Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation on the Whole Space with a Monotone Drift

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number. 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 $b\ne0$; $d_{E}$ is the Euclidean distance on $\mathbb{R}^{n}$.

Data. Let $P:\mathbb{R}^{n}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{n}$ with monotone gradient, that is,

$$\bigl(DP(x)-DP(y)\bigr)\cdot(x-y)\ge0\qquad\text{for all }x,y\in\mathbb{R}^{n}.$$

Let $\lambda\in\mathbb{R}$ be positive and let $\theta,\kappa\in\mathbb{R}$ be nonnegative. Let $g:\mathbb{R}^{n}\to\mathbb{R}$, 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

$$|g(x)-g(y)|\le\rho\bigl(\lVert x-y\rVert\bigr)\qquad\text{for all }x,y\in\mathbb{R}^{n}.$$

Let $M,\tau\in\mathbb{R}$ satisfy $0\le M$ and $0<\tau$, and let $r$ be the nonnegative square root of $4\tau M$. Let $F$, $F_{+}$ and $F_{-}$ be the penalty-drift Hamilton-Jacobi operators on $\mathbb{R}^{n}$ with potential $P$, discount $\lambda$, control cost $\theta$ and noise intensity $\kappa$, and with running costs $g$, $g+\rho(r)$ and $g-\rho(r)$ respectively, where $g\pm\rho(r)$ takes the value $g(x)\pm\rho(r)$ at $x$. Viscosity subsolutions and supersolutions on the open set $\mathbb{R}^{n}$ are those of Viscosity Subsolution and Supersolution of a Second-Order Equation, and semiconvexity is on $\mathbb{R}^{n}$, a convex subset of itself directly from that definition.

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

$$\Bigl\{\,u(y)-\tfrac{1}{2\tau}\lVert x-y\rVert^{2}\ :\ y\in\mathbb{R}^{n}\Bigr\}$$

is nonempty and bounded above by $M$, 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 $\overline{w}:\mathbb{R}^{n}\to\mathbb{R}$. A point $y\in\mathbb{R}^{n}$ is called a maximiser at $x$ if $\overline{w}(x)=u(y)-\tfrac{1}{2\tau}\lVert x-y\rVert^{2}$. Then the following hold.

A1. (Maximisers)¶ For every $x\in\mathbb{R}^{n}$ there is a maximiser at $x$, and every maximiser $y$ at $x$ satisfies $\lVert x-y\rVert\le r$.

A2. (Bounds)¶ For every $x\in\mathbb{R}^{n}$, $u(x)\le\overline{w}(x)\le M$.

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

A4. (Lipschitz bound)¶ $\overline{w}$ is Lipschitz from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$, with the metric of The Absolute Value Metric on the Real Line, with constant $\tau^{-1}r$.

A5. (Subsolution property)¶ $\overline{w}$ is a viscosity subsolution of $F_{+}$ on $\mathbb{R}^{n}$.

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

$$\Bigl\{\,v(y)+\tfrac{1}{2\tau}\lVert x-y\rVert^{2}\ :\ y\in\mathbb{R}^{n}\Bigr\}$$

is nonempty and bounded below by $-M$, 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 $\underline{w}:\mathbb{R}^{n}\to\mathbb{R}$. A point $y\in\mathbb{R}^{n}$ is called a minimiser at $x$ if $\underline{w}(x)=v(y)+\tfrac{1}{2\tau}\lVert x-y\rVert^{2}$. Then the following hold.

B1. (Minimisers)¶ For every $x\in\mathbb{R}^{n}$ there is a minimiser at $x$, and every minimiser $y$ at $x$ satisfies $\lVert x-y\rVert\le r$.

B2. (Bounds)¶ For every $x\in\mathbb{R}^{n}$, $-M\le\underline{w}(x)\le v(x)$.

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

B4. (Lipschitz bound)¶ $\underline{w}$ is Lipschitz from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with constant $\tau^{-1}r$.

B5. (Supersolution property)¶ $\underline{w}$ is a viscosity supersolution of $F_{-}$ on $\mathbb{R}^{n}$.