[Paper Review] Paraproducts, rough paths and controlled distributions

Massimiliano Gubinelli, Peter Imkeller+1 more|arXiv (Cornell University)|Oct 9, 2012

Stochastic processes and financial applications27 references22 citations

TL;DR

This paper introduces a novel theory for products of distributions in Rd with Besov regularity using paradifferential calculus and rough path techniques, enabling the solution of multi-dimensional SPDEs with rough space-time noise, including a Burgers-type equation and a 2D non-linear heat equation with rough coefficients.

ABSTRACT

We propose a theory of products of distributions in Rd with Besov regularity using techniques of paradifferential calculus and ideas from the theory of controlled rough paths. We apply this theory to solve a multi-dimensional Burgers type SPDE with rough space-time white noise, and a two-dimensional non-linear heat equation with rough space dependence.

Motivation & Objective

• To develop a rigorous framework for multiplying distributions with Besov regularity in Rd.
• To address the challenge of defining products of distributions when standard pointwise multiplication fails.
• To apply the theory to solve SPDEs with rough space-time white noise.
• To extend the applicability of controlled rough path theory to distributional products.
• To provide a unified approach for non-linear SPDEs with low-regularity coefficients.

Proposed method

• Utilizes paradifferential calculus to decompose and control singular distributional products.
• Applies techniques from controlled rough paths to handle irregularity in space and time.
• Introduces a notion of controlled distributions in the context of Besov spaces.
• Establishes a product structure compatible with the algebraic and analytic properties of Besov spaces.
• Employs a dyadic decomposition and paraproduct estimates to manage singularities.
• Relies on a priori estimates and fixed-point arguments to prove existence and uniqueness of solutions.

Experimental results

Research questions

• RQ1 How can products of distributions with Besov regularity be consistently defined when classical multiplication fails?
• RQ2 Can the theory of controlled rough paths be extended to handle distributional products in a function space setting?
• RQ3 What conditions ensure the existence and uniqueness of solutions to SPDEs with rough space-time noise?
• RQ4 How can non-linear SPDEs with low-regularity coefficients be solved using distributional product structures?
• RQ5 What is the role of Besov regularity in enabling the analysis of rough SPDEs?

Key findings

• A consistent product structure for distributions in Besov spaces is established using paradifferential decomposition.
• The theory enables the solution of a multi-dimensional Burgers-type SPDE with rough space-time white noise.
• A two-dimensional non-linear heat equation with rough space dependence admits a solution under the proposed framework.
• The method provides a systematic way to handle singular products arising in SPDEs with low regularity.
• The approach unifies paradifferential calculus and rough path techniques in the context of distributional products.
• Existence and uniqueness of solutions are proven via fixed-point arguments in appropriate Besov-type function spaces.