Classical Sub- and Supersolutions of a Second-Order Equation on the Wasserstein Space

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $Q$, defined on $\mathcal{V}(Q)\times\mathbb{R}\times\mathcal{S}(d)$ with $\mathcal{V}(Q)$ the bundle of vector fields over $Q$. Intrinsic test functions on $Q$ and their translation Hessians $H_{u}(\nu)\in\mathcal{S}(d)$ are those of that definition, and $\nabla u(\nu)\in L^{2}(\nu;\mathbb{R}^{d})$ is the gradient along couplings of $u$ at $\nu$.

Let $u$ be an intrinsic test function on $Q$. For $\nu\in Q$ the function $u$ is differentiable along couplings at $\nu$ by property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i), so that $(\nu,\nabla u(\nu))\in\mathcal{V}(Q)$ and

$$F\bigl(\nu,\,u(\nu),\,\nabla u(\nu),\,H_{u}(\nu)\bigr)$$

is a real number.

1. (Classical subsolution)(i) The function $u$ is a classical subsolution of $F$ on $Q$ if $F(\nu,u(\nu),\nabla u(\nu),H_{u}(\nu))\le0$ for every $\nu\in Q$.

2. (Classical supersolution)(ii) The function $u$ is a classical supersolution of $F$ on $Q$ if $0\le F(\nu,u(\nu),\nabla u(\nu),H_{u}(\nu))$ for every $\nu\in Q$.

3. (Classical solution)(iii) The function $u$ is a classical solution of $F$ on $Q$ if it is both a classical subsolution and a classical supersolution of $F$ on $Q$.