Proof of Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field for adding, multiplying and scaling equalities and inequalities (among them $0<1$, the positivity of $\lambda_{0}^{-1}$ and $\lambda_{0}\lambda_{0}^{-1}=1$), and claims 1, 3 and 6 of Properties of the Absolute Value in an Ordered Field, are used without further mention.

Step 0: the operator and the properties used. Let $F$ be the Langevin Hamilton-Jacobi operator with common noise, with potential $V$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$. By The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §operator, $F$ is the Hamilton-Jacobi operator with common noise and penalty drift of the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ with discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$, a second-order equation operator over $\mathcal{D}_{\Sigma}$, with $\delta$-shifts relative to that pair; and by The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation and The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation, a viscosity subsolution, supersolution or solution of the equation of the statement is a function $\mathcal{D}\to\mathbb{R}$ that is a viscosity subsolution, supersolution or solution of $F$ relative to the pair, in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution. We record the following properties.

(P1) 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, the pair is a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$; in particular $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair.

(P2) 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 §coercive, the pair is Wasserstein-coercive and $\mathcal{D}$ has the map property.

(P3) By The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §closed, the pair has closed score along couplings.

(P4) By The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §regular, the pair has regular penalised maxima.

(P5) 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 §growth, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

(P6) $F$ is degenerate elliptic, by The Hamilton-Jacobi Operator with Common Noise and Penalty Drift is Degenerate Elliptic, whose hypotheses hold: the pair is a penalty pair by (P1), $\lambda_{0}$ and $\theta$ are positive, $p\in\mathbb{N}$ and $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$, $g$ is a function $\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and $F$ is the operator named there for these data.

(P7) $F$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs. This is The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair §conclusion, applied to the pair (a penalty pair by (P1)), with $\lambda_{0}$, with $\theta$ (which satisfies $0<\theta\le1$), with $p$, $\Gamma$, $g$ and $F$; we discharge its hypotheses one by one. The hypothesis (Convexity) is The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §convex. The hypothesis (Semicontinuity) is (P5).

The trace as a finite sum of entries. For $k\in[p]$ let $\gamma_{k}\in\mathbb{R}^{d}$ be the $k$th row of $\Gamma$, the point whose $i$th coordinate is $\gamma_{k,i}=\Gamma_{ki}$ for $i\in[d]$. Let $\mu\in\mathcal{D}$; then $H_{\mathcal{E}}(\mu)\in\mathcal{S}(d)$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §hessian. Apply The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §rows with its matrix $A$ taken to be $\Gamma$ and $X=H_{\mathcal{E}}(\mu)$; the letters $m$ and $p$ of that lemma are its own dimensions, and here its $m$ is read as our $p$ and its $p$ as our $d$, so that its rows $a_{k}$ are our $\gamma_{k}$ (its hypotheses $1\le m$ and $1\le p$ hold because the matrix sets of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, to which $\Gamma$ belongs, are formed only for dimensions at least $1$). Together with claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, applied with $n=d$, $M=H_{\mathcal{E}}(\mu)$ and $w=z=\gamma_{k}$ for each $k\in[p]$, and the commutativity and associativity of multiplication in $\mathbb{R}$, it gives

Put $s_{\Gamma}=\sum_{k=1}^{p}\sum_{i=1}^{d}\sum_{l=1}^{d}|\gamma_{k,i}|\,|\gamma_{k,l}|$, a real number that does not depend on $\mu$ and is nonnegative by claim 5 of Properties of Finite Sums (applied to the innermost sums first), each summand being a product of nonnegative numbers.

For (Growth), let $C$ be the constant of 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 §growth, so that $M_{2}(\mu)\le C(1+|\mathcal{E}(\mu)|)$ and $|\mathrm{tr}\,H_{\mathcal{E}}(\mu)|\le C(1+|\mathcal{E}(\mu)|)$ for every $\mu\in\mathcal{D}$. Let $\mu\in\mathcal{D}$ and write $t_{\mu}=C(1+|\mathcal{E}(\mu)|)$. For $i,l\in[d]$, 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 §entries gives $|H_{\mathcal{E}}(\mu)_{il}|\le\mathrm{tr}\,H_{\mathcal{E}}(\mu)\le|\mathrm{tr}\,H_{\mathcal{E}}(\mu)|\le t_{\mu}$. By claim 4 of Properties of the Absolute Value in an Ordered Field, used twice, and claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $|\gamma_{k,i}|\,|\gamma_{k,l}|$, every summand of (T) satisfies

Claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers bounds the absolute value of each of the three nested sums in (T) by the sum of the absolute values of its summands, claim 1 of that lemma carries these bounds through the enclosing sums, and claim 3 of Properties of Finite Sums (applied to each of the three sums) takes out the factor $t_{\mu}$; so

Put $C'=|C|(1+s_{\Gamma})$. Since $0\le1+|\mathcal{E}(\mu)|$, $C\le|C|$, $s_{\Gamma}C\le s_{\Gamma}|C|$, and $|C|\le|C|(1+s_{\Gamma})$ and $s_{\Gamma}|C|\le|C|(1+s_{\Gamma})$ (as $0\le s_{\Gamma}$ and $0\le|C|$), we obtain $M_{2}(\mu)\le C'(1+|\mathcal{E}(\mu)|)$ and $|\mathrm{tr}(\Gamma^{\top}\Gamma H_{\mathcal{E}}(\mu))|\le C'(1+|\mathcal{E}(\mu)|)$ for every $\mu\in\mathcal{D}$, which is the hypothesis with the constant $C'$.

For (Hessian continuity), let $R$ be positive and $S_{R}=\{\mu\in\mathcal{D}:|\mathcal{E}(\mu)|\le R\}$. For $i,l\in[d]$ the restriction of $\mu\mapsto H_{\mathcal{E}}(\mu)_{il}$ to $S_{R}$ is continuous 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 §entries, so for $k\in[p]$ the restriction of $\mu\mapsto\gamma_{k,i}\gamma_{k,l}\,H_{\mathcal{E}}(\mu)_{il}$ to $S_{R}$ is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space (the multiple $cf$ with $c=\gamma_{k,i}\gamma_{k,l}$), taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ with the subset $S_{R}$. A finite sum of functions continuous on $S_{R}$ is continuous on $S_{R}$, by induction on the number of summands along the recursion of claim 1 of Properties of Finite Sums, each step being claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space for $f+g$. Applying this to the three nested sums of (T), the restriction of $\mu\mapsto\mathrm{tr}(\Gamma^{\top}\Gamma H_{\mathcal{E}}(\mu))$ to $S_{R}$ is continuous.

For (Running cost), for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ we have $|g(\nu)|\le b\le|b|$ and $0\le|b|$, so $g$ is bounded with bound $|b|$; and $g$ is uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line by hypothesis, which is the uniform continuity required there.

Part 1 (Comparison). Let $u$ and $v$ be as in clause 1, and let $b_{u},b_{v}\in\mathbb{R}$ satisfy $u(\mu)\le b_{u}$ and $b_{v}\le v(\mu)$ for every $\mu\in\mathcal{D}$. By Step 0, $u$ is a viscosity subsolution and $v$ a viscosity supersolution of $F$ relative to the pair. We apply A Comparison Principle for Viscosity Solutions on the Wasserstein Space §comparison to the pair, which is a Wasserstein-coercive penalty pair by (P1) and (P2), has closed score along couplings by (P3), and whose penalty domain $\mathcal{D}$ has the map property by (P2); to the operator $F$, a second-order equation operator over $\mathcal{D}_{\Sigma}$ with $\delta$-shifts relative to the pair, which is locally strictly proper and satisfies the shift-coercivity, shift-semicontinuity and second-order structure conditions by (P7); and to $u$, $v$, $b=b_{u}$ and $b'=b_{v}$. It gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$.

Part 2 (Existence). Put $c_{-}=-\lambda_{0}^{-1}b$ and $c_{+}=\lambda_{0}^{-1}b$, so that $\lambda_{0}c_{-}=-b$ and $\lambda_{0}c_{+}=b$.

Constant test functions. Let $c\in\{c_{-},c_{+}\}$. Let $\phi_{c}:\mathbb{R}^{d}\to\mathbb{R}$ be the constant function with value $c$. It is the function $Q$ of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity with $n=d$, $M=0_{d}\in\mathcal{S}(d)$, $q=0_{\mathbb{R}^{d}}$ and constant $c$, because $0_{d}z=0_{\mathbb{R}^{d}}$ and $z\cdot0_{\mathbb{R}^{d}}=0_{\mathbb{R}^{d}}\cdot z=0$ for $z\in\mathbb{R}^{d}$; so by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic it is of class $C^{2}$ on $\mathbb{R}^{d}$ with $D\phi_{c}(z)=0_{\mathbb{R}^{d}}$ and $D^{2}\phi_{c}(z)=0_{d}$ for every $z\in\mathbb{R}^{d}$. Let $\chi_{c}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be the constant function with value $c$; then $\chi_{c}(\mu)=\phi_{c}(m(\mu))$ for every $\mu$, with $m(\mu)$ the mean. By The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean, applied with the subset $\mathcal{D}$ and $\phi=\phi_{c}$, $\chi_{c}$ is an intrinsic test function on $\mathcal{D}$, with $\nabla\chi_{c}(\mu)$ equal, for $\mu\in\mathcal{D}$, to the class in $L^{2}(\mu;\mathbb{R}^{d})$ of the constant map with value $0$, and with $H_{\chi_{c}}(\mu)=0_{d}$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. That class is the zero vector $0_{\mu}$ of $L^{2}(\mu;\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors §classes, applied, as in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, on the probability space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$. Since $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by (P1), Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction shows that $\chi_{c}$ is also an intrinsic test function on $\mathcal{D}_{\Sigma}$, with the same gradients along couplings and translation Hessians.

The operator at a constant. Let $\nu\in\mathcal{D}_{\Sigma}$. By The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §operator,

By The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §rows, applied as in (P7) with $X=0_{d}\in\mathcal{S}(d)$, $\mathrm{tr}(\Gamma^{\top}\Gamma0_{d})=\sum_{k=1}^{p}\gamma_{k}\cdot(0_{d}\gamma_{k})$; each summand is $\gamma_{k}\cdot0_{\mathbb{R}^{d}}=0$, since $0_{d}z=0_{\mathbb{R}^{d}}$ and $z\cdot0_{\mathbb{R}^{d}}=0$ for $z\in\mathbb{R}^{d}$ as recorded above, so $\mathrm{tr}(\Gamma^{\top}\Gamma0_{d})=0$ by claim 3 of Properties of Finite Sums with the factor $0$. By Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu and The Space of Square-Integrable Random Vectors §inner-product, $L^{2}(\nu;\mathbb{R}^{d})$ is a real inner product space with inner product $\langle\cdot,\cdot\rangle_{\nu}$, norm $\lVert\cdot\rVert_{\nu}$ and zero vector $0_{\nu}$, so $\langle\Sigma(\nu),0_{\nu}\rangle_{\nu}=0$ and $\lVert0_{\nu}\rVert_{\nu}=0$ by Elementary Identities in a Real Inner Product Space §zero. Hence

Since $|g(\nu)|\le b$, we have $-b\le g(\nu)\le b$. For $c=c_{-}$ the value is $-b-g(\nu)\le0$, and for $c=c_{+}$ it is $b-g(\nu)\ge0$. As $\nu\in\mathcal{D}_{\Sigma}$ was arbitrary, $\chi_{c_{-}}$ is a classical subsolution and $\chi_{c_{+}}$ a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$.

Growth. Let $w_{-},w_{+}:\mathcal{D}\to\mathbb{R}$ be the restrictions of $\chi_{c_{-}}$ and $\chi_{c_{+}}$ to $\mathcal{D}$. For $c\in\{c_{-},c_{+}\}$ the restriction of $\chi_{c}$ satisfies $\chi_{c}(\mu)\le c$ and $c\le\chi_{c}(\mu)$ for every $\mu\in\mathcal{D}$, so by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, applied to the pair, which is a Wasserstein-coercive penalty pair by (P1) and (P2), with bound $c$, it has penalty-subordinate growth from above and from below.

Viscosity sub- and supersolution. We apply Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions to the pair, which is a penalty pair by (P1) with regular penalised maxima by (P4) and with $\mathcal{E}$ lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by (P5), and to $F$, a second-order equation operator over $\mathcal{D}_{\Sigma}$ that is degenerate elliptic by (P6). With the intrinsic test function $\chi_{c_{-}}$ on $\mathcal{D}$, whose restriction $w_{-}$ to $\mathcal{D}$ has penalty-subordinate growth from above and from below, and which is a classical subsolution of $F$ on $\mathcal{D}_{\Sigma}$, Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions §subsolution shows that $w_{-}$ is a viscosity subsolution of $F$ relative to the pair. With $\chi_{c_{+}}$, which has the same two growth properties and is a classical supersolution of $F$ on $\mathcal{D}_{\Sigma}$, Classical Sub- and Supersolutions of a Degenerate Elliptic Equation on the Wasserstein Space are Viscosity Sub- and Supersolutions §supersolution shows that $w_{+}$ is a viscosity supersolution of $F$ relative to the pair.

Order. Let $\nu\in\mathcal{D}$. Then $0\le|g(\nu)|\le b$, so $0\le b$ and $0\le\lambda_{0}^{-1}b$, whence $w_{-}(\nu)=-\lambda_{0}^{-1}b\le0\le\lambda_{0}^{-1}b=w_{+}(\nu)$.

Perron's method. We apply Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution to the pair, a Wasserstein-coercive penalty pair by (P1) and (P2) with regular penalised maxima by (P4), whose penalty domain has the map property by (P2); to $F$, degenerate elliptic by (P6); with $f=w_{-}$, a viscosity subsolution with penalty-subordinate growth from below; with the supersolution written $g$ there taken to be $w_{+}$, a viscosity supersolution with penalty-subordinate growth from above; and with $w_{-}(\nu)\le w_{+}(\nu)$ for every $\nu\in\mathcal{D}$, as just shown. Let $u:\mathcal{D}\to\mathbb{R}$ be the function defined there. By Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution §solution, $u$ is a viscosity solution of $F$ relative to the pair, hence, by Step 0, a viscosity solution of the equation; and by Perron's Method on the Wasserstein Space: Existence of a Viscosity Solution Between a Subsolution and a Supersolution §bounds, $w_{-}(\nu)\le u(\nu)\le w_{+}(\nu)$, that is $-\lambda_{0}^{-1}b\le u(\nu)\le\lambda_{0}^{-1}b$, for every $\nu\in\mathcal{D}$.

Part 3 (Uniqueness and continuity). By Step 0, a bounded viscosity solution of the equation is a bounded viscosity solution of $F$ relative to the pair. We apply Uniqueness and Continuity on Energy Sublevel Sets of Bounded Viscosity Solutions on the Wasserstein Space to the pair, a Wasserstein-coercive penalty pair by (P1) and (P2) with closed score along couplings by (P3), whose penalty domain has the map property by (P2), and to $F$, a second-order equation operator over $\mathcal{D}_{\Sigma}$ with $\delta$-shifts relative to the pair that is locally strictly proper and satisfies the shift-coercivity, shift-semicontinuity and second-order structure conditions by (P7). If $u$ and $v$ are bounded viscosity solutions of the equation, then $u(\mu)=v(\mu)$ for every $\mu\in\mathcal{D}$ by Uniqueness and Continuity on Energy Sublevel Sets of Bounded Viscosity Solutions on the Wasserstein Space §uniqueness. If $u$ is a bounded viscosity solution of the equation and $c\in\mathbb{R}$, then by Uniqueness and Continuity on Energy Sublevel Sets of Bounded Viscosity Solutions on the Wasserstein Space §continuity the restriction of $u$ to $\{\mu\in\mathcal{D}:\mathcal{E}(\mu)\le c\}$ is uniformly continuous for $W_{2}$ restricted to that set and the metric of The Absolute Value Metric on the Real Line, which is the uniform continuity asserted in clause 3.