The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes

In the setting of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential, let $g:H^{-1}\to\mathbb{R}$ be a running cost and let $N\in\mathbb{N}$, with $m$, $\kappa_{N}$, the coordinate map $\iota_{N}$ and the Riccati potential $\Phi_{N}$ as fixed there. Traces of matrices, squared norms and halves are written as in The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space.

1. (The Galerkin operators)¶ For a family $b:\mathbb{Z}^{n}\to\mathbb{R}$ (the counterterm), the Galerkin operator at cutoff $N$ with counterterm $b$ is the second-order equation operator on $\mathbb{R}^{m}$

$$\mathcal{F}^{b}_{N}(z,r,p,X)=\gamma\,r-\tfrac{\nu}{2}\operatorname{tr}(X)+\tfrac12\lVert p\rVert^{2}+\sum_{i=1}^{m}\mu_{\kappa_{N}(i)}\,z_{i}\,p_{i}-\beta\sum_{i=1}^{m}\bigl(z_{i}^{2}-b(\kappa_{N}(i))\bigr)-g(\iota_{N}z),$$

for $z,p\in\mathbb{R}^{m}$, $r\in\mathbb{R}$ and $X\in\mathcal{S}(m)$.

2. (Wick-ordered and bare)¶ The Wick-ordered Galerkin operator is $\mathcal{F}_{N}=\mathcal{F}^{c}_{N}$, the counterterm being the family $k\mapsto c_{k}$ of free-field variances; the bare Galerkin operator is $\mathcal{F}^{0}_{N}$, the counterterm being the zero family.

3. (The shifted Galerkin operator)¶ The shifted Galerkin operator $\mathcal{F}^{\sharp}_{N}$ is the penalty-drift Hamilton-Jacobi operator on $\mathbb{R}^{m}$ with potential $\Phi_{N}$, which is of class $C^{2}$ by The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §functions, discount $\gamma$, control cost $1$, noise intensity $\nu$ and running cost $g\circ\iota_{N}$:

$$\mathcal{F}^{\sharp}_{N}(z,r,p,X)=\gamma\,r+\tfrac12\lVert p\rVert^{2}+D\Phi_{N}(z)\cdot p-\tfrac{\nu}{2}\operatorname{tr}(X)-g(\iota_{N}z).$$

4. (The equations)¶ The Galerkin Hamilton-Jacobi-Bellman equation at cutoff $N$ with counterterm $b$, for a function $U:\mathbb{R}^{m}\to\mathbb{R}$, is

$$\gamma\,U-\tfrac{\nu}{2}\operatorname{tr}(D^{2}U)+\tfrac12\lVert DU\rVert^{2}+\sum_{i=1}^{m}\mu_{\kappa_{N}(i)}\,z_{i}\,\partial_{i}U-\beta\sum_{i=1}^{m}\bigl(z_{i}^{2}-b(\kappa_{N}(i))\bigr)=g(\iota_{N}z)\qquad(z\in\mathbb{R}^{m}),$$

with $\partial_{i}U$ the $i$-th component of $DU$, that is, $\mathcal{F}^{b}_{N}(z,U(z),DU(z),D^{2}U(z))=0$ for every $z\in\mathbb{R}^{m}$ when $U$ is of class $C^{2}$; the shifted Galerkin equation is $\mathcal{F}^{\sharp}_{N}(z,U(z),DU(z),D^{2}U(z))=0$ for every $z\in\mathbb{R}^{m}$. Their viscosity subsolutions and supersolutions are those of Viscosity Subsolution and Supersolution of a Second-Order Equation for the operators $\mathcal{F}^{b}_{N}$ and $\mathcal{F}^{\sharp}_{N}$ on the open set $\mathbb{R}^{m}$, and a viscosity solution is a function that is both a viscosity subsolution and a viscosity supersolution.