[Paper Review] On stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

Hao Tang|arXiv (Cornell University)|Feb 20, 2020

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes local existence, blow-up criteria, and global existence for stochastic Euler-Poincaré equations driven by pseudo-differential and multiplicative noise. It introduces novel cancellation properties for pseudo-differential operators that enable energy estimates, and proves continuous dependence on initial data and stability of exit times under noise perturbations, extending results to a broader class of stochastic fluid models beyond transport noise.

ABSTRACT

In this paper we focus on the stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise. We first establish two new cancellation properties on pseudo-differential operators, which play a key role in energy estimate. Then, we obtain results on local solution, blow-up criterion and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data are also studied for the multiplicative noise case.

Motivation & Objective

• To extend the analysis of stochastic fluid equations beyond transport noise to include general pseudo-differential operators.
• To establish local and global existence of solutions for the stochastic Euler-Poincaré system with multiplicative noise.
• To analyze the interplay between exit time stability and continuous dependence on initial data in the presence of multiplicative noise.
• To develop new cancellation properties for pseudo-differential operators that are essential for energy estimates in the stochastic setting.

Proposed method

• Derives two new cancellation identities for pseudo-differential operators that control singularities in energy estimates.
• Applies a cut-off approximation scheme to construct local solutions and derive a priori estimates in Sobolev spaces.
• Uses Itô and Stratonovich calculus to handle the stochastic term, transforming the equation into Itô form for analysis.
• Employs commutator estimates and operator norm bounds for pseudo-differential operators in Sobolev spaces to control nonlinearities.
• Leverages boundedness and commutativity conditions on operator symbols to control error terms in the approximation and stability analysis.
• Applies Lemma A.7 and A.8 to bound commutators involving pseudo-differential operators and multiplication operators, ensuring regularity in the energy estimates.

Experimental results

Research questions

• RQ1 How can cancellation properties for pseudo-differential operators be established to control energy estimates in stochastic fluid equations?
• RQ2 Under what conditions does the stochastic Euler-Poincaré equation with multiplicative pseudo-differential noise admit local or global solutions?
• RQ3 How does the noise structure affect the continuous dependence of solutions on initial data?
• RQ4 What is the relationship between exit time stability and solution regularity in the presence of multiplicative noise?

Key findings

• Two new cancellation properties for pseudo-differential operators are derived, which are crucial for bounding energy growth in the stochastic setting.
• Local existence and blow-up criteria are established for the stochastic Euler-Poincaré equation with multiplicative pseudo-differential noise.
• Global existence is proven under suitable conditions on the noise and initial data, extending known results for deterministic and transport-noise cases.
• The paper establishes continuous dependence of solutions on initial data, with stability of exit times under perturbations.
• The analysis confirms that the noise-induced singularities in the Itô formula can be controlled via operator commutator estimates and symbol regularity.
• The framework generalizes previous results on transport noise by allowing a broader class of pseudo-differential noise operators, including non-local and non-transport terms.