[Paper Review] Quasi-invariant gaussian measures for one dimensional Hamiltonian PDE's

Nikolay Tzvetkov|arXiv (Cornell University)|Jun 11, 2015

Advanced Mathematical Physics Problems20 references22 citations

TL;DR

This paper establishes the quasi-invariance of Gaussian measures supported on high-order Sobolev spaces under the flow of one-dimensional Hamiltonian PDEs, specifically generalized Benjamin-Bona-Mahony (BBM) equations. By combining higher-order pseudo-energy estimates with Cameron-Martin-type arguments, it proves that these measures remain quasi-invariant despite the absence of high-order conservation laws, extending known results beyond KdV and Benjamin-Ono equations.

ABSTRACT

We prove the quasi-invariance of gaussian measures (supported by functions of increasing Sobolev regularity) under the flow of one dimensional Hamiltonian PDE's such as the regularized long wave (BBM) equation.

Motivation & Objective

• To investigate the transport of Gaussian measures with high Sobolev regularity under Hamiltonian PDE flows, particularly when high-order conservation laws are absent.
• To extend the theory of measure quasi-invariance beyond equations like KdV and Benjamin-Ono, which rely on special conservation laws.
• To analyze the behavior of generalized BBM-type equations with dispersive regularization and weak nonlinearity.
• To determine whether Gaussian measures on high-regularity Sobolev spaces remain quasi-invariant under such flows.
• To explore the limits of existing quasi-invariance techniques in the absence of strong conservation laws.

Proposed method

• Uses a generalized BBM equation with parameterized dispersion to model one-dimensional Hamiltonian PDEs with slow oscillations and weak nonlinearity.
• Defines Gaussian measures μ_s on Sobolev spaces H^s with zero mean, constructed via random Fourier series with decay |n|^{-(s+γ/2)}.
• Applies higher-order pseudo-energy estimates to control the regularity of solutions and ensure global existence in C(R; H^σ) for σ ≥ γ/2.
• Reduces the quasi-invariance problem to analyzing the Cameron-Martin property of the shift induced by the nonlinear flow.
• Employs the Banach-Steinhaus theorem or explicit construction to identify a direction k in H^{s+γ/2} such that the shift f(t) is not in the Cameron-Martin space.
• Uses orthogonality of Gaussian coefficients to show that the image measure is singular with respect to the original measure when the shift lies outside the Cameron-Martin space.

Experimental results

Research questions

• RQ1 Can Gaussian measures supported on high-regularity Sobolev spaces remain quasi-invariant under the flow of Hamiltonian PDEs lacking high-order conservation laws?
• RQ2 What conditions on the dispersion and nonlinearity allow for the preservation of measure quasi-invariance in the absence of strong a priori bounds?
• RQ3 How does the interplay between linear dispersion and nonlinear smoothing affect the transport of Gaussian measures in 1D Hamiltonian PDEs?
• RQ4 To what extent can the Cameron-Martin argument be adapted to nonlinear flows that are not affine transformations?
• RQ5 Can the quasi-invariance result be extended to other Hamiltonian PDEs such as the Klein-Gordon equation or 2D models?

Key findings

• The flow of the generalized BBM equation (1.4) is globally well-posed in H^σ for σ ≥ γ/2 and γ > 1, ensuring a continuous dynamical system on Sobolev spaces.
• Gaussian measures μ_s on H^s (s ≥ 1) are quasi-invariant under the flow of the generalized BBM equation for γ > 1.
• The image measure under the flow is singular with respect to the original measure when the shift f(t) lies outside the Cameron-Martin space H^{s+γ/2}.
• This singularity is established by showing that f(t) ∉ H^{s+γ/2} for t ≠ 0, implying the existence of a direction k such that the inner product ∑|n|^{2(s+γ/2)} ŝf(t)(n) ̄k̂(n) diverges.
• The proof relies on the fact that the Gaussian series ∑|n|^{s+γ/2} g_n(ω) ̄k̂(n) is almost surely finite, while the shifted variable fails this condition.
• The result demonstrates that quasi-invariance holds even without high-order conservation laws, provided the nonlinearity and dispersion are sufficiently regularizing.