Mean-Field-Side Data for the Van Trees Assembly: Regularity of the Drift Jacobian and Fluctuation Covariance Along a Trajectory Pair, the Fundamental-Solution Bound, the Profile Response and Its Restriction, and the Observation Information Matrix of the Fluctuation LQG Data

Data. Let $l\ge2$, $m\ge1$ and $\tilde{l}\ge1$ be natural numbers, let $\mathcal{A}\subseteq\mathbb{R}^{m}$ be a nonempty subset of Euclidean space, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B\ge0$, let $(U,V,\bar\beta)$ be a twice continuously differentiable extension of $\beta$ with derivative bound $K\ge0$, with the identification of $\mathbb{R}^{l}\times\mathbb{R}^{m}$ with $\mathbb{R}^{l+m}$, the coordinates $x_{1},\dots,x_{l+m}$ of a point $x=(\Sigma,\alpha)$ and the partial derivatives $\partial_{i}$, all as in Twice Continuously Differentiable Extension of a Transition-Rate Family, let $\bar{b}$ be the extended aggregate state drift of $(U,V,\bar\beta)$, and let $\Theta$ be the aggregate fluctuation covariance of $\beta$, an assignment of a real matrix $\Theta(\Sigma,\alpha)$ with $l$ rows and $l$ columns to each point of $\Delta^{l}\times\mathcal{A}$, $\Delta^{l}$ being the probability simplex. Let $\tilde\beta$ be an observation-rate family on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}\ge0$, let $\tilde{b}$ be its aggregate observation drift, let $(\tilde{U},\bar{\tilde\beta})$ be a twice continuously differentiable extension of $\tilde\beta$ with derivative bound $\tilde{K}\ge0$, and let $\bar{\tilde{b}}$ be its extended aggregate observation drift, with the partial derivatives $\partial_{\gamma}$ ($\gamma\in\{1,\dots,l\}$) on the open set $\tilde{U}\subseteq\mathbb{R}^{l}$. Assume

(OC) there is a real number $\underline{b}>0$ with $\tilde{b}^{\upsilon}(\Sigma)\ge\underline{b}$ for all $\Sigma\in\Delta^{l}$ and all $\upsilon\in\{1,\dots,\tilde{l}\}$.

Let $T>0$ be a real number, let $(S,A)$ be a mean-field trajectory pair for $\beta$ with horizon $T$, with values $S_{t}\in\Delta^{l}$ and $A_{t}\in\mathcal{A}$, let $s$ be a real number with $0<s\le T$ (the intermediate time), and let $\lambda$ assign to each $u\in[0,T]$ a vector $\lambda(u)\in\mathbb{R}^{l}$ with continuous components on $[0,T]$ (the profile).

Conventions. Continuity of a real-valued function on a subinterval $I$ of the real line means continuity on $I$ relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line; for maps between subsets of Euclidean spaces, continuity at a point is that of Continuity at a Point for Maps Between Euclidean Spaces, and the two notions agree for real-valued maps by claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions. A real-valued function on $[0,T]$ or on $[0,s]$ is measurable when it is measurable with respect to the trace Borel $\sigma$-algebra of that interval and the Borel $\sigma$-algebra of the real line, and bounded measurable when moreover it is bounded; a map into $\mathbb{R}^{l}$ is (bounded) measurable when its components are. Integrals $\int_{[0,u]}\cdot\,dr$ are Lebesgue integrals over compact intervals, taken componentwise for vector-valued integrands and equal to $0$ when $u=0$. Write $|\cdot|$ for the Euclidean norm, $x\cdot y$ for the dot product, $Mz$ for the matrix-vector product, $MM'$ for the product of real matrices, $M^{\top}$ for the transpose, $I_{n}$ for the identity matrix and $M^{-1}$ for the inverse of an invertible real square matrix; matrix entries are indexed with the row index first, and $\mathbf{1}_{\{\upsilon=\upsilon'\}}$ denotes the indicator of the condition in braces ($1$ if it holds, $0$ otherwise). The class $C^{k}$ of a map on an open subset of a Euclidean space is that of C^k Maps on a Euclidean Open Set. Finite sums over a finite index set are those of Sum over a Finite Index Set, independent of the enumeration by A Sum over a Finite Index Set Does Not Depend on the Enumeration.

The copy-side objects. For a transition label $c=(\sigma_{0},\gamma_{0})$ ($\sigma_{0}\ne\gamma_{0}$ in $\{1,\dots,l\}$) let $v_{c}=\delta_{\gamma_{0}}-\delta_{\sigma_{0}}\in\mathbb{R}^{l}$, where $\delta_{1},\dots,\delta_{l}$ are the standard basis vectors of $\mathbb{R}^{l}$ as in Probability Simplex, let $\psi_{c}:U\times V\to\mathbb{R}$, $\psi_{c}(\Sigma,\alpha)=\Sigma^{\sigma_{0}}\bar\beta(\sigma_{0},\gamma_{0},\Sigma,\alpha)$, be the label rate, and, wherever the partial derivatives exist, let $g^{c}(\Sigma,\alpha)=(\partial_{1}\psi_{c},\dots,\partial_{l}\psi_{c})(\Sigma,\alpha)\in\mathbb{R}^{l}$ be the state gradient and $\mathcal{E}(\Sigma,\alpha)$ the drift Jacobian, the real matrix with $l$ rows and $l$ columns and entries $\mathcal{E}^{\gamma\eta}(\Sigma,\alpha)=\sum_{c\in\mathcal{L}}v_{c}^{\gamma}\,\partial_{\eta}\psi_{c}(\Sigma,\alpha)$, the sum running over the set $\mathcal{L}$ of all transition labels; these are the objects so named in Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect, whose defining formulas involve only the extension $(U,V,\bar\beta)$. For $x\in\Delta^{l}$ and $\upsilon\in\{1,\dots,\tilde{l}\}$ let $g_{\upsilon}(x)=(\partial_{1}\bar{\tilde{b}}^{\upsilon}(x),\dots,\partial_{l}\bar{\tilde{b}}^{\upsilon}(x))\in\mathbb{R}^{l}$ be the observation gradient and let $\tilde{D}(x)$ be the observation information matrix, the real matrix with $l$ rows and $l$ columns and entries

$$\tilde{D}^{\gamma\delta}(x)=\sum_{\upsilon=1}^{\tilde{l}}\frac{\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(x)\,\partial_{\delta}\bar{\tilde{b}}^{\upsilon}(x)}{\tilde{b}^{\upsilon}(x)}\qquad(\gamma,\delta\in\{1,\dots,l\}),$$

the partial derivatives existing by claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift and the denominators being positive by (OC); these are the objects so named in the setting of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound, whose defining formulas involve only $\bar{\tilde{b}}$ and $\tilde{b}$.

The fluctuation LQG matrices. For $t\in[0,T]$ let $\mathcal{E}_{t}$, $\Theta^{\star}_{t}$, $\tilde{\mathcal{E}}_{t}$ and $\tilde{\Theta}^{\star}_{t}$ be the real matrices with entries

$$(\mathcal{E}_{t})_{\gamma\delta}=\partial_{\delta}\bar{b}^{\gamma}(S_{t},A_{t}),\qquad (\Theta^{\star}_{t})_{\gamma\delta}=\Theta^{\gamma\delta}(S_{t},A_{t}),\qquad (\tilde{\mathcal{E}}_{t})_{\upsilon\gamma}=\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(S_{t}),\qquad (\tilde{\Theta}^{\star}_{t})_{\upsilon\upsilon'}=\mathbf{1}_{\{\upsilon=\upsilon'\}}\,\tilde{b}^{\upsilon}(S_{t}),$$

for $\gamma,\delta\in\{1,\dots,l\}$ and $\upsilon,\upsilon'\in\{1,\dots,\tilde{l}\}$, the partial derivatives of $\bar{b}$ existing by claim (i) of Regularity and Derivative Bounds of the Extended Aggregate State Drift; these are, by their defining formulas, the state matrix, state noise covariance, observation matrix and observation noise covariance of the fluctuation LQG data in every instance of that definition whose transition-rate family, observation-rate family and both extensions are the present ones and whose pair $(S,A)$ is the present pair. The matrix $\tilde{\Theta}^{\star}_{t}$ is invertible (claim 5); put $\tilde{D}_{t}=\tilde{\mathcal{E}}_{t}^{\top}\bigl((\tilde{\Theta}^{\star}_{t})^{-1}\tilde{\mathcal{E}}_{t}\bigr)$. Notational cautions: $A$ with a time subscript is the control component of the pair, and the coefficient matrix of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution, written $A$ there, is always written $\mathcal{E}_{r}$ or $\mathcal{E}^{\star}_{u}$ here; $\delta$ as a matrix index (in $\tilde{D}^{\gamma\delta}$, $(\mathcal{E}_{t})_{\gamma\delta}$ and similar entries) is unrelated to the basis vectors $\delta_{1},\dots,\delta_{l}$, which occur only inside $v_{c}$; $\tilde{D}(x)$ with a point of the simplex as argument is the copy-side matrix, $\tilde{D}_{t}$ with a time subscript the LQG matrix; $\mathcal{E}(\Sigma,\alpha)$ with two arguments is the drift Jacobian, $\mathcal{E}_{t}$ the state matrix; $V$ is the control-side open set of the extension; $K$ and $\tilde{K}$ are derivative bounds; $\Lambda$ and $\mathsf{M}$ below are bounds for the profile and its response; $\mathcal{A}$ is the control set while $\mathcal{A}_{s}(\lambda)$ in claim 6 is a real number; and the label rates $\psi_{c}$ are unrelated to the profile response $\psi_{\lambda}$ of claim 4.

Then the following hold.

1. (Regularity of the label rates and the drift Jacobian.) For every label $c$ the label rate $\psi_{c}$ is of class $C^{2}$ on $U\times V$; consequently $g^{c}$ and $\mathcal{E}$ are defined at every point of $U\times V$, every component of $g^{c}$ and every entry $\mathcal{E}^{\gamma\eta}$ is of class $C^{1}$, in particular continuous, on $U\times V$, and

$$\mathcal{E}^{\gamma\eta}(x)=\partial_{\eta}\bar{b}^{\gamma}(x)\qquad(x\in U\times V,\ \gamma,\eta\in\{1,\dots,l\}).$$

2. (Continuity along the pair.) The map $t\mapsto(S_{t},A_{t})$ is continuous at every point of $[0,T]$ as a map from $[0,T]\subseteq\mathbb{R}$ into $\mathbb{R}^{l+m}$, with values in $\Delta^{l}\times\mathcal{A}\subseteq U\times V$, and $t\mapsto S_{t}$ likewise into $\mathbb{R}^{l}$, with values in $\Delta^{l}\subseteq\tilde{U}$. For all $\gamma,\delta,\eta\in\{1,\dots,l\}$ and $\upsilon\in\{1,\dots,\tilde{l}\}$ the functions

$$t\mapsto\mathcal{E}^{\gamma\eta}(S_{t},A_{t})=(\mathcal{E}_{t})_{\gamma\eta},\qquad t\mapsto\Theta^{\gamma\delta}(S_{t},A_{t})=(\Theta^{\star}_{t})_{\gamma\delta},\qquad t\mapsto\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(S_{t})=(\tilde{\mathcal{E}}_{t})_{\upsilon\gamma},\qquad t\mapsto\tilde{b}^{\upsilon}(S_{t})$$

are continuous on $[0,T]$, the last one with values in $[\underline{b},\infty)$; their restrictions to $[0,s]$, as well as the components of the restrictions of $S$ and $A$ to $[0,s]$, are continuous on $[0,s]$, hence bounded measurable.

3. (The fundamental solution on $[0,s]$.) The assignment $u\mapsto\mathcal{E}^{\star}_{u}=\mathcal{E}(S_{u},A_{u})$ ($u\in[0,s]$), which by claims 1 and 2 satisfies $\mathcal{E}^{\star}_{u}=\mathcal{E}_{u}$, has continuous entries on $[0,s]$, so that the setting of Variation of Constants with Bounded Measurable Forcing and the Two-Parameter Fundamental Solution is instantiated with $k=l$, horizon $s$ and coefficient matrix $\mathcal{E}^{\star}$; let $\Phi^{\mathcal{E}}(t,u)$ ($t,u\in[0,s]$) be its two-parameter fundamental solution. There is a real number $\bar\Phi\ge0$ with

$$|\Phi^{\mathcal{E}}(t,u)\,y|\le\bar\Phi^{2}\,|y|\qquad(t,u\in[0,s],\ y\in\mathbb{R}^{l}).$$

4. (The profile response.) There is a real number $\Lambda\ge0$ with $|\lambda(u)|\le\Lambda$ for every $u\in[0,T]$. There is exactly one bounded measurable map $\psi_{\lambda}:[0,T]\to\mathbb{R}^{l}$ such that

$$\psi_{\lambda}(u)=\int_{[0,u]}\bigl(\mathcal{E}_{r}\,\psi_{\lambda}(r)+\Theta^{\star}_{r}\,\lambda(r)\bigr)\,dr\qquad(u\in[0,T]);$$

its components are continuous on $[0,T]$, and there is a real number $\mathsf{M}\ge0$ with $|\psi_{\lambda}(u)|\le\mathsf{M}$ for every $u\in[0,T]$. Let $\varpi$ and $\bar\psi$ be the restrictions of $\lambda$ and of $\psi_{\lambda}$ to $[0,s]$, with values $\varpi_{u}=\lambda(u)$ and $\bar\psi_{u}=\psi_{\lambda}(u)$. Then $\varpi$ and $\bar\psi$ have continuous, hence bounded measurable, components on $[0,s]$, $|\varpi_{u}|\le\Lambda$ and $|\bar\psi_{u}|\le\mathsf{M}$ for $u\in[0,s]$,

$$\bar\psi_{u}=\int_{[0,u]}\bigl(\mathcal{E}(S_{r},A_{r})\,\bar\psi_{r}+\Theta(S_{r},A_{r})\,\varpi_{r}\bigr)\,dr\qquad(u\in[0,s]),$$

and $\bar\psi$ is the only bounded measurable map $[0,s]\to\mathbb{R}^{l}$ satisfying this equation.

5. (The observation information matrix.) For every $t\in[0,T]$ the matrix $\tilde{\Theta}^{\star}_{t}$ is invertible, its inverse being the matrix with entries $\mathbf{1}_{\{\upsilon=\upsilon'\}}/\tilde{b}^{\upsilon}(S_{t})$; $\tilde{D}_{t}$ is a real matrix with $l$ rows and $l$ columns with entries

$$(\tilde{D}_{t})_{\gamma\delta}=\sum_{\upsilon=1}^{\tilde{l}}\frac{\partial_{\gamma}\bar{\tilde{b}}^{\upsilon}(S_{t})\,\partial_{\delta}\bar{\tilde{b}}^{\upsilon}(S_{t})}{\tilde{b}^{\upsilon}(S_{t})}=\tilde{D}^{\gamma\delta}(S_{t}),$$

so that $\tilde{D}_{t}=\tilde{D}(S_{t})$ and $(\tilde{D}_{t})_{\gamma\delta}=(\tilde{D}_{t})_{\delta\gamma}$; and for every $z\in\mathbb{R}^{l}$,

$$z\cdot(\tilde{D}_{t}z)=\sum_{\upsilon=1}^{\tilde{l}}\frac{(g_{\upsilon}(S_{t})\cdot z)^{2}}{\tilde{b}^{\upsilon}(S_{t})}\ \ge\ 0 .$$

Moreover, for all $\gamma,\delta$, the function $t\mapsto(\tilde{D}_{t})_{\gamma\delta}$ is continuous on $[0,T]$, with $|(\tilde{D}_{t})_{\gamma\delta}|\le\tilde{l}\,(\tilde{B}+\tilde{K})^{2}/\underline{b}$.

6. (The information functional and its two-integral form.) The functions $r\mapsto\lambda(r)\cdot(\Theta^{\star}_{r}\lambda(r))$ and $r\mapsto\psi_{\lambda}(r)\cdot(\tilde{D}_{r}\psi_{\lambda}(r))$ are continuous and nonnegative on $[0,T]$, the second with values at most $\tilde{l}\,l^{2}\,(\tilde{B}+\tilde{K})^{2}\mathsf{M}^{2}/\underline{b}$. Consequently

$$\mathcal{A}_{s}(\lambda)=\int_{[0,s]}\Bigl(\lambda(r)\cdot\bigl(\Theta^{\star}_{r}\lambda(r)\bigr)+\psi_{\lambda}(r)\cdot\bigl(\tilde{D}_{r}\psi_{\lambda}(r)\bigr)\Bigr)\,dr$$

is a well-defined real number, and

$$\mathcal{A}_{s}(\lambda)=\int_{[0,s]}\varpi_{r}\cdot\bigl(\Theta(S_{r},A_{r})\varpi_{r}\bigr)\,dr+\int_{[0,s]}\bar\psi_{r}\cdot\bigl(\tilde{D}(S_{r})\bar\psi_{r}\bigr)\,dr,$$

both integrands being continuous, nonnegative and bounded on $[0,s]$, both integrals being nonnegative, and hence $\mathcal{A}_{s}(\lambda)\ge0$. (The first integral is the profile energy $\mathcal{P}$ of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data in any instance of its setting with horizon $s$, comparison pair $(S,A)|_{[0,s]}$ and profile $\varpi$.)