Malliavin–Stein Analysis

• Malliavin–Stein analysis is a probabilistic framework that merges Malliavin calculus with Stein's method to provide explicit error estimates in limit theorems.
• It constructs infinitesimal exchangeable pairs through diffusive perturbations, leveraging Markov semigroups, generators, and carré du champ operators for rigorous analysis.
• The approach yields quantitative convergence rates for applications ranging from random matrices to geometric eigenfunctions, ensuring practical error bounds in complex models.

Malliavin–Stein analysis is a probabilistic framework that synthesizes the Malliavin calculus of diffusion generators with Stein's method for quantitative limit theorems, particularly the normal (and other classical) approximations. Central to this approach is the construction of infinitesimal exchangeable pairs through perturbation by Markovian or diffusive dynamics, yielding explicit error bounds in Wasserstein or Kolmogorov distance for the convergence of functionals of stochastic processes, random matrices, and geometric measures. The method leverages the infinitesimal generator, reversibility, and carré du champ structures associated with Markov semigroups, extending efficiently to manifold-valued models, Witten Laplacians, compact Lie groups, and circular ensembles.

1. Markov Semigroups, Generators, and Carré du Champ

Let $(X_t)_{t\ge0}$ be a stationary, reversible Markov process on a state space $S$ with invariant measure $\mu$. The Markov semigroup $(P_t)_{t\ge0}$ acts on bounded measurable functions $f : S \to \mathbb{R}$ as

$$P_t f(x) = \mathbb{E}\big[ f(X_t) \mid X_0 = x \big].$$

The infinitesimal generator $L$ is defined by

$$Lf(x) = \lim_{t \to 0^+} \frac{P_t f(x) - f(x)}{t},$$

with $L$ self-adjoint in $L^2(\mu)$ under reversibility. The carré du champ operator $\Gamma$ is the bilinear map

$$\Gamma(f, g)(x) = \tfrac12 \left[ L(fg) - fLg - gLf \right](x),$$

which characterizes the local covariance structure of the process. For vector-valued $F$, $\Gamma(F, F)$ is the matrix of entries $\Gamma(F_i, F_j)$. These operators encode the diffusion and fluctuation behavior fundamental to Malliavin–Stein analysis (Grzybowski et al., 29 Sep 2025, Du, 2020).

2. Infinitesimal Exchangeable Pairs via Diffusive Perturbation

Given a smooth $F : S \to \mathbb{R}^d$, the construction of infinitesimal exchangeable pairs proceeds by setting $W = F(X_0)$ and $W' = F(X_t)$ for small $t>0$. Exchangeability, ensured by the reversibility of the Markov process or the structure of the diffusion on manifolds, is established by

$(X, X_t) \stackrel{d}{=} (X_t, X),$

and thus $(W, W')$ is an exchangeable pair.

The crucial infinitesimal expansions, by Taylor/Itô, are

$\mathbb{E}[W' - W \mid X_0] = t L F(X_0) + o(t),$

$$\mathbb{E}[(W' - W)(W' - W)^T \mid X_0] = 2t\,\Gamma(F, F)(X_0) + o(t).$$

On a Riemannian manifold $(M,g)$, equipped with the Witten Laplacian $L = \Delta_w = \Delta - \nabla H \cdot \nabla$, the diffusion process $dU_t$ on the orthonormal frame bundle and $X_t = \pi(U_t)$ realizes these expansions for geometric functionals (Du, 2020).

3. Multivariate Normal Approximation Theorems

Under the existence of an invertible deterministic matrix $\Lambda$, positive semidefinite $\Sigma$, and small remainder fields $E_1$, $E_2$, the regression and conditional covariance take the form: $L F(X_0) = -\Lambda F(X_0) + E_1(X_0) \tag{R}$

$$\Gamma(F, F)(X_0) = \Lambda \Sigma + E_2(X_0) \tag{C}$$

Imposing further conditions—centeredness $\mathbb{E} F(X_0) = 0$, a Lindeberg-type moment condition, and finite $\mathbb{E} \| E_1 \|$, $\mathbb{E} \| E_2 \|_{HS}$—one obtains for $Z_\Sigma \sim N(0, \Sigma)$ the Wasserstein-$L^1$ bound: $$d_W(F(X_0), Z_\Sigma) \le \| \Lambda^{-1} \|_{op} \Big( \mathbb{E} \| E_1 \| + \| \Sigma^{-1/2} \|_{op} \mathbb{E} \| E_2 \|_{HS} \Big)$$ and for smooth $g \in C^2(\mathbb{R}^d)$,

$$\big| \mathbb{E} g(F(X_0)) - \mathbb{E} g(Z_\Sigma) \big| \le \| \Lambda^{-1} \|_{op} \Big( \| \nabla g \|_{\infty} \mathbb{E} \| E_1 \| + \tfrac12 \| D^2 g \|_{HS, \infty} \mathbb{E} \| E_2 \|_{HS} \Big).$$

The proof, via the solution to the Stein equation for the Gaussian and exchangeability-based identities, exploits the generator structure and delivers explicit control of the error in terms of the remainders (Grzybowski et al., 29 Sep 2025).

4. Applications to Random Matrices, Eigenfunctions, Spherical and Circular Ensembles

Random Matrices

For the GUE, define $A \in M_n^{sa}(\mathbb{C})$, with eigenvalues $\lambda_1, ..., \lambda_n$. For polynomial $f$, the linear statistic $$S_n = \sum_{i=1}^n f(\lambda_i) = \operatorname{tr} f(A)$$ is analyzed by constructing the Ornstein–Uhlenbeck diffusion with $L = \Delta - \langle X, \nabla \cdot \rangle$. The centered statistics

$$F_p(A) = \operatorname{tr} T_p \left( \frac{A}{2\sqrt{n}} \right) - \mathbb{E} \operatorname{tr} T_p \left( \frac{A}{2\sqrt{n}} \right)$$

satisfy the regression/covariance requirements with

$$\Lambda = \operatorname{diag}(1, ..., d), \quad \Sigma = \tfrac14 \operatorname{diag}(1^2, ..., d^2),$$

and error $O(n^{-1})$. Accordingly, a quantitative version of Johansson's theorem with optimal $1/n$ rate is obtained: $$d_W \Big( (F_1, ..., F_d), (Z_1/2, \sqrt{2} Z_2/2, ..., \sqrt{d} Z_d/2) \Big ) = O(n^{-1})$$ where $Z_1, ..., Z_d$ are independent standard Gaussians (Grzybowski et al., 29 Sep 2025).

Eigenfunctions and Geometry

Let $(M,g)$ be a Riemannian manifold, $L = \Delta$ or $L = \Delta_w$, and $\{f_i\}$ a family of $L^2(M, e^{-H} dx)$-orthonormal eigenfunctions of $\Delta_w$. For $W = (f_1(X), ..., f_k(X))$, the infinitesimal exchangeable pair analysis yields normal approximation in Wasserstein distance for $W$ (Du, 2020).

On the sphere $\mathbb{S}^{n-1}$, for $f(x) = x_1$, the expansions yield

$d_{TV}(x_1, Z) = O(n^{-1}),$

recovering the infinitesimal CLT with improved quantitative error.

Circular Ensembles and Haar Trace Statistics

For unitary $U \sim \mathrm{Haar}(U(n))$, applying this framework with the exponential Stein operator to $W = \frac{1}{k} |\operatorname{Tr} U^k|^2$ gives

$$d_{\mathrm{Kol}} \left( \frac{1}{k} |\operatorname{Tr}(U^k)|^2, \mathrm{Exp}(1) \right) = O\left( n^{-1/2} k^{3/2} \right).$$

The same structure applies to the circular $\beta$-ensemble and generalizes exponential limit theorems to functionals of eigenangles (Du, 2020).

5. Diffusion on Manifolds and Witten Laplacians

In geometric probability, the diffusion process on a Riemannian manifold $(M,g)$ with potential $H$ is constructed via the solution to the SDE on the orthonormal frame bundle: $$dU_t = \sum_{i=1}^n H_i(U_t) \circ dB^i_t - (\nabla H)^{\text{lift}}(U_t) dt, \quad X_t = \pi(U_t),$$ yielding the generator $L = \Delta_w$. The induced exchangeable pairs $(W, W_t)$, for $W = f(X)$ and $W_t = f(X_t)$, are analyzed through their small-time expansions, enabling normal approximation theorems for geometric eigenfunction statistics, perturbed by Brownian motion rather than deterministic flows (Du, 2020).

The key feature of the diffusion-based construction, compared to discrete or deterministic perturbation approaches, is that the generator $L$ arises directly in the first-order expansion, and the method can incorporate curvature, drift, and manifold structure seamlessly.

6. Connections, Extensions, and Methodological Comparisons

Approach	Generator Type	Exchangeability Construction
Discrete Markov	Markov chain	One-step, reversible chain
Diffusion–Stein	Diffusion $L$	Small-time (microscopic) stochastic perturb.
Geometric flows	Drift / ODE	Deterministic perturbations (e.g. rotations)

The diffusion perturbation approach integrates smoothly with the infinitesimal Stein method, circumventing heavy higher-moment combinatorics and extending the reach of exchangeable-pair techniques to Witten Laplacians, compact group measures, and ensembles with nontrivial drift terms. A plausible implication is the unification and extension of quantitative CLT techniques in high-dimensional probability under a single analytic framework, driven by the principle "perturb by the diffusion whose generator characterizes your target distribution" (Du, 2020).

7. Quantitative Error Bounds and Significance

The Malliavin–Stein framework enables fully quantitative convergence rates (e.g., $O(n^{-1})$ in Wasserstein for GUE linear statistics, $O(n^{-1})$ in total variation for spherical coordinates, and $O(n^{-1/2} k^{3/2})$ in Kolmogorov for Haar trace statistics), with explicit dependence on operator norms and the Hilbert–Schmidt norm of remainders. These rates recover and refine classical results, including Johansson's theorem and the Meckes infinitesimal CLT, providing explicit constants and adaptable methodology across algebraic, geometric, and random matrix models (Grzybowski et al., 29 Sep 2025, Du, 2020).