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

In the setting of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, let $C_{g},\ell_{g}\in\mathbb{R}$ be nonnegative and let $d_{H}$ be the distance of $H=H^{-3}(\mathbb{T}^{n})$, restricted to $H^{-1}=D(A)$, and let $g:H^{-1}\to\mathbb{R}$ satisfy

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

Renormalised viscosity sub-, super- and solutions are those of Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function for the running cost $g$, and $u_{0}$ is the free solution. Let $\mathcal{C}$ be the set of functions $u:H^{-1}\to\mathbb{R}$ for which $u-u_{0}$ is bounded and Lipschitz from $(H^{-1},d_{H})$ to $\mathbb{R}$. Then the following hold.

1. (Existence)¶ There is a renormalised viscosity solution $u\in\mathcal{C}$ with

$$|u(x)-u_{0}(x)|\le\frac{C_{g}}{\gamma}\quad\text{and}\quad|(u-u_{0})(x)-(u-u_{0})(y)|\le\Bigl(\frac{\ell_{g}}{\gamma}+\frac{2C_{g}}{\gamma}+1\Bigr)|x-y|_{H}\qquad\text{for all }x,y\in H^{-1}.$$

2. (Comparison)¶ If $u_{1},u_{2}\in\mathcal{C}$, $u_{1}$ is a renormalised viscosity subsolution and $u_{2}$ is a renormalised viscosity supersolution, then $u_{1}(x)\le u_{2}(x)$ for every $x\in H^{-1}$.

3. (Uniqueness)¶ There is exactly one renormalised viscosity solution in $\mathcal{C}$.

4. (The free solution)¶ If $g$ is the zero function, then $u_{0}$ is the renormalised viscosity solution in $\mathcal{C}$.