Rough analysis and stochastic partial differential equations - Jin, R (2024)
Rough analysis involves the study of systems governed by (partial) differential equations driven by irregular (’rough’) signals. This thesis explores advancements in pathwise Itô formula with arbitrary rough signal, singulars stochastic partial differential equations (SPDEs) and its applications. In the first part of the research, with Prof. Rama Cont,we extend the Itô calculus to paths with finite p-variation, where p is any positive real number. This generalization builds upon previous work that handled integer values of p, introducing a fractional Itô calculus in a purely pathwise setting. The new framework includes the local Caputo fractional derivative and provides a comprehensive Itô formula applicable to a broader class of paths. The second part of the research delves into singular SPDEs, focusing on applications in superprocess and fluid dynamics. In collaboration with Prof. Nicolas Perkowski, we investigate the compact support property of rough super Brownian motion and extend the model to include branching random walks with infinite variance off-spring distributions. Furthermore, we address fractional stochastic Landau-Lifshitz Navier-Stokes equations, demonstrating existence and uniqueness of energy solutions, and exploring the large-scale behavior of the systems in the super critical cases.