Random models for singular SPDEs

Technical Summary: Random models for singular SPDEs

Problem Statement
The paper addresses the convergence of the BHZ (Bruned-Hairer-Zambotti) renormalized model associated with the generalized (KPZ) equation on the one-dimensional torus. The equation is given by:
$$(\partial_t - \partial_x^2)u = f(u)\zeta + g(u)(\partial_x u)^2$$
where $\zeta$ is a spacetime white noise with parabolic Hölder regularity $-3/2 - \kappa$. The equation is "singular" because the nonlinearity involves products of distributions (e.g., $f(u)\zeta$ and $(\partial_x u)^2$) that are ill-defined in the classical sense.

The core difficulty lies in proving that the sequence of models constructed from regularized noise $\zeta_\epsilon$ converges to a limiting model as $\epsilon \to 0$. Standard approaches, such as the full BPHZ renormalization machinery developed by Chandra and Hairer [13], rely on complex inductive arguments involving cumulants of the noise. The authors aim to provide a proof of convergence that avoids the "full strength" of these general BPHZ techniques, specifically for the generalized KPZ equation.

Methodology
The authors employ a strategy based on chaos decomposition (Stroock's formula) and a refined analysis of Feynman diagrams (specifically "mirror graphs").

1. Chaos Decomposition:
   Instead of relying on the full algebraic machinery of BPHZ, the authors decompose the $L^2(\Omega)$ norm of the difference between renormalized models using Stroock's formula. For a symbol $\tau$ with $|\tau|_\zeta$ noise symbols, the norm is bounded by a sum of expectations of Malliavin derivatives:
   $$\|\langle \Pi^\epsilon_z \tau - \Pi^{\epsilon'}_z \tau, \phi^\lambda_z \rangle\|_{L^2(\Omega)} \lesssim \sum_{n=0}^{|\tau|_\zeta-1} \| \mathbb{E}[\langle d^n(\Pi^\epsilon_z \tau - \Pi^{\epsilon'}_z \tau), \phi^\lambda_z \rangle] \| + \| \langle d^{|\tau|_\zeta}(\Pi^\epsilon_z \tau - \Pi^{\epsilon'}_z \tau), \phi^\lambda_z \rangle \|$$
   A crucial observation is that for the generalized KPZ equation, the regularity structure contains trees with at most four noise symbols. This limits the order of derivatives required and simplifies the analysis compared to general singular SPDEs.

2. Mirror Graphs and Taylor Remainders:
   The expectations of the Malliavin derivatives are represented as integrals over "mirror graphs." These graphs encode the structure of iterated integrals involving the noise and its derivatives.

   • The authors utilize a generalization of the convergence theorem for Feynman diagrams by Hairer and Quastel [21], refined by Bruned and Nadeem [11].
   • Unlike the original Hairer-Quastel criterion, which requires Taylor expansions to be performed around a single point for all edges, the Bruned-Nadeem refinement allows expansions around edge-dependent points.
   • This flexibility is essential because certain graphs arising in the KPZ analysis (specifically those with two incoming edges on a subdivergent diagram) fall outside the scope of the original criterion.

3. Elementary Changes and Stability:
   The proof relies on the concept of "elementary changes." The authors demonstrate that by shifting the base point of a Taylor expansion or changing the order of expansion, one can transform a graph that satisfies the convergence conditions into the target graph through a finite sequence of steps. They prove that these local changes preserve the two sets of conditions required for convergence:

   • Integrability/Power Counting (Condition 3.2): Ensures the integral is well-defined.
   • Recentering (Condition 3.3): Ensures the correct scaling behavior with respect to the test function parameter $\lambda$.

Key Contributions

• Direct Convergence Proof: The paper provides a proof of the convergence of the BHZ renormalized model for the generalized KPZ equation that does not require the full inductive BPHZ renormalization framework.
• Utilization of Low Noise Count: The authors exploit the specific combinatorial property of the generalized KPZ equation where trees in the regularity structure have at most four noise symbols. This reduces the complexity of the chaos decomposition significantly.
• Application of Refined Diagrammatic Criteria: The work successfully applies the Bruned-Nadeem refinement of the Hairer-Quastel theorem to handle specific graph topologies (involving subdivergences with two incoming edges) that were previously problematic for the standard criteria.
• Telescoping Sum Renormalization: The authors explicitly demonstrate how the renormalization procedure (extraction-contraction of subtrees) corresponds to telescoping sums of Taylor remainders in the mirror graphs, establishing a direct link between the algebraic renormalization map and the analytic estimates.

Results
The main result is Corollary 8, which establishes the uniform estimate for the BHZ model $\Pi^\epsilon$:
$$\|\langle \Pi^\epsilon_z \tau, \phi^\lambda_z \rangle\|_{L^2(\Omega)} \lesssim \lambda^{|\tau|}$$
for all symbols $\tau$ of negative degree, uniformly in $\epsilon$. Furthermore, the paper establishes the convergence of the sequence of models:
$$\|\langle \Pi^\epsilon_z \tau - \Pi^{\epsilon'}_z \tau, \phi^\lambda_z \rangle\|_{L^2(\Omega)} \lesssim o_{\epsilon \vee \epsilon'}(1) \lambda^{|\tau|}$$
This confirms the existence of a limit model for the generalized KPZ equation.

Significance and Claims
The authors claim that their approach offers a "convenient" and potentially simpler alternative to the general BPHZ machinery for specific equations where the noise count is low.

• Scope: The method is explicitly stated to apply to equations where diagrams contain only one subdivergence, such as the generalized KPZ equation and potentially 2D/3D stochastic Yang-Mills dynamics.
• Limitations: The authors modestly note that if one must face "nasty or overlapping subdivergences," the approach would need to be extended, which they describe as "achievable but far from being trivial."
• Contribution to Theory: The work serves as a bridge between the abstract algebraic renormalization (BPHZ) and concrete analytic estimates, demonstrating that for certain classes of equations, the full machinery of BPHZ is not strictly necessary if one leverages the specific combinatorial constraints of the noise structure.

The paper does not claim to solve the general problem of singular SPDEs for arbitrary noise structures or overlapping divergences, but rather provides a streamlined proof for the specific case of the generalized KPZ equation using chaos decomposition and refined diagrammatic bounds.