Stochastic Quantization: Invariant Measures and Convergence Theorems

Stochastic quantization constructs a Euclidean field as an invariant law of a Markov evolution, but the formal Langevin equation is usually singular. A theorem must define the renormalized equation, its function space, lifetime, approximation scheme, convergence topology, and the precise invariant measure. Dynamics and static measure construction are related conclusions, not synonyms.

Required background. Interacting measures, stability, and Wick ordering supplies the target law and counterterms; thermodynamic limits, correlation decay, and phase control separates finite volume from infinite volume.

Helpful background. Gaussian Euclidean fields as measures supplies white noise and the linear field; domains, signatures, supports, and regularity helps read convergence in negative Hölder spaces.

The renormalized dynamic Φ⁴₃ equation

On the three-torus, the formal stochastic quantization equation is

$$\partial_t\Phi=\Delta\Phi-a\Phi-\lambda\Phi^3+\xi,$$

where $\xi$ is space-time white noise. The linear solution already has spatial regularity below $-1/2$, so $\Phi^3$ is not classically defined. A lattice approximation with mesh $\varepsilon$ is instead

$$\partial_t\Phi_\varepsilon=\Delta_\varepsilon\Phi_\varepsilon+ (C_\lambda(\varepsilon)-a)\Phi_\varepsilon-\lambda\Phi_\varepsilon^3+\xi_\varepsilon,$$

with $C_\lambda(\varepsilon)$ containing divergences proportional to $\lambda\varepsilon^{-1}$ and $-\lambda^2\log\varepsilon$. These are part of the definition of the approximating family.

Hairer and Matetski prove that if the initial data converge in the discrete version of $\mathcal C^\eta$, $\eta>-2/3$, then for every $\alpha<-1/2$ the renormalized lattice solutions converge in probability, up to converging maximal lifetimes, in a weighted parabolic Hölder topology. This is Hairer and Matetski 2018, Theorem 1.1, pp. 4–5. The mechanism combines convergence of renormalized stochastic models with continuity of the regularity-structure fixed-point map.

Invariant measure and globality

At each lattice spacing the finite-dimensional gradient SDE has the lattice $\Phi^4_3$ Gibbs measure $\mu_\varepsilon$ as a reversible invariant law. For positive mass and sufficiently small coupling, these measures are tight in $\mathcal C^\alpha$, $\alpha<-1/2$, and converge to the constructive $\Phi^4_3$ measure; Hairer and Matetski 2018, Propositions 7.7–7.8, pp. 48–49. Starting the approximations in equilibrium and passing both the processes and one-time laws to the limit proves invariance. Their Corollary 1.3, p. 5, gives a global reversible Markov process for almost every initial condition under the limiting measure.

This is the first application returned to rigorous status, construction, and open problems: dynamic $\Phi^4_3$ on $\mathbb T^3$, a named counterterm prescription, convergence in negative Hölder spaces, and invariance of the finite-volume Euclidean measure. It does not establish an infinite-volume relativistic theory, OS reflection positivity of the time-evolution law, or convergence for every deterministic initial distribution.

An independent finite-dimensional check explains the sign. If $dX_t=-\nabla V(X_t)dt+\sqrt2,dB_t$, then the Fokker–Planck adjoint annihilates $e^{-V}dx$. For the lattice field, the discrete action plays $V$. If the counterterm in the drift is not the derivative of the counterterm in that action, the claimed invariant density fails this direct calculation.

Subcriticality and adversarial failure

Parabolic scaling gives white noise regularity just below $-(d+2)/2$ and the stochastic convolution regularity just below $(2-d)/2$. In $d=3$ this is just below $-1/2$; regularity structures can renormalize the finitely many divergent products of the cubic equation. In $d=4$ the equation is critical, so the same subcritical fixed-point theorem does not apply. Replacing $\mathbb T^3$ by four dimensions while retaining the three-dimensional counterterms invalidates both model convergence and tightness.

Changing the noise to a rougher distribution can cause the same failure even in three dimensions. The surviving statement is existence of each smoothed SDE. It gives neither a cutoff-independent SPDE nor an invariant continuum measure.

The converse boundary is strict: an invariant measure does not imply convergence from arbitrary initial data, uniqueness of invariant measures, ergodicity, or a spectral gap for the Markov generator. Conversely, local solution convergence up to explosion does not prove globality or invariance.

There are two logically independent limit passages in the equilibrium argument. Process convergence controls finite time windows in a path-space topology; static tightness controls the one-time laws. To pass stationarity, test a bounded continuous functional of fields at finitely many times, use stationarity of every approximation, and then take the process limit. To identify the stationary marginal with the desired Euclidean field, separately use convergence of the lattice Gibbs measures. If either step is missing, one may have a limiting dynamics with an unidentified invariant law, or a constructed static measure with no associated global Markov process. This separation is especially important when stopping times appear in the local convergence theorem.

Exercises

1. Stationary density. Verify that $\rho=e^{-V}$ satisfies $\nabla\cdot(\rho\nabla V)+\Delta\rho=0$.

Solution

$\nabla\rho=-\rho\nabla V$ and $\Delta\rho=-\nabla\cdot(\rho\nabla V)$, so the two terms cancel. Normalizability of $e^{-V}$ is still required.

2. Why negative regularity? For $\alpha<-1/2$, explain why convergence in $\mathcal C^\alpha$ does not give pointwise convergence.

Solution

Negative Hölder–Besov spaces consist of distributions tested against rescaled smooth functions. Point evaluation is not continuous there, so only smeared fields and distributional operations supplied by the theorem are defined.

References

• Hairer, Martin. “A Theory of Regularity Structures.” Inventiones Mathematicae 198 (2014): 269–504. DOI.
• Hairer, Martin, and Konstantin Matetski. “Discretisations of Rough Stochastic PDEs.” Annals of Probability 46 (2018): 1651–1709. DOI. Open PDF.