Sign Reversal for Second-Order Equations on the Wasserstein Space: Subsolutions of F are Supersolutions of its Reversal

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 $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, and let $F$ be a second-order equation operator over $\mathcal{D}_{\Sigma}$, with the bundle $\mathcal{V}(\mathcal{D}_{\Sigma})$ of vector fields over $\mathcal{D}_{\Sigma}$ and, for each positive $\delta\in\mathbb{R}$, the $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to that pair. For $\nu\in\mathcal{D}_{\Sigma}$ and $q\in L^{2}(\nu;\mathbb{R}^{d})$, $-q$ denotes the real multiple $(-1)q$ in that real Hilbert space; for $Y$ in the set $\mathcal{S}(d)$ of symmetric real $d\times d$ matrices, $-Y$ denotes the real multiple $(-1)Y$ of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, which belongs to $\mathcal{S}(d)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure and satisfies $\lVert -Y\rVert=\lVert Y\rVert$ for the norm fixed there, by claim 5 of Properties of the Norm of a Symmetric Real Matrix applied with the multiplier $-1$, whose absolute value is $1$ by claim 2 of Properties of the Absolute Value in an Ordered Field and Absolute Value in an Ordered Field, since $0\le1$ by claim 1 of Elementary Arithmetic in an Ordered Field; and for $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, $-u$ denotes the function with value $-u(\nu)$ at $\nu$. Test functions on $\mathcal{P}_{2}(\mathbb{R}^{d})$, their intrinsic gradients and translation Hessians, the $\delta$-envelopes of a function and degenerate ellipticity are those of the definitions cited, as are the classical subsolutions, supersolutions and solutions of an operator on a subset of $\mathcal{P}_{2}(\mathbb{R}^{d})$ and the viscosity subsolutions, supersolutions and solutions relative to a penalty pair. In this statement the letter $q$ denotes a vector field; the dimension written $q$ in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces and in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §background is not used. Define

Then the following hold.

1. (Operator)¶ $\tilde{F}$ is a second-order equation operator over $\mathcal{D}_{\Sigma}$, and applying the same construction to $\tilde{F}$ returns $F$.

2. (Ellipticity)¶ $\tilde{F}$ is degenerate elliptic if and only if $F$ is degenerate elliptic.

3. (Shifts)¶ For every positive $\delta\in\mathbb{R}$, every $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, every $r\in\mathbb{R}$ and every $Y\in\mathcal{S}(d)$,

4. (Classical solutions)¶ Let $u$ be a test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $u$ is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical supersolution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$; $u$ is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical subsolution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$; and $u$ is a classical solution of $F$ on $\mathcal{D}_{\Sigma}$ if and only if $-u$ is a classical solution of $\tilde{F}$ on $\mathcal{D}_{\Sigma}$.

5. (Viscosity solutions)¶ Let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$. Then $u$ is a viscosity subsolution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity supersolution of $\tilde{F}$ relative to the penalty pair; $u$ is a viscosity supersolution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity subsolution of $\tilde{F}$ relative to the penalty pair; and $u$ is a viscosity solution of $F$ relative to the penalty pair if and only if $-u$ is a viscosity solution of $\tilde{F}$ relative to the penalty pair. In each equivalence the local boundedness required of $u$ by the one definition holds if and only if that required of $-u$ by the other holds.