A Priori Fourth-Moment Bound for the State Fluctuation Process

Adopt the setting of the fluctuation processes of the controlled $N$-agent dynamics: a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$, a nonempty subset of Euclidean space $\mathbb{R}^m$, and rate bound $B$, an observation-rate family $\tilde{\beta}$, a horizon $T>0$, an $N$-agent driving system $(\Omega,\mathcal{F},P)$, an observation-driven control policy $h$ which is $\mathcal{A}$-valued, a solution on $[0,T]$ with regular event $\Omega_0$, empirical state measure $\Sigma_t$, and control $\alpha_t$, a mean-field trajectory pair $(S,A)$ for $\beta$ with horizon $T$, and the fluctuation processes $\mathfrak{s}_t=\sqrt{N}(\Sigma_t-S_t)$ and $\mathfrak{a}_t=\sqrt{N}(\alpha_t-A_t)$. Assume that $\beta$ admits a twice continuously differentiable extension $(U,V,\bar{\beta})$ with derivative bound $K$, and assume that $\mathcal{A}$ is convex; the bound $\Lambda$ below is warranted by clauses (i) and (ii) of the regularity and derivative bounds of the extended aggregate state drift. Let $b$ be the aggregate state drift of $\beta$, let $M_t=(M^\gamma_t)_{\gamma\in\{1,\dots,l\}}$ be as in the martingale decomposition, write $\mathbb{E}$ for the expectation, $|\cdot|$ for the Euclidean norm (Euclidean distance to the origin), and $\mathbf{1}_{\Omega_0}$ for the function equal to $1$ on $\Omega_0$ and $0$ off $\Omega_0$, set $g_s=\sqrt{N}\,(b(\Sigma_s,\alpha_s)-b(S_s,A_s))$ as in the a priori second-moment bound, set

and let $c_M=6\,l^2\,\big(2(l-1)\big)^4$ be the constant of the moment bounds for the aggregate compensated counters.

(a) (Well-definedness.) The maps $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{s}_t(\omega)|^4$, $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|\mathfrak{a}_t(\omega)|^4$, and $(t,\omega)\mapsto\mathbf{1}_{\Omega_0}(\omega)|g_t(\omega)|^4$ are measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$, by the joint measurability of the state and control together with measurability of sequentially continuous functions of measurable Euclidean maps. Consequently, by the Tonelli theorem, the functions $t\mapsto\mathbb{E}[|\mathfrak{s}_t|^4]$, $t\mapsto\mathbb{E}[|\mathfrak{a}_t|^4]$, and $t\mapsto\mathbb{E}[|g_t|^4]$ (unchanged by the $\mathbf{1}_{\Omega_0}$ modification, $\Omega_0$ having probability $1$) are measurable $[0,\infty]$-valued functions on $[0,T]$, with $\mathbb{E}[|\mathfrak{s}_t|^4]\le16\,N^2$ and $\mathbb{E}[|g_t|^4]\le256\,l^2\,N^2\,(l-1)^4\,B^4$ finite for every $t$, and the Lebesgue integral

is well defined with value in $[0,\infty]$ (the subscripted $\mathcal{A}_4$ is distinct from the control energy $\mathcal{A}_2$ of the second-moment bound and from the control set $\mathcal{A}$).

(b) (Three-term estimate.) For every $t\in[0,T]$, with the right-hand side valued in $[0,\infty]$,

(c) (A priori bound.) For every $t\in[0,T]$, with the exponential function,

where the right-hand side is interpreted as $+\infty$ when $\mathcal{A}_4=+\infty$. In particular, if $\mathcal{A}_4<\infty$, then $\sup\{\mathbb{E}[|\mathfrak{s}_t|^4]:t\in[0,T]\}$ is finite, with a bound depending on $N$ only through $\mathbb{E}[|\mathfrak{s}_0|^4]$ and $\mathcal{A}_4$.