Proof of Lipschitz Dependence of the Bounded Viscosity Solution of the Langevin Hamilton-Jacobi Equation with a Density Cost on the Running Cost

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. Put $s=\lambda_{0}^{-1}k$. Let $F$ and $F'$ be the Langevin Hamilton-Jacobi operators with common noise and density cost with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$, integrand $\Phi$, and running costs $g$ and $g'$ respectively; they are second-order equation operators over $\mathcal{D}_{\Sigma}$ by that clause. By The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space §equation, a viscosity solution, subsolution or supersolution of the equation with running cost $g$ (resp. $g'$) is a function on $\mathcal{D}$ that is a viscosity solution, subsolution or supersolution of $F$ (resp. $F'$) relative to the pair; and the pair is a penalty pair by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair.

Step 1 (The two operators). Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$. In the formula of The Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost on the Wasserstein Space §operator only the terms $\lambda_{0}r$ and $-g(\nu)$ depend on the value slot or on the running cost, the others being the same for $F$ and $F'$; as $\lambda_{0}(r+s)=\lambda_{0}r+k$,

Both right-hand sides are nonnegative, since $\pm\bigl(g(\nu)-g'(\nu)\bigr)\le|g(\nu)-g'(\nu)|\le k$ (claims 2 and 3 of Properties of the Absolute Value in an Ordered Field). Hence

Step 2 ($u\le u'+s$). By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution, $u'$ is a viscosity supersolution of $F'$ relative to the pair. By Adding a Constant to a Viscosity Subsolution or Supersolution on the Wasserstein Space §supersolution, applied with $F'$ in the role of the operator named $F$ there, $F$ in the role of the operator named $F'$ there, the function $u'$ and the constant $s$, whose hypothesis is the first inequality of (1), the function $u'+s$ is a viscosity supersolution of $F$, i.e. of the equation with running cost $g$. It is bounded below, by $-c+s$. Likewise $u$ is a viscosity subsolution of $F$, i.e. of the equation with running cost $g$, bounded above by $c$ (claim 6 of Properties of the Absolute Value in an Ordered Field). The running cost $g$ is uniformly continuous and bounded by $b$, and the remaining parameters are as in Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution; its clause Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution §comparison, applied with the subsolution $u$ and the supersolution $u'+s$, gives $u(\mu)\le u'(\mu)+s$ for every $\mu\in\mathcal{D}$.

Step 3 ($u'\le u+s$). Exchanging the roles of $g,b,F,u$ and $g',b',F',u'$ in Step 2, and using the second inequality of (1), Adding a Constant to a Viscosity Subsolution or Supersolution on the Wasserstein Space §supersolution shows that $u+s$ is a viscosity supersolution of $F'$, bounded below, and Well-Posedness of the Langevin Hamilton-Jacobi Equation with Common Noise and a Density Cost: Existence and Uniqueness of a Bounded Viscosity Solution §comparison, applied with the running cost $g'$ (uniformly continuous and bounded by $b'$), the subsolution $u'$ and the supersolution $u+s$, gives $u'(\mu)\le u(\mu)+s$ for every $\mu\in\mathcal{D}$.

Step 4 (Conclusion). For $\mu\in\mathcal{D}$, Steps 2 and 3 give $-s\le u(\mu)-u'(\mu)\le s$, that is $|u(\mu)-u'(\mu)|\le s=\lambda_{0}^{-1}k$ by claim 6 of Properties of the Absolute Value in an Ordered Field.