A pathological set regarding the propagation of almost sure properties of Gaussian measures

Technical Summary: A Pathological Set Regarding the Propagation of Almost Sure Properties of Gaussian Measures

Problem Statement
This paper addresses a fundamental tension between topological and measure-theoretic properties in the dynamics of the three-dimensional periodic defocusing cubic nonlinear wave equation (NLW) on $\mathbb{T}^3$:
$$\partial_{tt}u - \Delta u + u^3 = 0, \quad u(x,0)=0, \quad \partial_t u(x,0)=g(x).$$
While previous work by Gunaratnam, Oh, Tzvetkov, and Weber (2022) established that Gaussian measures supported on high-regularity Sobolev spaces are quasi-invariant under the NLW flow—implying that "almost sure" regularity properties are preserved for random initial data—the author investigates whether this preservation holds for all initial data in the support of these measures. Specifically, the paper seeks to identify a dense set of initial velocities for which the solution instantaneously loses the $W^{\sigma,p}$ regularity, despite the initial data belonging to a dense subset of $W^{\sigma-1,p}$.

Methodology
The approach is deterministic and relies on a multi-step construction involving linear analysis, periodization, and nonlinear control:

1. Linear Counterexample on $\mathbb{R}^d$: The author constructs a specific family of distributions $F_\lambda$ on Euclidean space $\mathbb{R}^d$ ($d \ge 2$) defined via Fourier coefficients $g_\lambda(\xi) = \langle \xi \rangle^{-\sigma+1} h(\xi) |\xi|^{-\lambda}$. Here, $h$ is a smooth cutoff function vanishing near the origin. The analysis leverages known $L^p$ derivative loss results for the linear wave equation (Peral-Miyachi estimates) to show that for specific ranges of $p$ and $\lambda$, the linear evolution $S(t)(0, F_\lambda)$ fails to remain in $W^{\sigma,p}(\mathbb{R}^d)$. The proof involves detailed asymptotic analysis of Bessel functions and the inverse Fourier transform of homogeneous distributions to identify singularities at $|x|=t$ and $x=0$.

2. Periodization to $\mathbb{T}^d$: Using the Poisson summation formula, the Euclidean counterexample $F_\lambda$ is periodized to define $f_\lambda$ on the torus $\mathbb{T}^d$. The author proves that the non-propagation of regularity observed in $\mathbb{R}^d$ persists in the periodic setting for $t$ in a small neighborhood of zero. This relies on showing that the singularities of the periodized sum are dominated by the singularity of the original term near the origin, while the contributions from other lattice points remain smooth or bounded.

3. Extension to the Nonlinear Flow: To extend the result from the linear flow to the cubic NLW, the author employs a fixed-point argument in suitable Sobolev spaces. The core task is to bound the Duhamel integral term:
   $$\int_0^t \frac{\sin((t-\tau)|D|)}{|D|} u^3(\tau) \, d\tau$$
   in the space $L^\infty_T W^{\sigma,p}(\mathbb{T}^3)$. By utilizing Strichartz estimates adapted to the periodic setting via finite speed of propagation arguments, the author demonstrates that this nonlinear term remains bounded in $W^{\sigma,p}$ provided the initial data lies in a sufficiently regular space (specifically $H^{\sigma+1/2}$). Since the linear part of the solution loses $W^{\sigma,p}$ regularity while the nonlinear part preserves it, the total solution must lose regularity.

Key Results

• Theorem 1.2 (Linear Case): For $d \ge 2$, $p \neq 2$, and specific ranges of $\lambda$ and $\sigma$, the linear evolution of the constructed initial data $f_\lambda$ does not belong to $W^{\sigma,p}(\mathbb{T}^d)$ for any $t \neq 0$ (within a small time interval). The conditions for this loss of regularity depend on the relationship between $p$ and the dimension $d$ (e.g., $p \ge \frac{2d}{d-1}$ or $p < \frac{2d}{d+1}$).
• Theorem 1.1 (Nonlinear Case): For $d=3$, $p \ge 3$, and $\lambda \in (3 - 3/p, 3)$, the set $S_\lambda$ (defined as smooth functions plus scaled copies of $f_\lambda$) is a dense subset of $W^{\sigma-1,p}(\mathbb{T}^3)$. For any $g \in S_\lambda$, the global solution $u$ to the NLW exists in $C(\mathbb{R}, H^{\sigma+1/2})$ but satisfies $u(\cdot, t) \notin W^{\sigma,p}(\mathbb{T}^3)$ for any $t \in (-1, 1) \setminus \{0\}$.
• Proposition 2.3 (Almost Sure Propagation): In contrast to the deterministic pathological set, the paper confirms that for Gaussian measures $\vec{\mu}_s$ with $s > \sigma + 3/2$, the solution remains in $W^{\sigma,p}$ almost surely. This relies on the quasi-invariance of the measure (Theorem 2.1) and Fernique's theorem.

Significance and Claims
The paper claims to provide a "complementary result" to the quasi-invariance of Gaussian measures. Its primary significance lies in demonstrating that the propagation of almost sure regularity properties is not a topological property of the flow. Specifically:

• The set of initial data for which regularity is not preserved is dense in the relevant Sobolev space $W^{\sigma-1,p}$.
• This creates a sharp contrast: while a randomization of initial data (Gaussian measure) ensures regularity propagation with probability 1, there exists a dense set of deterministic initial data where this regularity is lost instantaneously.
• The work highlights the "pathological" nature of certain deterministic solutions in the context of probabilistic well-posedness, showing that the "good" behavior is measure-theoretic rather than topological.

The author explicitly states that the time constraints ($|t| < 1$ or $|t| < 1/2$) are technical limitations arising from the finite speed of propagation arguments used in the periodization step, but these constraints do not diminish the core finding that regularity ceases to exist instantaneously. The paper does not propose new applications or future experimental proposals, focusing strictly on the deterministic analysis of these specific counterexamples.