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The da Prato–Debussche lift

The problem

The tree machinery requires subcriticality: \(\beta_0 > -\text{order}\). The dynamical \(\Phi^4\) models violate it —

\[(\partial_t - \Delta)\,\varphi = -\varphi^3 + \xi,\]

with spacetime white noise has \(\beta_0 = -2-\kappa\) in \(d=2\) and \(\beta_0 = -\tfrac52-\kappa\) in \(d=3\), both \(\le -2\). Feeding these to renormalize() raises a ValueError naming the fix.

The change of variables

da Prato–Debussche: split off the rough part. Let \(X = L^{-1}\xi\) (the stochastic convolution) and solve for the remainder \(v = u - X\). Substituting \(u = X + v\) (tex 2026–2034):

\[(\partial_t - \Delta)\, v = -v^3 - 3v^2 X - 3v X^2 - X^3.\]

The powers \(X^k\) are ill-defined for white noise, but their Wick versions \(:\!X^k\!:\) are honest random distributions of regularity \((k \cdot \alpha_X)^-\), where \(\alpha_X = \beta_0 + \text{order}\) is the Schauder gain. The remainder equation is driven by the family of noises \(:\!X\!:, :\!X^2\!:, :\!X^3\!:\), and its worst noise is regular enough that the equation is subcritical — the standard machinery applies.

Usage

from counterterms import SPDE, Noise, Parabolic, Unknown, daprato_lift, kappa
from sympy import Rational

u  = Unknown("u", 3)
xi = Noise("xi", regularity=Rational(-5, 2) - kappa)     # Φ⁴₃
phi43 = SPDE(noises=[xi], operator=Parabolic(dim=3), unknown=u,
             rhs=xi.symbol - u.field ** 3)

lifted = daprato_lift(phi43)      # the v-equation, with noises X1, X2, X3 = :X:, :X²:, :X³:
eq = lifted.renormalize()         # the renormalized remainder equation

For \(\Phi^4_3\) the lift produces the remainder right-hand side \(-v^3 - 3X_1 v^2 - 3X_2 v - X_3\) with \(X_k \in \mathcal{C}^{-k/2 - k\kappa}\) — matching the paper’s display line for line — and the renormalized family reproduces the literature’s structure: the divergences are mass terms (\(\propto v\), two independent constants in \(d=3\)), the gradient-type terms vanish.

What the lift does and does not claim

  • Applies to: supercritical equations with additive noise and polynomial nonlinearity — the class where finitely many Wick powers absorb the roughness. Non-polynomial supercritical equations (sine-Gordon needs Wick exponentials) are rejected.
  • Wick powers as independent symbols. The lifted noises \(X_1, X_2, X_3\) are treated as independent noises with the right regularities. The free-constant family is exactly right; but the canonical constants for \(\Phi^4\) would need the true joint law of the \(:\!X^k\!:\) (one underlying Gaussian), which the symbolic scheme does not model. Compare canonical \(\Phi^4\) output by structure, not by value.
  • If the lift is insufficient (the remainder still supercritical), the downstream subcriticality check rejects it — you never get silent wrong output.