The Observation-Centred Fluctuation at an Intermediate Time: Restriction of the Mean-Field Flow and of the Realized Control to a Shorter Horizon, Representation through the Aggregate State and the Record Prefix, and Transport of Its Joint Law with the Record to the Synthetic Copy

Data. Adopt the Data paragraph of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set (its comparison data and its path data are not used here): natural numbers $N\ge1$, $l\ge2$, $\tilde{l}\ge1$ and $m\ge1$; a real number $\Lambda\ge0$ and an affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$, nonempty, convex and compact for the topology of the Euclidean distance, and Lipschitz constant $\Lambda$; its transition-rate family $\beta$ of claim 2 of the affine rate family lemma, with the rate bound $B$ defined there and its aggregate state drift $b$; the control bound $\sup_{a\in\mathcal{A}}|a|$ of that lemma, written $R$ there and $R^{\mathcal{A}}$ here (the letter $R$ being reserved for the clock horizon of claim 5); the point $a_{0}\in\mathcal{A}$ fixed there; an observation-rate family $\tilde{\beta}$ on $l$ states with $\tilde{l}$ channels and rate bound $\tilde{B}$; a real number $T>0$; an $N$-agent driving system $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$ (extended to $[0,\infty]$-valued measurable maps as their integrals with respect to $P$); an $\mathcal{A}$-valued observation-driven control policy $h=(h_{k})_{k\ge0}$ with horizon $T$, control dimension $m$ and $\tilde{l}$ channels; a solution of the controlled $N$-agent dynamics on $[0,T]$ for these data, with regular event $\Omega_{0}$, state processes $\sigma^{i}$, occupation indicators $\eta^{i,\gamma}_{t}$, control process $\alpha$, empirical state measure $\Sigma$ and observation record $W$ with values in the observation record space $(\mathbf{R},\mathcal{R})=(\mathbf{R}(T,\tilde{l}),\mathcal{R}(T,\tilde{l}))$; the realized control $\hat{\alpha}:[0,T]\times\Omega\to\mathbb{R}^{m}$ of claim 2 of the realized-control lemma, formed from a family $(\tau_{j},\upsilon_{j})_{j\ge1}$ furnished by its claim 1, the metric written $\rho$ there and the dense sequence in $L^{2}([0,T];\mathbb{R}^{m})$ from which it is formed being fixed as in the adopted setting; and the record-frozen control paths $a^{r}:[0,T]\to\mathbb{R}^{m}$ ($r\in\mathbf{R}$) of $h$.

Adopt from claim 2 of the flow stability lemma, whose setting is instantiated by $(\beta_{0},\beta_{1})$ and the horizon $T$, the set $\mathcal{U}_{\mathcal{A}}$ of $\mathcal{A}$-valued controls, its admissible representatives, and the mean-field flow $\mathsf{S}(x,\xi)$ (written $S(x_{0},\xi)$ there) with values $\mathsf{S}_{t}(x,\xi)\in\Delta^{l}$ ($t\in[0,T]$), defined for $x$ in the probability simplex $\Delta^{l}$ and $\xi\in\mathcal{U}_{\mathcal{A}}$: for every admissible representative $u$ of $\xi$, $\mathsf{S}(x,\xi)$ is the map $S^{u}$ furnished by claim 1 of the existence and uniqueness theorem applied to the horizon $T$, the initial value $x$ and the control $u$. Fix a point $z_{0}\in\Delta^{l}$, the base point, and put, for $\omega\in\Omega$ and $t\in[0,T]$,

the realized mean-field flow and the observation-centred fluctuation, where $\hat{\alpha}(\omega)$ denotes the path $t\mapsto\hat{\alpha}(t,\omega)$ and the element of $\mathcal{U}_{\mathcal{A}}$ it represents, of which it is an admissible representative, by claim 3 of the realized-control lemma.

The intermediate time and the restricted solution. Fix a real number $s$ with $0<s\le T$. Write $(\mathbf{R}_{s},\mathcal{R}_{s})=(\mathbf{R}(s,\tilde{l}),\mathcal{R}(s,\tilde{l}))$ for the observation record space with horizon $s$ and $\tilde{l}$ channels; let $\pi_{s}:\mathbf{R}\to\mathbf{R}_{s}$ be the prefix map when $s<T$ and the identity map of $\mathbf{R}$ when $s=T$; and put $W^{(s)}=\pi_{s}\circ W$. When $s<T$, let $h^{(s)}$ be the truncated policy and let the restricted solution be the solution on $[0,s]$ furnished by claim 2 of that lemma; when $s=T$, put $h^{(T)}=h$ and let the restricted solution be the given solution. By claims 1 and 2 of that lemma (trivially when $s=T$), $h^{(s)}$ is an $\mathcal{A}$-valued observation-driven control policy with horizon $s$, control dimension $m$ and $\tilde{l}$ channels, and the restricted solution is a solution of the controlled $N$-agent dynamics on $[0,s]$ for the policy $h^{(s)}$, on the same driving system and for the same rate families and control set, with regular event $\Omega_{0}$, control process $(\alpha_{t})_{t\in[0,s]}$, empirical state measure $(\Sigma_{t})_{t\in[0,s]}$ and observation record $W^{(s)}$. Fix a sequence in $L^{2}([0,s];\mathbb{R}^{m})$ whose set of terms is dense there (one exists, as shown in the proof of claim 1); when $s=T$, take it to be the sequence already fixed in the adopted setting. Assume that reconstruction data are fixed for the driving system and the policy $h^{(s)}$ with horizon $s$, as in the setting of the observation-filtration lemma; they enter claims 1--4 only through the measurability of $W^{(s)}$ and are needed in substance only in claim 5.

With the instantiations of claim 1, let $\hat{\alpha}^{(s)}:[0,s]\times\Omega\to\mathbb{R}^{m}$ be the realized control of the restricted solution, that is, the map of claim 2 of the realized-control lemma with horizon $s$, policy $h^{(s)}$ and the same point $a_{0}$, formed from a family furnished by its claim 1 for the restricted solution; let $a^{(s),r}:[0,s]\to\mathbb{R}^{m}$ ($r\in\mathbf{R}_{s}$) be the record-frozen control paths of $h^{(s)}$; let $\mathcal{U}^{(s)}_{\mathcal{A}}$ and $\mathsf{S}^{(s)}(x,\xi)$, with values $\mathsf{S}^{(s)}_{t}(x,\xi)$ ($t\in[0,s]$), be the set of $\mathcal{A}$-valued controls and the mean-field flow of claim 2 of the flow stability lemma with horizon $s$; let

be the record-frozen flow of claim 4 of the record-frozen control lemma with horizon $s$ and base point $z_{0}$, with components $\Phi^{(s),r,\gamma}_{t}$; and put $\Phi^{(s)}_{t}(\omega)=\mathsf{S}^{(s)}_{t}(z_{0},\hat{\alpha}^{(s)}(\omega))$ for $\omega\in\Omega$ and $t\in[0,s]$.

Conventions. $\mathcal{B}_{[0,T]}$, $\lambda_{[0,T]}$, $\mathcal{B}_{[0,s]}$ and $\lambda_{[0,s]}$ are the trace Borel $\sigma$-algebras and restricted Lebesgue measures on $[0,T]$ and on $[0,s]$; $u|_{[0,s]}$ denotes the restriction of a map $u$ on $[0,T]$ to $[0,s]$; $|\cdot|$ is the Euclidean norm, $x\cdot y$ the dot product, and $\sqrt{N}$ the nonnegative square root; $\mathbb{G}_{N}\subseteq\Delta^{l}$ is the aggregate lattice with $N$ agents and $\mathcal{P}(\mathbb{G}_{N})$ the $\sigma$-algebra of all its subsets; $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of the real line and $\otimes$ the product $\sigma$-algebra; a real-valued map on a measurable space is measurable when it is measurable with respect to the named $\sigma$-algebra and $\mathcal{B}(\mathbb{R})$; a random variable on a probability space is a real-valued map measurable with respect to its $\sigma$-algebra, square-integrable in the sense of that definition when so stated; and the image measure of a probability measure under a measurable map is that of claim 1 of the image measure lemma. Notational cautions: $s$ is the fixed intermediate time, and time and integration variables are written $t$ or $v$, the letter $u$ being reserved for control maps; the base point $z_{0}\in\Delta^{l}$ plays the role of the point $x_{0}$ of The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record and is in general different from the point $x_{0}\in\mathbb{G}_{N}$ of claim 5, which is the initial aggregate state of the copy setting; $\rho$ denotes the weak metric on the control set formed from the dense sequence of the horizon in force ($T$ in the adopted setting, $s$ in the horizon-$s$ instances of claim 1), and neither metric enters any claim; the reference measures of $(\mathbf{R},\mathcal{R})$ and $(\mathbf{R}_{s},\mathcal{R}_{s})$ are written $\varrho$ and $\varrho_{s}$ (the latter is the measure written $\rho$ in Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record), and neither plays a role except through the copy of claim 5; the letters $\mathsf{S}$ and $\mathsf{S}^{(s)}$ are reserved for the mean-field flows, the parameter lattice written $\mathsf{S}$ in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record not being used here; the smoothing parameter $\eta$ of claim 5 is a real number occurring only as the subscript of the Gaussian smoothing weight of the copy, the occupation indicators $\eta^{i,\gamma}_{t}$ always carrying superscripts; $\alpha$ is the control process of the solution and $\hat{\alpha}$ the realized control, neither being related to a vector of cell coefficients; the comparison map written $A$ in claim 1 always carries a time subscript, the bare letters $A$, $A'$ denoting sets; the record-frozen control paths $a^{r}$ of $h$ are carried only as part of the adopted Data and enter no claim; the letter $\Pi$ of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record denotes a path-valued map; and in claim 5 the second coordinate $\theta$ of a point of $\Omega^{\sharp}$ ranges over $\mathbb{R}^{d}$, the parameter coordinate map written $\Theta$ in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record not being used here.

Then the following hold.

1. (Instantiation at horizon $s$.) The numbers $N$, $l$, $\tilde{l}$, $m$, $\Lambda$, the family $(\beta_{0},\beta_{1})$ with its control set $\mathcal{A}$, the point $a_{0}$, the family $\tilde{\beta}$, the horizon $s$, the driving system $(\Omega,\mathcal{F},P)$, the policy $h^{(s)}$, the restricted solution and the fixed dense sequence in $L^{2}([0,s];\mathbb{R}^{m})$ satisfy the hypotheses of the Data paragraph of The Realized Control as the Record-Frozen Control at the Observation Record, and Measurability of the Path-and-Record Closeness Set with $s$ in the role of $T$, in particular those of The Realized Control of the Controlled N-Agent Dynamics as a Random Element of the Control Set with horizon $s$, and together with the comparison data $S^{*}_{t}=0_{\mathbb{R}^{l}}$ ($t\in[0,s]$), $K^{*}=0$ and $A_{t}=a_{0}$ ($t\in[0,s]$) and the path data $E=\{0_{\mathbb{R}^{l}}\}$, where $0_{\mathbb{R}^{l}}$ is the origin, they instantiate the whole setting of that lemma with horizon $s$ (these comparison and path data enter none of the present claims); the family $(\beta_{0},\beta_{1})$ and the horizon $s$ satisfy the hypotheses of Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls; and these together with $\tilde{l}$, the policy $h^{(s)}$, the point $z_{0}$ in the role of $x_{0}$ and the fixed dense sequence satisfy the hypotheses of The Record-Frozen Control as a Measurable Map into the Weakly Metrized Control Set, and the Record-Frozen Mean-Field Flow: Joint Measurability, Flow Equation, and Measurability in the Record; and the data of the dynamics, the given solution, the time $s$ and the fixed reconstruction data satisfy the hypotheses of The Observation Filtration of a Solution of the Controlled N-Agent Dynamics is Generated, up to Null Sets, by the Observation Record up to that Time. Consequently $\hat{\alpha}^{(s)}$, the paths $a^{(s),r}$, the flow $\mathsf{S}^{(s)}$, the record-frozen flow $\Phi^{(s),r}$ and $\Phi^{(s)}$ are defined; for every $t\in[0,s]$ and every $\gamma\in\{1,\dots,l\}$ the map $r\mapsto\Phi^{(s),r,\gamma}_{t}$ on $\mathbf{R}_{s}$ is measurable with respect to $\mathcal{R}_{s}$; and $W^{(s)}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{R}_{s}$.

2. (Restriction of the mean-field flow.) Let $x\in\Delta^{l}$ and let $u:[0,T]\to\mathcal{A}$ be a map whose components are measurable with respect to $\mathcal{B}_{[0,T]}$. Then the components of $u|_{[0,s]}$ are measurable with respect to $\mathcal{B}_{[0,s]}$, and, writing $S^{u}$ for the map furnished by claim 1 of Existence and Uniqueness of the Generalized Mean-Field Trajectory for a Measurable Control for the horizon $T$, the initial value $x$ and the control $u$, and $S^{(s),u|_{[0,s]}}$ for the map furnished by the same claim for the horizon $s$, the initial value $x$ and the control $u|_{[0,s]}$,

Consequently, for every $\xi\in\mathcal{U}_{\mathcal{A}}$ and every admissible representative $u$ of $\xi$, the map $u|_{[0,s]}$ is square-integrable on $[0,s]$ in the sense of claim 1 of The Lebesgue Space of Square-Integrable Vector-Valued Functions on a Compact Interval, its class $[u|_{[0,s]}]$ belongs to $\mathcal{U}^{(s)}_{\mathcal{A}}$ with $u|_{[0,s]}$ as an admissible representative, and

3. (Restriction of the realized control and of the realized flow; the record-frozen flow at the record prefix.) For every $\omega\in\Omega$ and every $t\in[0,s]$,

Moreover, for every $\omega\in\Omega_{0}$ and every $t\in[0,s]$,

4. (Representation through the aggregate state and the record prefix.) $\Sigma_{t}(\omega)\in\mathbb{G}_{N}$ for every $\omega\in\Omega$ and every $t\in[0,T]$, and $|x-y|\le2$ for all $x,y\in\Delta^{l}$. For $t\in[0,s]$ define

Then each component of $\Psi_{t}$ is measurable with respect to $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$, $|\Psi_{t}(x,r)|\le2\sqrt{N}$ for all $(x,r)$, and

Moreover, $|X'_{t}(\omega)|\le2\sqrt{N}$ for every $\omega\in\Omega$ and every $t\in[0,T]$; for every $t\in[0,s]$ each component of $\Phi_{t}$ and of $X'_{t}$ is a random variable on $(\Omega,\mathcal{F},P)$, and, for every $\mathbf{c}\in\mathbb{R}^{l}$, each component of $X'_{t}$ and the random variable $\mathbf{c}\cdot X'_{t}$ are square-integrable, being bounded by $2\sqrt{N}$ and by $2\sqrt{N}|\mathbf{c}|$ respectively; and for $t\in[0,s]$ and $\mathbf{c}\in\mathbb{R}^{l}$ the map $\omega\mapsto(\mathbf{c}\cdot X'_{t}(\omega),W^{(s)}(\omega))$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$.

5. (Transport of the joint law to the synthetic copy at horizon $s$.) Assume in addition that the setting of Law Identity between the N-Agent Aggregate Path with Its Observation Record and the Synthetic Copy's Regularised Path with Its Record is instantiated with the horizon $s$ in the role of its horizon $T$: with the numbers $N$, $l$, $m$, $\tilde{l}$, the control set $\mathcal{A}$, the rate families $\beta$ (with rate bound $B$) and $\tilde{\beta}$ (with rate bound $\tilde{B}$), the record space $(\mathbf{R}_{s},\mathcal{R}_{s})$ with its reference measure, the policy $h^{(s)}$ with its record-frozen control paths $a^{(s),r}$, a point $x_{0}\in\mathbb{G}_{N}$, a real number $R>NBs$, and the remaining data of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (its clock cells, index set $\mathsf{L}$, dimension $d$, smoothing parameter $\eta\in(0,1]$, driving variables, cell-count vector and copy clocks on a probability space $(\Omega^{\flat},\mathcal{F}^{\flat},P^{\flat})$), together with the setting of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood adopted there, all chosen for that horizon, so that its regularised recursion paths $\bar{\Sigma}^{\sharp,r}_{t}(\omega^{\flat})$ ($r\in\mathbf{R}_{s}$, $t\in[0,s]$), its synthetic copy $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $\Omega^{\sharp}=\Omega^{\flat}\times\mathbb{R}^{d}\times\mathbf{R}_{s}$ and its record $\mathsf{D}(\omega^{\flat},\theta,r)=r$ are defined; and with the driving system $(\Omega,\mathcal{F},P)$, the restricted solution and the fixed reconstruction data in the roles of the solution and the reconstruction data of that lemma. Assume moreover that $P(\Sigma_{0}=x_{0})=1$. Put $\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}(\omega^{\flat},\theta,r)=\bar{\Sigma}^{\sharp,r}_{s}(\omega^{\flat})$ and

the recentred copy endpoint. Then $\bar{\Sigma}^{\sharp,\mathsf{D}}_{s}$ takes its values in $\mathbb{G}_{N}$ and the map $(\bar{\Sigma}^{\sharp,\mathsf{D}}_{s},\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{P}(\mathbb{G}_{N})\otimes\mathcal{R}_{s}$; each component of $X''_{s}$ is a random variable on $(\Omega^{\sharp},\mathcal{F}^{\sharp},\mu^{\sharp})$ with $|X''_{s}|\le2\sqrt{N}$ everywhere; and for every $\mathbf{c}\in\mathbb{R}^{l}$ the random variable $\mathbf{c}\cdot X''_{s}$ is square-integrable, the map $(\mathbf{c}\cdot X''_{s},\mathsf{D})$ is measurable with respect to $\mathcal{F}^{\sharp}$ and $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$, and

In particular, for every map $F:\mathbb{R}\times\mathbf{R}_{s}\to[0,\infty]$ measurable with respect to $\mathcal{B}(\mathbb{R})\otimes\mathcal{R}_{s}$ in the sense of Lebesgue Integral of a Nonnegative Measurable Function,

the two sides being the integrals of nonnegative measurable functions with respect to $P$ and to $\mu^{\sharp}$.

Remark (not part of the claims; the setting mentioned here is not assumed). This lemma is intended for use in the setting of Asymptotic Lower Bound for the Recentred N-Agent Cost by the Fluctuation LQG Value, under Injection Certificates, where, for a stationary mean-field triple (whose co-state is written $P$ there, unrelated to the probability measure $P$ of the present Data) and with $z_{0}$ taken to be the value at time $0$ of its first component, $\Phi$ and $X'$ are the realized mean-field flow adopted there and the observation-centred fluctuation defined there, and $X''_{s}$ corresponds, through the coordinate maps of the copy, to the recentred endpoint $X''$ of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter for the copy with horizon $s$ and base point $z_{0}$.