Localized Joint Coercivity of the Recentred N-Agent Cost Integrand under a Positive-Definite Fluctuation Hessian

Adopt the setting and notation of the first-order expansion lemma for the recentred $N$-agent cost: the transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ and rate bound $B$; the observation-rate family $\tilde{\beta}$; the horizon $T>0$; the $N$-agent driving system; the $\mathcal{A}$-valued observation-driven control policy $h$; the solution on $[0,T]$ with regular event $\Omega_{0}$, empirical state measure $\Sigma_{t}$ and control $\alpha_{t}$; the mean-field trajectory pair $(S,A)$; the fluctuation processes $\mathfrak{s}_{t}=\sqrt{N}(\Sigma_{t}-S_{t})$ and $\mathfrak{a}_{t}=\sqrt{N}(\alpha_{t}-A_{t})$; the population cost data $(L,G)$; the twice continuously differentiable extension $(U,V,\bar{\beta})$ of $\beta$ with derivative bound $K$ and extended aggregate state drift $\bar{b}$; the twice continuously differentiable extension $(U_{c},\bar{L},\bar{G})$ of $(L,G)$ with second-derivative bound $K_{c}$; the stationary co-state $P$ with its bound $C_{P}$, so that $(S,A,P)$ is a stationary mean-field triple; the mean-field Hamiltonian $\mathcal{H}_{t}(\Sigma,\alpha)=\bar{L}(\Sigma,\alpha)-\sum_{\delta=1}^{l}P^{\delta}_{t}\bar{b}^{\delta}(\Sigma,\alpha)$ and its state derivative coefficients $\partial_{\gamma}\mathcal{H}_{t}$; the abbreviation $y_{t}=\Sigma_{t}-S_{t}$; and the quantities

$$\mathcal{D}_{t}=\mathcal{H}_{t}(\Sigma_{t},\alpha_{t})-\mathcal{H}_{t}(S_{t},A_{t})-\sum_{\gamma=1}^{l}\partial_{\gamma}\mathcal{H}_{t}(S_{t},A_{t})\,y^{\gamma}_{t},\qquad \mathcal{D}_{G}=\bar{G}(\Sigma_{T})-\bar{G}(S_{T})-\sum_{\gamma=1}^{l}\partial_{\gamma}\bar{G}(S_{T})\,y^{\gamma}_{T}.$$

Write $d$ for the Euclidean distance, $|\cdot|$ for the Euclidean norm, $\Delta^{l}$ for the probability simplex, and $\sqrt{\cdot}$ for the nonnegative square root. Assume, as there, hypothesis (A): $\mathcal{A}$ is compact and convex. Adopt the fluctuation Hessian coefficients $H_{ij}(t)$ ($i,j\in\{1,\dots,l+m\}$) and the terminal coefficients $F_{\gamma\delta}=\partial_{\delta}\partial_{\gamma}\bar{G}(S_{T})$ of these data, and the moduli of continuity $\omega_{L}$, $\omega_{b}$ and $\omega_{G}$ of the second-order expansion theorem for the $N$-agent cost, which by part (a) of that theorem are nondecreasing functions from $[0,\infty)$ to $[0,\infty)$, bounded by $2K_{c}$, $6\,l\,K$ and $2K_{c}$ respectively, and such that for every real $\varepsilon>0$ there is a real $\eta>0$ with $\omega_{L}(u)\le\varepsilon$, $\omega_{b}(u)\le\varepsilon$ and $\omega_{G}(u)\le\varepsilon$ for all $u\in[0,\eta]$.

For $t\in[0,T]$ write $\mathfrak{z}_{t}=(\mathfrak{s}_{t},\mathfrak{a}_{t})$ for the $\mathbb{R}^{l+m}$-valued map whose first $l$ components are those of $\mathfrak{s}_{t}$ and whose last $m$ components are those of $\mathfrak{a}_{t}$, so that $|\mathfrak{z}_{t}|^{2}=|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}$. For $t\in[0,T]$ and $\omega\in\Omega$ write, as in that theorem,

$$\rho_{t}=d\bigl((\Sigma_{t},\alpha_{t}),(S_{t},A_{t})\bigr)=N^{-1/2}\bigl(|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}\bigr)^{1/2},\qquad d(\Sigma_{T},S_{T})=N^{-1/2}|\mathfrak{s}_{T}| .$$

Hypothesis (JC) (Local joint coercivity of the cost). There is a real number $c_{J}>0$ such that

$$\tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}H_{ij}(t)\,w^{i}w^{j}\ \ge\ c_{J}\,|w|^{2}\qquad\text{for every }t\in[0,T]\text{ and every }w\in\mathbb{R}^{l+m},$$

and the terminal coefficients are positive semidefinite:

$$\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}F_{\gamma\delta}\,v^{\gamma}v^{\delta}\ \ge\ 0\qquad\text{for every }v\in\mathbb{R}^{l}.$$

Then the following hold.

1. (Second-order remainder bound.) For every $t\in[0,T]$ and every $\omega\in\Omega$,

$$\Bigl|\,N\,\mathcal{D}_{t}-\tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}H_{ij}(t)\,\mathfrak{z}^{i}_{t}\,\mathfrak{z}^{j}_{t}\,\Bigr|\ \le\ \tfrac{1}{2}\,(l+m)\,\bigl(\omega_{L}(\rho_{t})+C_{P}\,\omega_{b}(\rho_{t})\bigr)\,\bigl(|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}\bigr),$$

and for every $\omega\in\Omega$,

$$\Bigl|\,N\,\mathcal{D}_{G}-\tfrac{1}{2}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}F_{\gamma\delta}\,\mathfrak{s}^{\gamma}_{T}\,\mathfrak{s}^{\delta}_{T}\,\Bigr|\ \le\ \tfrac{1}{2}\,l\,\omega_{G}\bigl(d(\Sigma_{T},S_{T})\bigr)\,|\mathfrak{s}_{T}|^{2}.$$

This conclusion does not use (JC).

2. (A coercivity radius.) There is a real number $\rho^{*}>0$ such that

$$\tfrac{1}{2}\,(l+m)\,\bigl(\omega_{L}(u)+C_{P}\,\omega_{b}(u)\bigr)\ \le\ \tfrac{c_{J}}{2}\qquad\text{for every }u\in[0,\rho^{*}].$$

3. (Localized joint coercivity.) Let $\rho^{*}$ be as in claim 2. For every $t\in[0,T]$ and every $\omega\in\Omega$ with $\rho_{t}(\omega)\le\rho^{*}$,

$$N\,\mathcal{D}_{t}\ \ge\ \frac{c_{J}}{2}\,\bigl(|\mathfrak{s}_{t}|^{2}+|\mathfrak{a}_{t}|^{2}\bigr).$$

In particular $N\mathcal{D}_{t}\ge\tfrac{c_{J}}{2}|\mathfrak{a}_{t}|^{2}$ there, with no deficit in the state fluctuation.

4. (Terminal bound.) For every $\omega\in\Omega$,

$$N\,\mathcal{D}_{G}\ \ge\ -\,\tfrac{1}{2}\,l\,\omega_{G}\bigl(d(\Sigma_{T},S_{T})\bigr)\,|\mathfrak{s}_{T}|^{2};$$

consequently, for every real $\epsilon>0$ there is a real $\rho^{*}_{G}>0$ such that $N\mathcal{D}_{G}\ge-\epsilon\,|\mathfrak{s}_{T}|^{2}$ at every $\omega\in\Omega$ with $d(\Sigma_{T},S_{T})\le\rho^{*}_{G}$.

5. (Joint coercivity implies control-block coercivity.) Hypothesis (H1) of the quadratic growth lemma for the mean-field Hamiltonian holds with the constant $r=c_{J}$; that is, writing $R_{t}$ for the matrix defined there,

$$h\cdot R_{t}h\ \ge\ c_{J}\,|h|^{2}\qquad\text{for every }t\in[0,T]\text{ and every }h\in\mathbb{R}^{m}.$$

In particular, if in addition hypothesis (U) of that lemma holds, its conclusion (d) furnishes a real $r_{0}>0$ with $\mathcal{H}_{t}(S_{t},a)-\mathcal{H}_{t}(S_{t},A_{t})\ge r_{0}|a-A_{t}|^{2}$ for all $t$ and all $a\in\mathcal{A}$, so that the conclusions of the first-order expansion lemma requiring (H1) are available under (JC) as well.