The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost

In the settings of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation and The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential, let $C_{g},\ell_{g}\in\mathbb{R}$ be nonnegative and let $g:H^{-1}\to\mathbb{R}$ satisfy

$$|g(x)|\le C_{g}\quad\text{and}\quad|g(x)-g(y)|\le\ell_{g}\,|x-y|_{H}\qquad\text{for all }x,y\in H^{-1}.$$

Let $u_{0}$ be the free solution, and let $u:H^{-1}\to\mathbb{R}$ be the renormalised viscosity solution for the running cost $g$ for which $u-u_{0}$ is bounded and Lipschitz from $(H^{-1},d_{H})$ to $\mathbb{R}$; it exists by 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 is unique by 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 $N\in\mathbb{N}$ let $m$, $\mathrm{pr}_{N}$ and $\tilde{u}_{N}$ be as in The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential, let $\mathcal{F}_{N}$ be the Wick-ordered Galerkin operator for $g$, with viscosity solutions on $\mathbb{R}^{m}$ as in The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation, and put $u_{0,N}(x)=\tilde{u}_{N}(\mathrm{pr}_{N}x)$ for $x\in H^{-1}$. Then the following hold.

1. (Galerkin solutions)¶ For every $N\in\mathbb{N}$ there is exactly one viscosity solution $U_{N}$ of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ for which $U_{N}-\tilde{u}_{N}$ is bounded.

2. (Bounds uniform in the cutoff)¶ With $U_{N}$ as in clause 1, put $u_{N}(x)=U_{N}(\mathrm{pr}_{N}x)$ for $x\in H^{-1}$. For every $N\in\mathbb{N}$ and all $x,y\in H^{-1}$,

$$|u_{N}(x)-u_{0,N}(x)|\le\frac{C_{g}}{\gamma}\quad\text{and}\quad\bigl|(u_{N}-u_{0,N})(x)-(u_{N}-u_{0,N})(y)\bigr|\le\Bigl(\frac{\ell_{g}}{\gamma}+\frac{2C_{g}}{\gamma}+1\Bigr)|x-y|_{H}.$$

3. (Uniform convergence on bounded sets of $H^{-2}$)¶ For all real numbers $R>0$ and $\varepsilon>0$ there is $N_{0}\in\mathbb{N}$ such that

$$\bigl|(u_{N}-u_{0,N})(x)-(u-u_{0})(x)\bigr|\le\varepsilon$$

for every $N\in\mathbb{N}$ with $N\ge N_{0}$ and every $x\in H^{-1}$ with $|x|_{V}\le R$.

4. (Pointwise convergence)¶ For every $x\in H^{-1}$ the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ converges to $u(x)$.