Octonion random functions and integration of stochastic PDEs

Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space

We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general Lévy processes in this context.

Pathwise Itô Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients

This paper introduces the path derivatives, in the spirit of Dupire's functional Itô calculus, for the controlled paths in the rough path theory with possibly non-geometric rough paths. The theory allows us to deal with rough integration and rough PDEs in the same manner as standard stochastic calculus. We next study rough PDEs with coefficients depending on the rough path itself, which corresponds to stochastic PDEs with random coefficients. Such coefficients is less regular in the time variable and is not covered in the existing literature. The results are useful for studying viscosity solut…

随机偏微分方程导论

An Introduction to Stochastic PDEs

本文基于系列讲座介绍随机偏微分方程基本理论，以测度论、泛函分析和概率论为基础，重点探讨由加性噪声驱动的半线性抛物型问题，将其视为无限维空间中的随机发展方程以简化随机积分问题。

带记忆的分段确定性过程的有限差分方法

A finite difference method for Piecewise Deterministic Processes with memory

针对带记忆的分段确定性过程的Liouville-Master方程，提出结合迎风格式和直接求积的一阶有限差分方法，并证明数值解向平衡态的收敛性。

A class of supercritical/critical singular stochastic PDEs: existence, non-uniqueness, non-Gaussianity, non-unique ergodicity

We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }θ+ u\cdot\nablaθ= -ν(-Δ)^{γ/2}θ+ ζ,\qquad u=\nabla^{\perp}(-Δ)^{-1}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, with $ν\geq 0$, $γ\in [0,3/2)$ and $ζ\in B^{-2+κ}_{\infty,\infty}(\mathbb{T}^{2})$ for some $κ>0$. This covers the case of $ζ= (-Δ)^{α/2}ξ$ for $α<1$ and $ξ$ a spatial white noise on $\mathbb{T}^{2}$. Depending on the relation between $γ$ and $α$, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures \cite{Hai14}. Based on purely analy…

A theory of regularity structures

We introduce a new notion of "regularity structure" that provides an algebraic framework allowing to describe functions and / or distributions via a kind of "jet" or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication,…