Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number and let $D\subseteq\mathbb{R}^{n}$ be nonempty, open and convex. We use the notation of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space for traces, squared norms and halves, and more generally write $\tfrac{a}{b}$ for $ab^{-1}$ when $a,b\in\mathbb{R}$ and $b\ne0$.

Data. Let $P$ be a penalty on $D$ with monotone gradient, that is,

Let $\lambda\in\mathbb{R}$ be positive, let $\theta,\kappa\in\mathbb{R}$ be nonnegative, and let $\varepsilon,C\in\mathbb{R}$ satisfy $0<\varepsilon\le1$ and the dissipation inequality

Let $g:D\to\mathbb{R}$, write $[0,\infty)=\{s\in\mathbb{R}:0\le s\}$, and let $\rho:[0,\infty)\to[0,\infty)$ be nondecreasing (that is, $\rho(s)\le\rho(s')$ whenever $0\le s\le s'$) with

Let $M,\eta,\tau\in\mathbb{R}$ satisfy $0\le M$, $0<\eta$, $\theta\eta\le\varepsilon$ and $0<\tau$.

Derived quantities. The function $P$ attains a least value on $D$, which we denote by $p_{0}$: choosing $x_{0}\in D$, the set $L_{0}=\{x\in D:P(x)\le P(x_{0})\}$ contains $x_{0}$ and is compact by the sublevel property of a penalty; $P$ is of class $C^{2}$, hence of class $C^{1}$ by claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, hence continuous at every point of $D$, hence lower semicontinuous on $D$ by claim 2 of Semicontinuity Under Negation and Characterization of Continuity and on $L_{0}$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions; so by claim 2 of Semicontinuous Functions Attain Their Extrema on a Compact Set there is $x_{*}\in L_{0}$ with $P(x_{*})\le P(x)$ for all $x\in L_{0}$, while every $x\in D\setminus L_{0}$ satisfies $P(x_{*})\le P(x_{0})<P(x)$; thus $p_{0}=P(x_{*})$. For $x\in D$ the real number $2\tau\bigl(2M+\eta(P(x)-p_{0})\bigr)$ is nonnegative, and we let $r(x)$ be its nonnegative square root, so that $0\le r(x)$ and

Put

both nonnegative because $0\le\theta\eta\le\varepsilon$ gives $\tfrac{1}{2}\le1-\tfrac{\theta\eta}{2\varepsilon}$, and let $g_{1},g_{2}:D\to\mathbb{R}$ be given by

Let $F$, $F_{1}$ and $F_{2}$ be the penalty-drift Hamilton-Jacobi operators on $D$ with potential $P$, discount $\lambda$ and noise intensity $\kappa$, and with control cost and running cost $(\theta,g)$, $(\theta_{1},g_{1})$ and $(\theta_{2},g_{2})$ respectively.

Part A (sup-convolution of a subsolution). Let $u:D\to\mathbb{R}$ be a viscosity subsolution of $F$ on $D$ with $|u(x)|\le M$ for every $x\in D$. For $x\in D$ the set

is nonempty and bounded above by $M-\eta p_{0}$ (each element is at most $u(y)-\eta P(y)\le M-\eta p_{0}$), so it has a least upper bound because the real numbers are Dedekind complete; let $\overline{w}(x)$ be that least upper bound, which defines $\overline{w}:D\to\mathbb{R}$. A point $y\in D$ is called a maximiser at $x$ if $\overline{w}(x)=u(y)-\eta P(y)-\tfrac{1}{2\tau}\lVert x-y\rVert^{2}$. Then the following hold.

A1. (Maximisers)¶ For every $x\in D$ there is a maximiser at $x$, and every maximiser $y$ at $x$ satisfies $\lVert x-y\rVert\le r(x)$.

A2. (Bounds)¶ For every $x\in D$,

A3. (Semiconvexity)¶ $\overline{w}$ is semiconvex on $D$ with constant $\tau^{-1}$.

A4. (Gradient bound)¶ Let $x\in D$ be such that, for every $i\in\{1,\dots,n\}$, the partial derivative of $\overline{w}$ with respect to the $i$th variable exists at $x$; by claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique this is the case whenever $\overline{w}$, regarded as a map into $\mathbb{R}^{1}$, is differentiable at $x$. Then the gradient satisfies $D\overline{w}(x)=\tau^{-1}(y-x)$ for every maximiser $y$ at $x$, and

A5. (Transfer between points)¶ For all $x,y\in D$,

A6. (Subsolution property)¶ $\overline{w}$ is a viscosity subsolution of $F_{1}$ on $D$.

Part B (inf-convolution of a supersolution). Let $v:D\to\mathbb{R}$ be a viscosity supersolution of $F$ on $D$ with $|v(x)|\le M$ for every $x\in D$. For $x\in D$ the set

is nonempty and bounded below by $-M+\eta p_{0}$, so it has a greatest lower bound, namely the additive inverse of the least upper bound (which exists by Dedekind completeness) of the set of additive inverses of its elements; let $\underline{w}(x)$ be that greatest lower bound, which defines $\underline{w}:D\to\mathbb{R}$. A point $y\in D$ is called a minimiser at $x$ if $\underline{w}(x)=v(y)+\eta P(y)+\tfrac{1}{2\tau}\lVert x-y\rVert^{2}$. Then the following hold.

B1. (Minimisers)¶ For every $x\in D$ there is a minimiser at $x$, and every minimiser $y$ at $x$ satisfies $\lVert x-y\rVert\le r(x)$.

B2. (Bounds)¶ For every $x\in D$,

B3. (Semiconcavity)¶ The function $-\underline{w}:D\to\mathbb{R}$, whose value at $x$ is $-\underline{w}(x)$, is semiconvex on $D$ with constant $\tau^{-1}$.

B4. (Gradient bound)¶ Let $x\in D$ be such that, for every $i\in\{1,\dots,n\}$, the partial derivative of $\underline{w}$ with respect to the $i$th variable exists at $x$ (in particular, by claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique, whenever $\underline{w}$ is differentiable at $x$). Then $D\underline{w}(x)=\tau^{-1}(x-y)$ for every minimiser $y$ at $x$, and

B5. (Transfer between points)¶ For all $x,y\in D$,

B6. (Supersolution property)¶ $\underline{w}$ is a viscosity supersolution of $F_{2}$ on $D$.