Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information

We work in the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation and in the setting of Second-Order Equations on Euclidean Open Sets, whose clauses are used with $q=d$; where The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space and Viscosity Subsolution and Supersolution of a Second-Order Equation speak of a dimension $n$, they are applied with $n=d$. To avoid clashes of letters: $\mu$ always denotes a probability measure on $\mathbb{R}^{d}$ and never a scalar; the letter $P$ is reserved for the probability measure of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures, and the potential called $P$ in The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space is called $U$ here; the discount $\lambda$ below is a real number, Lebesgue measure on $\mathbb{R}^{d}$ being always written with its subscript, $\lambda_{d}$, as in Euclidean Space and Lebesgue Measure: Standing Notation §measure. The space $L^{2}(\mu;\mathbb{R}^{d})$, its inner product $\langle\cdot,\cdot\rangle_{\mu}$ and the convention that a class is written like any of its representatives are those of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu; Borel maps and integrable functions are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; continuity of a function on a subset of $\mathbb{R}^{d}$ and convexity of a set and semiconvexity of a function are as fixed in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation.

Data. Let $d\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{d}$ be open and convex, and let $U:D\to\mathbb{R}$ be of class $C^{2}$ on $D$, with gradient $DU(x)$ at $x\in D$; for instance $U$ may be a penalty on $D$. Let $\lambda,\kappa\in\mathbb{R}$ be positive, let $\theta'\in\mathbb{R}$ be nonnegative, and put $a=\tfrac{\kappa}{2}$, a positive real number. The sets $\mathcal{D}_{U,a}$ and $\mathcal{D}^{\Sigma}_{U,a}$, the maps $\bar{U}$ and $\nabla U$ and the relative score $\Sigma_{U,a}(\mu)=\nabla U+a\,\xi_{\mu}$ of $\mu\in\mathcal{D}^{\Sigma}_{U,a}$ are those of The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set (clauses relative free energy and relative score), $\xi_{\mu}$ being the score of $\mu$. Let $\tilde{g}:D\to\mathbb{R}$ be bounded, and let $\bar{g}:\mathbb{R}^{d}\to\mathbb{R}$, equal to $\tilde{g}$ on $D$ and to $0$ off $D$, be Borel. Let $F$ be the penalty-drift Hamilton-Jacobi operator on $D$ with potential $U$, discount $\lambda$, control cost $\theta'$, noise intensity $\kappa$ and running cost $\tilde{g}$, that is

with $\operatorname{tr}$ the trace; its viscosity subsolutions and supersolutions on $D$ are those of Viscosity Subsolution and Supersolution of a Second-Order Equation.

Let $w:D\to\mathbb{R}$ be bounded. Let $E_{w}$ be the set of those $x\in D$ at which $w$ is differentiable. For $x\in E_{w}$ the derivative matrix $J$ of $w$ at $x$, a real matrix with one row and $d$ columns, is unique, all partial derivatives of $w$ exist at $x$ and $J$ has entries $J_{1i}=\partial_{i}w(x)$, by claims 2 and 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique; we write $Dw(x)\in\mathbb{R}^{d}$ for the gradient of $w$ at $x$, so that $Jh=Dw(x)\cdot h$ for $h\in\mathbb{R}^{d}$. Let $\nabla w:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be the map equal to $Dw(x)$ at $x\in E_{w}$ and to $0_{\mathbb{R}^{d}}$ at every $x\in\mathbb{R}^{d}\setminus E_{w}$, and let $\bar{w}:\mathbb{R}^{d}\to\mathbb{R}$ be the map equal to $w$ on $D$ and to $0$ off $D$. Suppose that there are real numbers $A\ge0$, $B\ge0$ and $p_{0}$ (for instance $p_{0}=\min_{D}U$ when this minimum exists) such that

Then the following hold.

1. (Borel measurability)¶ If $w$ is continuous on $D$, then $E_{w}\in\mathcal{B}(\mathbb{R}^{d})$, and the maps $\nabla w$ and $\bar{w}$ are Borel.

2. (Integrability)¶ If $w$ is continuous on $D$ and $\mu\in\mathcal{D}_{U,a}$, then

so that the class of $\nabla w$ belongs to $L^{2}(\mu;\mathbb{R}^{d})$ and is again written $\nabla w$, and the function $\lambda\bar{w}+\tfrac{\theta'}{2}\lVert\nabla w\rVert^{2}-\bar{g}$ is integrable with respect to $\mu$.

A function semiconvex on $D$ is continuous on $D$ by Local Lipschitz Bound and Continuity for a Semiconvex Function on an Open Convex Set §continuity, and so is its negative; hence, under the hypotheses of clause 3 or of clause 4 below, clauses 1 and 2 apply to $w$ and every $\mu\in\mathcal{D}^{\Sigma}_{U,a}\subseteq\mathcal{D}_{U,a}$, and the left-hand sides below are defined.

3. (Subsolutions)¶ Suppose that $w$ is semiconvex on $D$ with some constant $K\ge0$ and is a viscosity subsolution of $F$ on $D$. Then for every $\mu\in\mathcal{D}^{\Sigma}_{U,a}$,

4. (Supersolutions)¶ Suppose that the function $-w:D\to\mathbb{R}$, $x\mapsto-w(x)$, is semiconvex on $D$ with some constant $K\ge0$ and that $w$ is a viscosity supersolution of $F$ on $D$. Then for every $\mu\in\mathcal{D}^{\Sigma}_{U,a}$,