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-closed 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, penalty-subordinate growth from above of a function on $\mathcal{D}$, and the $\delta$-envelopes $v^{-}_{\delta}$ of such a function, functions on $\mathcal{D}$, 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 $v:\mathcal{D}\to\mathbb{R}$ be a viscosity subsolution of $F$ relative to the penalty pair. Let $w:\mathcal{D}\to\mathbb{R}$ be given by

$$w(\nu)=\max\{\psi(\nu)-\lambda\,\mathcal{E}(\nu),\,v(\nu)\}\ \text{ if }W_{2}(\nu,\hat{\mu})<\gamma,\qquad w(\nu)=v(\nu)\ \text{ otherwise}.$$

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 (i), so that $F^{+}_{\lambda}(\nu,\psi(\nu),\nabla\psi(\nu),H_{\psi}(\nu))$ is a real number. Assume the following.

(Upper bound)(i) There is $b\in\mathbb{R}$ with $\psi(\nu)\le b$ for every $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<\gamma$.

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

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

$$F^{+}_{\lambda}\bigl(\nu,\psi(\nu),\nabla\psi(\nu),H_{\psi}(\nu)\bigr)\le0 .$$

Then the following hold.

1. (Growth and agreement)(iv) The function $w$ has penalty-subordinate growth from above, $v(\nu)\le w(\nu)$ for every $\nu\in\mathcal{D}$, and $w(\nu)=v(\nu)$ for every $\nu\in\mathcal{D}$ with $\tfrac{\gamma}{2}<W_{2}(\nu,\hat{\mu})$.

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

$$w^{-}_{\delta}(\nu)=\max\bigl\{\psi(\nu)-(\lambda+\delta)\,\mathcal{E}(\nu),\ v^{-}_{\delta}(\nu)\bigr\}\qquad\text{for every }\nu\in\mathcal{D}\text{ with }W_{2}(\nu,\hat{\mu})<\gamma,$$

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