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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $F$ be a second-order equation operator over $Q$, with the bundle $\mathcal{V}(Q)$ of vector fields over $Q$. Test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients $\nabla u(\nu)$ and their translation Hessians $H_{u}(\nu)$, an element of the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices, are those of that definition. In this definition test function always means a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$; the test functions $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test and their gradient maps $\nabla\psi$ are not used.

Let $u$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\nu\in Q$ the intrinsic gradient $\nabla u(\nu)$ lies in $L^{2}(\nu;\mathbb{R}^{d})$, so that $(\nu,\nabla u(\nu))\in\mathcal{V}(Q)$ and

is a real number.

1. (Classical subsolution)¶ 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)¶ 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)¶ 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 $Q$.