Closeness of the Realized Control and the Realized Mean-Field Flow on a High-Probability Event under the Cost Bound

Data. Adopt the setting, notation and standing hypotheses of the asymptotic lower bound theorem for the recentred $N$-agent cost, and with them those of the ledger lemma and of the block cascade lemma on which it rests. In particular: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with compact convex control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and control bound $R$, and its transition-rate family $\beta$ with aggregate state drift $b$, state-Lipschitz constant $\Lambda_{b}$ and constant $K_{1}$; the horizon $T>0$; the stationary mean-field triple $(S,A,P)$ with $x_{0}=S_{0}$ in the probability simplex $\Delta^{l}$, so that the standing hypothesis $S^{*}=S$ of the block cascade lemma holds; and the family, indexed by the natural numbers $N\ge1$, of solutions of the controlled $N$-agent dynamics on $[0,T]$, the $N$-th of them carried by an $N$-agent driving system $(\Omega,\mathcal{F},P)$ with regular event $\Omega_{0}$ and observation filtration $(\mathcal{G}_{t})_{t\in[0,T]}$, together with all the standing hypotheses imposed there, among them (I$'$) with the bound $\kappa^{\sharp}$ and (CB) with the bound $\mathcal{J}^{\sharp}$. As there, the probability of the driving system is written $P$, the stationary co-state always carrying a time subscript, and the superscript $N$ is suppressed throughout.

Adopt from the extended good-set stopping-time lemma the constant $K_{2}=2\sqrt{l}\,(l-1)K_{1}$; the realized control $\hat{\alpha}$ of the realized-control lemma; the mean-field flow $S(z_{0},\xi)$ of claim 2 of the flow stability lemma, defined for an initial state $z_{0}\in\Delta^{l}$ and an $\mathcal{A}$-valued control $\xi$; the realized mean-field flow $\Phi_{t}(\omega)=S_{t}(x_{0},\hat{\alpha}(\omega))$ and the deviation process $Y_{t}(\omega)=|\Phi_{t}(\omega)-S_{t}|$; the control energy process

of claim 5 of the progressive measurability lemma for the realized control; and the constant

of claim 4 of the extended good-set stopping-time lemma. None of $\hat{\alpha}$, $\Phi$, $Y$, $\mathcal{E}$, $C_{S}$ and none of the conclusions of that lemma's claim 4 involves the thresholds $\delta$, $\theta_{\mathrm{out}}$, $\varepsilon_{Y}$ and $c_{\mathcal{E}}$ fixed in its setting; wherever that claim is invoked below, those four thresholds may therefore be given the value $1$.

Adopt finally the notion of an admissible parameter vector $\pi=(T_{0},\varepsilon_{1},\lambda_{c},\lambda_{o},\delta_{\mathrm{cl}},q_{0},\epsilon,\eta,\varrho)$, the absorption condition of clause (a) of the lower bound theorem, and the real number $\mathcal{Z}^{\sharp}(\pi)$ defined there. For an admissible $\pi$ and a natural number $N\ge1$, write $K$ for the number of blocks, $t_{0},\dots,t_{K}$ for the grid, $L_{k}$ for the levels, $\sigma^{(k)}$ for the anchored good-set clocks, $G_{0},\dots,G_{K}$ for the good sets, $D_{0},\dots,D_{K-1}$ for the leave events, $\Delta_{k}\mathcal{E}=\mathcal{E}_{\min(\sigma^{(k)},t_{k+1})}-\mathcal{E}_{t_{k}}$ for the block energy increments, $\Lambda_{\star}$ for the constant, and

for the tracked energy, all of these being the objects the block cascade lemma forms from $\pi$ for the $N$-th solution; and write $\mathcal{P}=\sum_{k=0}^{K-1}P(D_{k})$ and $\Upsilon_{\mathrm{lev}}=\sum_{k=0}^{K-1}L_{k}^{-2}$ as in the ledger lemma. Here $\mathbb{E}$ is the expectation, $\mathbf{1}_{D}$ is the function equal to $1$ on a set $D$ and $0$ off it, $|\cdot|$ is the Euclidean norm, $\int_{[a,b]}\cdot\,ds$ is the Lebesgue integral over a compact interval, taken to be $0$ when $a=b$, $\exp$ is the exponential function and $x^{1/2}=\sqrt{x}$ is the nonnegative square root of a nonnegative real $x$.

Then the following hold.

1. (Energy accrued while tracked, and the escape probability.) Let $\pi$ be admissible and let $N\ge1$. Then

all three quantities being finite.

2. (Pathwise bounds carried by the control energy.) For every $N\ge1$, every $\omega\in\Omega$ and every $t\in[0,T]$,

Neither bound involves the parameter vector, and neither excludes an exceptional set.

3. (A parameter vector with a uniform tracked energy bound.) There are an admissible parameter vector $\pi_{\bullet}$ satisfying the absorption condition of clause (a) of the lower bound theorem, a real number $\mathcal{Z}^{\sharp}\ge0$ and a natural number $N_{1}$, all three determined by the common data and by the bounds $\kappa^{\sharp}$ and $\mathcal{J}^{\sharp}$ and not by the individual solutions, such that $\mathcal{Z}^{\sharp}=\mathcal{Z}^{\sharp}(\pi_{\bullet})$ and such that for every $N\ge N_{1}$ the tracked energy of the $N$-th solution formed from $\pi_{\bullet}$ satisfies $\mathcal{Z}\le\mathcal{Z}^{\sharp}$.

4. (The good event.) Let $\pi_{\bullet}$, $\mathcal{Z}^{\sharp}$ and $N_{1}$ be as in claim 3, let $\Lambda_{\star}$, $\Upsilon_{\mathrm{lev}}$, $K$ and, for each $N\ge1$, the good set $G_{K}$ be those formed from $\pi_{\bullet}$, and set

Let $\varepsilon_{\flat}$ be a real number with $0<\varepsilon_{\flat}\le1$ and let $N_{\flat}$ be a natural number with $N_{\flat}\ge N_{1}$ and $\varepsilon_{\flat}N_{\flat}\ge C_{\mathrm{esc}}$. Then for every $N\ge N_{\flat}$ the set

is an event belonging to $\mathcal{G}_{T}$; it satisfies

and for every $\omega\in\Omega^{\flat}_{N}$ and every $t\in[0,T]$,

The constants $C_{\mathrm{ctl}}$, $C_{\mathrm{flw}}$ and $C_{\mathrm{esc}}$ depend only on the common data and on the bounds $\kappa^{\sharp}$ and $\mathcal{J}^{\sharp}$; the tolerance $\varepsilon_{\flat}$ enters the three pathwise bounds only through the factor $\varepsilon_{\flat}^{-1}$, enters the probability bound directly, and enters the threshold $N_{\flat}$ through the requirement $\varepsilon_{\flat}N_{\flat}\ge C_{\mathrm{esc}}$. The three bounds of claim 4 hold at every time of $[0,T]$ simultaneously, and are of the orders $N^{-1}$, $N^{-1/2}$ and $N^{-1/2}$. No hypothesis bounding $\mathbb{E}[|\mathfrak{s}_{t}|^{p}]$ or $\mathbb{E}[|\mathfrak{a}_{t}|^{p}]$ uniformly in $N$ at positive times is used: the closeness of the realized control to $A$ and of the realized mean-field flow to $S$ is extracted from the cost bound (CB) together with the hypothesis (I$'$) on the initial fourth moment, through the block cascade, and only on the event $\Omega^{\flat}_{N}$, off which no bound on either quantity is asserted.