Sharp bounds for non-trace class noise and applications to SPDEs

Technical Summary: Sharp Bounds for Non-Trace Class Noise and Applications to SPDEs

Problem Statement
The paper addresses the convergence and regularity of Gaussian series of the form $g \sum_{n \ge 1} \gamma_n \mu_n f_n$ in the context of stochastic partial differential equations (SPDEs) driven by colored, non-trace class space-time noise. Here, $(\gamma_n)$ are independent standard Gaussian variables, $g$ is a spatial coefficient in $L^\eta(O)$, $(\mu_n)$ is a sequence of scalars representing the "coloring" of the noise, and $(f_n)$ is an orthonormal system in $L^2(O)$.

Existing literature often restricts analysis to two extremes: trace-class noise (where $\sum \mu_n^2 \|f_n\|_{L^\infty}^2 < \infty$) or space-time white noise (where $\mu_n \equiv 1$). The intermediate regime, where the noise is non-trace class but more regular than white noise, presents challenges. Previous results in this intermediate regime frequently relied on restrictive assumptions, such as requiring the orthonormal system $(f_n)$ to be eigenfunctions of the differential operator involved, or imposing additional summability conditions on eigenvalues that limit the range of admissible noise intensities. Furthermore, many existing bounds were not sharp with respect to the scaling invariance of the associated SPDEs, particularly regarding the integrability of the coefficient $g$.

Methodology
The authors employ a functional analytic approach centered on $\gamma$-radonifying operators and multilinear interpolation.

1. $\gamma$-Radonifying Operators: The convergence of the Gaussian series is reformulated as the boundedness of a multiplication operator $M_g$ composed with a noise operator $R_\mu$ mapping into $\gamma$-radonifying operator spaces, specifically $\gamma(L^2(O), \tilde{H}^{-s,q}(O))$. This allows the use of the ideal property of $\gamma$-spaces and square-function characterizations.
2. Weighted Sequence Spaces: A crucial innovation is the introduction of the weighted sequence space $\ell^\zeta(S_f)$, defined by the norm $\|\mu\|_{\ell^\zeta(S_f)} = (\sum |\mu_n|^\zeta \|f_n\|_{L^\infty}^2)^{1/\zeta}$. This space naturally interpolates between the trace-class condition ($\zeta=2$) and the boundedness condition ($\zeta=\infty$), explicitly encoding the $L^\infty$-growth of the orthonormal system.
3. $\gamma$-Young Inequality: The authors establish a new "Young inequality" for $\gamma$-radonifying operators involving convolutions with weak Lebesgue spaces ($L^{r,\infty}$). This is essential for handling the singularities of Bessel potential kernels in the critical (scaling-invariant) cases.
4. Multilinear Interpolation: The main results are derived by interpolating between two endpoint cases:
   • Case $\zeta=2$: Corresponds to trace-class noise, proven via Sobolev embeddings and the identification of $\gamma$-spaces with $L^2$-valued $L^\eta$ spaces.
   • Case $\zeta=\infty$: Corresponds to bounded noise, proven using the $\gamma$-Young inequality and properties of multiplication operators in Sobolev spaces.
     The interpolation is performed on the bilinear map $(g, \mu) \mapsto M_g R_\mu$.

Key Contributions and Results

• Sharp Sobolev Embeddings for Gaussian Series (Theorem 4.1): The paper establishes necessary and sufficient conditions for the series $g \sum \gamma_n \mu_n f_n$ to converge in Bessel potential spaces $\tilde{H}^{-s,q}(O)$. The conditions are:
  $$\frac{s}{d} + \frac{1}{q} \ge \frac{1}{\eta} + \frac{1}{2} - \frac{1}{\zeta} \quad \text{and} \quad \frac{1}{\eta} - \frac{1}{\zeta} < \frac{1}{2}.$$
  These bounds are sharp, meaning they capture the exact scaling invariance of the problem. The result holds for general orthonormal systems $(f_n)$ without requiring them to be eigenfunctions of a specific differential operator.

• Homogeneous Spaces (Theorem 4.2): For the whole space $O = \mathbb{R}^d$, the authors provide analogous results for homogeneous Sobolev spaces $\dot{H}^{-s,q}(\mathbb{R}^d)$, where the first condition holds with equality.

• Generalization to Fourier Multipliers (Theorem 4.9): The framework is extended to Gaussian noises defined via Fourier multipliers $T_m$ on $\mathbb{R}^d$, where the symbol $m$ belongs to $L^\zeta(\mathbb{R}^d)$. This covers Matérn random fields and other convolution-based noise structures.

• Optimal Regularity for Stochastic Heat Equations (Theorem 6.3): Applying the Gaussian series bounds to the stochastic heat equation $\partial_t u - \Delta u = g \partial_t W$, the authors derive optimal space-time regularity estimates for the mild solution $u$. Specifically, they prove:
  $$\mathbb{E}\|u\|_{L^p(0,T; H^{1-s,q}(O))}^p + \mathbb{E}\|u\|_{C([0,T]; B^{1-s-2/p}_{q,p}(O))}^p \lesssim \|\mu\|_{\ell^\zeta(S_f)}^p \mathbb{E}\|g\|_{L^p(0,T; L^\eta(O))}^p.$$
  These estimates improve upon existing literature by allowing for lower integrability of $g$ (smaller $\eta$) and covering a wider range of noise intensities $\zeta$ without assuming the noise is tied to the eigenbasis of the Laplacian.

• Necessity of Weights (Section 5): The paper demonstrates that using unweighted sequence spaces $\ell^\zeta$ (ignoring the $\|f_n\|_{L^\infty}$ growth) fails to capture the correct scaling for SPDEs, leading to suboptimal regularity results.

Significance and Claims
The authors claim that their results provide the first sharp characterization of the convergence of such Gaussian series in Bessel potential spaces for non-trace class noise. The significance lies in several specific improvements over the literature:

1. Sharpness: The conditions derived are necessary and sufficient, preserving the scaling invariance of the stochastic heat equation. This is critical for analyzing critical nonlinear SPDEs where the loss of even a small amount of regularity prevents the use of fixed-point arguments.
2. Generality of the Noise Structure: Unlike previous works that required $(f_n)$ to be eigenfunctions of the differential operator (e.g., the Dirichlet Laplacian), this approach works for any orthonormal system. This allows the results to be applied to operators in divergence form where eigenfunctions are not explicitly known.
3. Flexibility in Coefficients: The results allow for the coefficient $g$ to be in $L^\eta(O)$ with $\eta$ smaller than previously possible, and even improve the norm on the left-hand side of the estimates (e.g., to Besov spaces). This flexibility is highlighted as a key motivation for the work, specifically for future applications to nonlinear SPDEs where one wishes to avoid losing smoothness in Sobolev embeddings during fixed-point iterations.
4. Dimensional Independence: The bounds hold for any spatial dimension $d$, whereas previous conditions involving eigenvalue summability often restricted the dimension (e.g., requiring $d < 6$ for certain trace conditions).

The paper concludes that these sharp bounds serve as a foundational tool for the study of nonlinear SPDEs with non-trace class noise, a topic the authors indicate will be explored in detail in forthcoming work.