How the Wick-Square Problem Is Renormalised: Equipartition, the Corrector Identity and the Gaussian Penalty

This remark explains the results on the Wick-square problem in the setting of The Wick-Square Problem on the Torus: Standing Notation. Nothing here is used elsewhere; the stochastic language is heuristic, since the corpus carries no Hilbert-space-valued Wiener process and no verification theorem.

The problem. Read formally, each mode $x(k)$ follows $dx(k)=\bigl(-\mu_{k}x(k)+\alpha_{k}(t)\bigr)dt+\sqrt{\nu}\,dW_{k}$ with independent Brownian motions $W_{k}$, and a controller pays $\int_{0}^{\infty}e^{-\gamma t}\bigl(\beta\,{:}x^{2}{:}+g+\tfrac12\sum_{k}\alpha_{k}^{2}\bigr)dt$. The control, a shift of the drift, is charged its square-integrable norm, the Cameron-Martin norm of the noise. Cutting the modes to the cube $\Gamma_{N}$ gives the finite-dimensional equations The Cutoff Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem, Wick-Ordered and Bare, whose operators are The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm; removing the cutoff gives The Renormalised Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem, whose operator The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators is the limit of the cutoff operators wherever that limit exists.

The counterterm is equipartition. Without control, mode $k$ is an Ornstein-Uhlenbeck process whose stationary variance is the free-field variance $c_{k}=\nu/(2\mu_{k})$ of The Free-Field Variances of the Fourier Modes §variances: at temperature $\nu$ each mode's energy $\mu_{k}x(k)^{2}$ averages $\nu/2$. The bare cost $\beta\sum_{k}x(k)^{2}$ therefore has equilibrium mean $\beta\sum_{k}c_{k}$, which is infinite when $n\ge2$ by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent (an ultraviolet catastrophe). This is the divergence of the bare solutions in The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §bare. Wick ordering subtracts the equilibrium mean mode by mode, so that the free equilibrium becomes a state of zero cost.

The singular cost is the dissipation of the penalty. The key identity The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §corrector says that the free-field generator of the Gaussian penalty $P=\tfrac\beta2|x|_{H^{-1}}^{2}$ is exactly $-\beta\,{:}x^{2}{:}_{N}$ at every cutoff. By Dynkin's formula (heuristically, without cutoff), $\mathbb{E}\,P(X_{t})-P(x)=-\beta\,\mathbb{E}\int_{0}^{t}{:}X_{s}^{2}{:}\,ds$ along the free dynamics: the Wick square, which for $n\ge2$ is not a function on any Sobolev space (the constants $\sum_{k}c_{k}$ diverge by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent), is the rate at which the free field dissipates the $H^{-1}$-energy $P$, and along trajectories it is a boundary term plus a martingale. This is the Ito trick of Gubinelli and Jara (arXiv 1208.6551v2), a corrector in the sense of homogenisation. It singles out the free-field variances as counterterm: any other choice leaves a leftover constant, and The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm §counterterm shows the limit then fails to exist unless the difference is summable.

The renormalised equation. Substituting the identity into the cutoff operators removes the singular cost entirely (The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised), and in the limit (The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm §penalised)

$$F[\varphi]=\gamma\varphi-L(\varphi-P)+\tfrac12|D\varphi|^{2}-g .$$

The free generator acts on $\varphi-P$, never on $\varphi$ alone, so the natural test functions are the penalty plus a regular function (Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3). The penalty is fixed by the cost through the Poisson equation $LP=-\beta\,{:}x^{2}{:}$, not by the solution. Without it the limit exists only on the Wick domain (The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm §wick-only), which is dense (The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm §dense) but contains no ball (The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm §no-interior), so a viscosity notion could not be built on it. The solution of The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge differs from $P$ by a regular function: its high-mode curvature $q_{k}$ is $\beta/(2\mu_{k})$ up to order $\mu_{k}^{-2}$, by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile.

Thermodynamic reading. By the logarithmic transformation (Boue-Dupuis), the value of a control problem of this type is a free energy, $-\nu\log\mathbb{E}\exp\bigl(-\nu^{-1}\int\text{cost}\bigr)$, and the control cost is a relative entropy; Barashkov and Gubinelli (arXiv 1805.10814v2) use this variational structure for $\Phi^{4}_{3}$. The Gaussian penalty plays the part that the energy $\tfrac12|x|_{V}^{2}$ plays for an unbounded drift in The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds, and, by analogy, that the entropy plays for the heat Laplacian in Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space: in each case the penalty is the potential whose free dynamics produces the singular term.

Outlook. For a Wick polynomial cost $V$, such as the quartic cost of $\Phi^{4}$, the same rewriting holds with the corrector $P_{1}=-L^{-1}V$, computable chaos by chaos because $L$ is diagonal on Wiener chaos. The new term $\tfrac12|DP_{1}|^{2}$ may itself be singular, calling for a second corrector and new counterterms; the iterated correctors are expected to reproduce the tree expansions of Da Prato-Debussche and of paracontrolled calculus. Splitting the correctors into shells of modes gives scale-dependent penalties, the Polchinski renormalisation-group flow (Bauerschmidt, Bodineau and Dagallier, arXiv 2307.07619v2); for the Wick square this flow is Gaussian and exactly solvable.

The viscosity theory and the Galerkin limit. For a general running cost the renormalised equation is read in the viscosity sense of Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function §solution, tested by the penalty plus a regular function. For bounded running costs that are Lipschitz in the distance of $H^{-3}$ it has a solution (Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function §existence), and exactly one among the functions differing from the free solution by a bounded Lipschitz function (Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function §uniqueness). For such running costs, the finite-dimensional Wick-ordered Galerkin problems of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §wick have solutions differing from the free Galerkin quadratics by bounded functions (The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §galerkin), and these converge to the renormalised solution at every point (The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §pointwise); after the free Galerkin quadratics are subtracted, the convergence is uniform on the points of $H^{-1}$ bounded in $H^{-2}$ (The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §convergence). For $n\ge2$ the bare Galerkin solutions are unbounded, and converge only after the divergent constant $\beta\sum_{k\in\Gamma_{N}}c_{k}/\gamma$ is subtracted (The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge §bare); with any other counterterm $b$ the Galerkin solutions converge exactly when $k\mapsto c_{k}-b(k)$ is cube-summable, and the limit then shifts by $\frac{\beta}{\gamma}\sum_{k}(c_{k}-b(k))$ (The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge §forced).