The Bump Construction on the Wasserstein Space: the Local Maximum of a Viscosity Subsolution and a Penalised Intrinsic Test Function

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$ with regular penalised maxima, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$ that is degenerate elliptic, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair for each positive $\delta\in\mathbb{R}$. Intrinsic test functions, their gradients along couplings $\nabla\psi(\nu)$ and their translation Hessians $H_{\psi}(\nu)$, viscosity subsolutions of $F$ relative to the pair, the $\delta$-envelopes $v^{-}_{\delta}$, functions on $\mathcal{D}$, and that a function is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$ are those of the definitions cited; $\max\{a,b\}$ is the maximum of $a,b\in\mathbb{R}$.

Let $\lambda\in\mathbb{R}$ be positive, let $\hat{\mu}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\gamma\in\mathbb{R}$ be positive, let $\psi$ be an intrinsic test function on $\mathcal{D}$, and let $v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be a viscosity subsolution of $F$ relative to the penalty pair. Let $w:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be given by

For $\nu\in\mathcal{D}_{\Sigma}$ one has $\nu\in\mathcal{D}$ and $\nabla\psi(\nu)\in T_{\nu}\subseteq L^{2}(\nu;\mathbb{R}^{d})$ by property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, so that $F^{+}_{\lambda}(\nu,\psi(\nu),\nabla\psi(\nu),H_{\psi}(\nu))$ is a real number. Assume the following.

(Annulus condition)¶ $\psi(\nu)-\lambda\,\mathcal{E}(\nu)\le v(\nu)$ for every $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})<\gamma$.

(Test condition)¶ For every $\nu\in\mathcal{D}_{\Sigma}$ with $W_{2}(\nu,\hat{\mu})<\gamma$ and $v(\nu)<\psi(\nu)-\lambda\,\mathcal{E}(\nu)$,

Then the following hold.

1. (Local bounds)¶ The function $w$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$, $v(\nu)\le w(\nu)$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and $w(\nu)=v(\nu)$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ other than those $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})\le\tfrac{\gamma}{2}$.

2. (The $\delta$-envelope of $w$)¶ For every positive $\delta\in\mathbb{R}$,

and $w^{-}_{\delta}(\nu)=v^{-}_{\delta}(\nu)$ for every $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})$.

3. (The bump is a viscosity subsolution)¶ The function $w$ is a viscosity subsolution of $F$ relative to the penalty pair.