Mean field singular stochastic PDEs

1. Problem Statement

The paper addresses the mathematical analysis of systems of interacting fields evolving according to singular stochastic partial differential equations (SPDEs) of mean field type. The primary equation studied is:
$$(\partial_t - \Delta)u_i^t = f(u_i^t, \mu_n^t) \xi_i^t + g(u_i^t, \mu_n^t), \quad 1 \le i \le n$$
where:

• $u_i^t$ represents the $i$-th field on a 2-dimensional torus $\mathbb{T}^2$.
• $\xi_i^t$ are independent, identically distributed (i.i.d.) random spacetime distributions (noises), such as space-time white noise.
• $\mu_n^t = \frac{1}{n} \sum_{i=1}^n \delta_{u_i^t}$ is the empirical measure of the system.
• $f$ and $g$ are interaction functions. Crucially, $f$ depends on the measure $\mu_n^t$, making the equation a mean field system.

The Core Difficulty:
The equations are "singular" because the noise $\xi$ has low regularity (e.g., space-time white noise has Hölder regularity $\alpha - 2$ where $\alpha < 1$). The solution $u$ typically inherits the regularity of the noise plus the smoothing effect of the heat operator, resulting in regularity $\alpha$. However, the product $f(u, \mu) \xi$ involves multiplying two distributions with regularities $\alpha$ and $\alpha-2$. Since $\alpha + (\alpha - 2) < 0$ for $\alpha < 1$, this product is ill-defined in the classical sense.

Furthermore, the mean field nature introduces a specific complexity: the coefficient $f(u, \mu)$ depends on the law of the solution itself, creating a non-linearity that standard singular SPDE techniques (like Regularity Structures or standard Paracontrolled Calculus applied to fixed coefficients) do not immediately resolve.

2. Methodology

The authors develop a robust framework combining Paracontrolled Calculus with Mean Field theory.

A. Paracontrolled Calculus Framework

They utilize the language of paracontrolled distributions (introduced by Gubinelli, Imkeller, and Perkowski) to make sense of the ill-defined products.

• Paracontrolled Structure: A solution $u$ is decomposed as $u = u' \prec \Xi + u^\sharp$, where $\Xi$ is a reference distribution (related to the noise), $u'$ is a "Gubinelli derivative," and $u^\sharp$ is a smoother remainder.
• Enhanced Noise: To define the product $f(u, \mu)\xi$, the noise must be "enhanced" with additional stochastic objects (renormalized products) that do not exist pointwise but exist as limits in probability.

B. Mean Field Enhancement

The paper's novel contribution is the adaptation of paracontrolled calculus to the mean field setting.

• Product Space: The authors work on the product probability space $(\Omega \times \Omega, \mathcal{F} \otimes \mathcal{F}, \mathbb{P} \otimes \mathbb{P})$. They introduce an independent copy of the noise, denoted $\xi(\omega, \varpi) = \xi(\varpi)$.
• Mean Field Paracontrolled Structure: They define a random field $\phi$ to be $\omega$-paracontrolled by a reference field $X$ if:
  $$\phi(\omega) \approx (\delta_z \phi)(\omega) \prec X(\omega) + \mathbb{E}[(\delta_\mu \phi)(\omega, \cdot) \prec X(\cdot)]$$
  Here, $\delta_z \phi$ is the classical Gubinelli derivative, and $\delta_\mu \phi$ is a generalized derivative capturing the dependence on the measure argument. The expectation $\mathbb{E}$ averages over the independent copy.
• Renormalization: For pointwise interactions (Dirac kernel), they define a mean field enhanced noise $\hat{\xi}^+$ which includes:
  1. The noise $\xi$.
  2. The renormalized self-product $\xi \odot X$.
  3. The independent copy $\xi'$.
  4. The cross-product $\xi \odot X'$ (limit of $\xi_\epsilon \odot X'_\epsilon$).

C. Fixed Point Strategy

The solution is constructed via a two-step fixed point argument:

1. Linearization: Fix a probability measure $\mu$ (representing the law of the solution). Solve the singular SPDE with $\mu$ as a parameter.
2. Consistency: Show that the map $\mu \mapsto \mathcal{L}(u^\mu)$ (where $u^\mu$ is the solution for fixed $\mu$) has a unique fixed point. This requires proving Lipschitz continuity of the solution map with respect to the measure in the Wasserstein metric.

3. Key Contributions

1. Unified Framework for Mean Field Singular SPDEs:
   The paper provides the first robust setting to treat singular SPDEs where the coefficients depend on the law of the solution. It unifies the treatment of the system of $n$ particles and the limiting mean field equation.

2. Definition of Mean Field Paracontrolled Fields:
   They introduce a new definition of paracontrolled fields that explicitly accounts for the measure dependence. This involves defining a "mean field Gubinelli derivative" ($\delta_\mu$) and constructing the product $f(u, \mu)\xi$ using both the classical and mean-field components of the enhancement.

3. Handling Pointwise Interactions:
   While previous works often assumed regular interaction kernels (e.g., $C^2$ kernels), this paper handles the pointwise interaction case (Dirac delta kernel), which is physically more relevant but analytically much harder due to the singularity of the interaction.

4. Propagation of Chaos:
   They prove that as the number of particles $n \to \infty$, the empirical measure of the system converges to the law of the solution of the limiting mean field equation. This establishes Propagation of Chaos for singular SPDEs.

4. Main Results

• Well-posedness (Local and Global):

  • Theorem 1 (a): Under Lipschitz assumptions on $f$ and $g$, there exists a deterministic time $T > 0$ such that both the finite system (1.1) and the limiting mean field equation (1.3) have unique solutions in a specific space of paracontrolled distributions.
  • Global Existence: For additive noise ($f=1$), the solution is globally well-posed in time. For multiplicative noise, global existence is established using a priori estimates adapted from Shen, Zhu, and Zhu.

• Propagation of Chaos:

  • Theorem 1 (b): As $n \to \infty$, the law of any finite tuple of fields $(u_1, \dots, u_k)$ from the interacting system converges weakly to the product measure $\mathcal{L}(u)^{\otimes k}$, where $u$ is the solution to the mean field equation (1.3).
  • This result holds for both regular and pointwise interaction kernels.

• Renormalization Limits:
  The solutions are shown to be limits in probability of solutions to renormalized equations with smooth approximations of the noise. For example, the solution $u$ is the limit of $u_\epsilon$ solving:
  $$(\partial_t - \Delta)u_\epsilon = f(u_\epsilon, \mathcal{L}(u_\epsilon)) \xi_\epsilon - C_\epsilon (f'f)(u_\epsilon, \mathcal{L}(u_\epsilon)) + g(u_\epsilon, \mathcal{L}(u_\epsilon))$$
  where $C_\epsilon$ is a diverging renormalization constant.

5. Significance

• Bridging Two Fields: The work successfully bridges the gap between Mean Field Game/Systems theory (typically dealing with regular coefficients) and Singular SPDE theory (dealing with rough noise).
• Physical Relevance: The ability to handle pointwise interactions and non-Gaussian or general Gaussian noises makes the model applicable to real-world phenomena like interacting neuron models, polymer systems, and stochastic fluid models where noise is not necessarily semimartingale.
• Methodological Advancement: The introduction of the "mean field enhancement" of noise and the specific paracontrolled structure for measure-dependent coefficients opens the door for analyzing a broader class of non-linear, non-local stochastic systems that were previously intractable.
• Robustness: The framework is designed to be robust, allowing for a large class of noises beyond the standard space-time white noise, provided they can be enhanced in the appropriate sense.

In summary, Bailleul and Moench provide a rigorous mathematical foundation for understanding large systems of interacting particles driven by singular noise, proving that these systems converge to a deterministic limit equation (the mean field limit) and establishing the existence and uniqueness of solutions to these complex singular equations.