An optimal local theory for reaction-diffusion equations driven by non-trace-class noise

Technical Summary: An Optimal Local Theory for Reaction-Diffusion Equations Driven by Non-Trace-Class Noise

Problem Statement
The paper addresses the local well-posedness of stochastic reaction-diffusion equations (SRDEs) driven by multiplicative noise that may be non-trace-class (i.e., "rough" noise). The general system is given by:
$$du_i - \text{div}(a_i \cdot \nabla u_i) dt = [\text{div}(F_i(\cdot, u)) + f_i(\cdot, u)] dt + \sum_{j=1}^\ell g_i^j(\cdot, u) dW^j(t),$$
on a bounded domain $\mathcal{O} \subset \mathbb{R}^d$, with Dirichlet boundary conditions. The authors focus on the interaction between rough stochastic forcing and polynomial nonlinearities (of degree $\rho_1, \rho_2, \rho_3$).

A central difficulty in this setting is that the combination of rough noise and superlinear nonlinearities naturally leads to solutions with low spatial regularity. In many existing theories, well-posedness relies on strong coercivity or dissipativity assumptions, or requires the noise and initial data to be relatively regular. For weakly dissipative systems with rough noise and highly irregular initial data, a sufficiently flexible local theory has been lacking. The specific challenge is to identify the "critical" initial-data spaces—the largest natural classes of initial data for which the equation is locally well-posed—while handling noise that is not necessarily trace-class (e.g., space-time white noise in 1D or colored noise in higher dimensions).

Methodology
The authors develop a general local existence and uniqueness theory by combining two recent theoretical advancements:

1. Improved Abstract Theory: They utilize the abstract framework for stochastic evolution equations in critical spaces developed in [AV26]. This framework provides local well-posedness once deterministic and stochastic nonlinearities satisfy specific mapping estimates in scales of Sobolev-Besov spaces.
2. Sharp Noise Estimates: They employ sharp estimates for multiplication operators composed with non-trace-class covariance operators, as established in [AGV26]. These estimates allow for the treatment of noise terms where the covariance operator is not trace-class, provided the noise intensity satisfies specific summability conditions related to the spatial dimension and the noise coloration parameter $\zeta$.

The methodology involves:

• Scaling Analysis: Using parabolic scaling arguments to identify the critical growth rates of nonlinearities and the corresponding critical Sobolev indices for initial data.
• Function Space Framework: Working within a scale of Sobolev-Besov spaces ($H^{s,q}$ and $B^s_{q,p}$) with Dirichlet boundary conditions. The solutions are sought in spaces of distributions to accommodate low regularity.
• Maximal $L^p$-Regularity: The proofs rely on stochastic maximal $L^p$-regularity for divergence-form operators in extrapolation spaces, allowing for time-dependent and random coefficients.
• Instantaneous Regularization: A bootstrapping argument is used to show that solutions immediately enter smoother spaces (Hölder continuous) after $t=0$, even if the initial data is rough.

Key Contributions and Results

1. Optimal Local Well-Posedness:
   The main result (Theorem 2.9) establishes the existence and uniqueness of a maximal local solution for SRDEs with non-trace-class multiplicative noise and highly irregular initial data. The theory identifies the critical spaces of initial data, which are shown to be scaling-invariant. For polynomial nonlinearities with growth parameters $\rho_1, \rho_2, \rho_3$, the critical initial data space is identified as a Besov space $B^{\frac{d}{q} - \frac{2}{\rho}}_{q,p}(\mathcal{O})$ (or similar variants depending on the specific nonlinearity type).

2. Identification of Critical Spaces:
   The paper explicitly identifies the critical initial-data spaces for a wide range of nonlinearities, including:

   • Cubic non-divergence terms ($\rho_1=2$): Relevant for Allen-Cahn and Gray-Scott models.
   • Quadratic non-divergence terms ($\rho_1=1$): Relevant for Fisher-KPP and Lotka-Volterra systems.
   • Quadratic divergence terms ($\rho_2=1$): Relevant for Burgers-type equations.
     The results show that for these models, local well-posedness holds for initial data with negative Sobolev regularity (distributions), which is a significant improvement over previous results requiring $L^2$ or continuous initial data.

3. Instantaneous Parabolic Regularization:
   Theorem 2.12 proves that solutions instantaneously regularize. Even if the initial data $u_0$ is a distribution, the solution $u(t)$ becomes Hölder continuous in space and time for any $t > 0$. Specifically, $u \in C^{\theta_1, \theta_2}_{\text{loc}}((0, \sigma) \times \mathcal{O})$ almost surely, where the spatial Hölder exponent $\theta_2$ depends on the noise regularity ($\theta_2 < 1 - \frac{d}{2} + \frac{d}{\zeta}$).

4. Blow-up Criteria:
   The authors derive sharp blow-up criteria (Theorem 2.20 and Corollary 2.21). These criteria state that if a solution exists only up to a finite time $\sigma < \infty$, then the norm of the solution in the critical space (or a slightly subcritical space) must blow up as $t \to \sigma$. This reduces global well-posedness questions to the derivation of a priori bounds in these specific norms.

5. Positivity Preservation:
   Under natural structural assumptions on the nonlinearities (e.g., $f_i(\dots, 0, \dots) \geq 0$ and $g_i^j(\dots, 0, \dots) = 0$), the paper establishes that non-negative initial data leads to non-negative solutions (Theorem 2.24). This result holds even for rough initial data and noise, leveraging the instantaneous regularization to apply maximum principles.

6. Applications to Prototypical Models:
   The abstract theory is applied to several specific models:

   • Stochastic Allen-Cahn: Local well-posedness is established for initial data in $L^r(\mathcal{O})$ for $r \geq \max\{d, 1+\epsilon\}$.
   • Stochastic Fisher-KPP: Well-posedness is shown for $r \geq \max\{d/2, 1+\epsilon\}$ in dimensions $d \geq 3$ (critical case).
   • Stochastic Burgers: Critical well-posedness for initial data with regularity arbitrarily close to $-1$.
   • 1D Space-Time White Noise: In the highly singular regime ($d=1, \zeta=\infty$), the local theory is combined with existing global a priori results (from [DKZ19, FKN25]) to obtain global well-posedness for initial data in distribution spaces $B^{-2/(\rho+1)+\epsilon}_{q,p}(\mathcal{O})$.

Significance and Claims
The paper claims to provide a "general local existence and uniqueness theory" that removes several technical restrictions on the noise commonly found in the literature. Its primary significance lies in:

• Handling Rough Noise: It extends the theory to non-trace-class noise without requiring the noise basis to be eigenfunctions of the leading operator, offering a flexible framework for general orthonormal systems.
• Optimality: By identifying the scaling-critical spaces, the theory provides the "optimal" setting for local well-posedness, which is conceptually satisfying and forms the natural starting point for future global well-posedness arguments.
• Bridging Gaps: It addresses the lack of a flexible local theory for weakly dissipative systems with rough noise, a gap that has hindered progress on global existence for many physically relevant models (e.g., coupled systems like Gray-Scott or Lotka-Volterra with rough noise).
• Global Implications: While the paper focuses on local theory, it demonstrates how this local framework, combined with instantaneous regularization and blow-up criteria, can be used to extend global well-posedness results to cases with very low (negative) smoothness initial data, particularly in the 1D space-time white noise setting.

The authors note that while they establish local well-posedness and provide the machinery for global results (via a priori bounds), the actual derivation of global well-posedness for specific weakly dissipative systems in the rough noise regime remains a subject for future work, though they demonstrate its feasibility for specific 1D cases.