case when the branching mechanism has infinite variance. In both case, we can prove the compact support properties and the exponential persistence.

Generating high-fidelity time series data using generative adversarial networks (GANs) remains a challenging task, as it is difficult to capture the temporal dependence of joint probability distributions induced by time-series data. To this end, a key step is the development of an effective discriminator to distinguish between time series distributions. In this talk, I will introduce the so-called PCF-GAN, a novel GAN that incorporates the path characteristic function (PCF) as the principled representation of time series distribution into the discriminator to enhance its generative performance.  On the one hand, we establish theoretical foundations of the PCF distance by proving its characteristicity, boundedness, differentiability with respect to generator parameters, and weak continuity, which ensure the stability and feasibility of training the PCF-GAN. On the other hand, we design efficient initialisation and optimisation schemes for PCFs to strengthen the discriminative power and accelerate training efficiency. To further boost the capabilities of complex time series generation, we integrate the auto-encoder structure via sequential embedding into the PCF-GAN, which provides additional reconstruction functionality. Extensive numerical experiments on various datasets demonstrate the consistently superior performance of PCF-GAN over state-of-the-art baselines, in both generation and reconstruction quality. Joint work with Dr. Siran Li (Shanghai Jiao Tong Uni) and Hang Lou (UCL). Paper: [https://arxiv.org/pdf/2305.12511.pdf].