Proof of The Empirical State Measure Deviates from the Realized Mean-Field Flow by at Most the Noise Majorant

Throughout, "the envelope lemma" is the pre-stopping envelope lemma, "the tracking lemma" is the pathwise tracking lemma and "the restricted-moments lemma" is the restricted-moments lemma for the martingale part; all notation is that of the statement, hence that of the envelope lemma. The argument is the one by which the proof of claim 6 of the envelope lemma bounds the first three summands of its triangle inequality; it is isolated here because the envelope lemma states only the resulting bound on $|\mathfrak{s}_t|$.

Fix $\omega\in\Omega_0$ and $t\in[0,T]$. By clause (vii)(a) of the existence and uniqueness theorem, $\Sigma_0(\omega)$ lies in the probability simplex $\Delta^l$, and $\hat{\alpha}(\omega)\in\mathcal{U}_{\mathcal{A}}$ for every $\omega$ by claim 3 of the realized-control lemma; so $S_t(\Sigma_0(\omega),\hat{\alpha}(\omega))$ is defined by claim 2 of the flow stability lemma. Since $\Phi_t(\omega)=S_t(x_0,\hat{\alpha}(\omega))$, the triangle inequality for the Euclidean norm (claim 6 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) gives

$$\bigl|\Sigma_t(\omega)-\Phi_t(\omega)\bigr|\ \le\ \bigl|\Sigma_t(\omega)-S_t(\Sigma_0(\omega),\hat{\alpha}(\omega))\bigr|+\bigl|S_t(\Sigma_0(\omega),\hat{\alpha}(\omega))-S_t(x_0,\hat{\alpha}(\omega))\bigr| .$$

First summand. By claim 2 of the tracking lemma, whose flow $S^{\omega}$ is $S(\Sigma_0(\omega),\hat{\alpha}(\omega))$, the first summand is at most $|M_t(\omega)|+\Lambda_be^{\Lambda_bt}\mathcal{M}_t(\omega)$, hence at most $|M_t(\omega)|+\Lambda_be^{\Lambda_bT}\mathcal{M}_t(\omega)$, the exponential function being nondecreasing (claim 4 of the exponential properties theorem), $\Lambda_b=2\sqrt{l}(l-1)\bigl(B+\Lambda(1+R)\bigr)\ge0$ because $B\ge0$ and $R\ge0$ by claim 1 of the affine rate family lemma and $\Lambda\ge0$ is a Lipschitz constant, and $\mathcal{M}_t(\omega)\ge0$.

Second summand. By claim 4 of the flow stability lemma, applied with base pair $(x_0,\hat{\alpha}(\omega))$ and perturbed pair $(\Sigma_0(\omega),\hat{\alpha}(\omega))$ — the perturbed control equals the base control, so $g^{\gamma}_r(\xi')=0$ for every $\gamma$ and $r$ in the notation of claim 3 there, and the constant $G=0$ of claim 4 there (not the terminal cost of the adopted setting) is admissible — the second summand is at most $e^{\Lambda_bT}|\Sigma_0(\omega)-x_0|$. Since $S_0=x_0$, $|\Sigma_0(\omega)-x_0|=N^{-1/2}|\mathfrak{s}_0(\omega)|$.

Passage to the majorant. By the supremum lemma, $\overline{M}^{\gamma}(\omega)=\sup_{r\in[0,T]}|M^{\gamma}_r(\omega)|$ for each $\gamma$, because $\omega\in\Omega_0$; hence $|M^{\gamma}_t(\omega)|\le\overline{M}^{\gamma}(\omega)$ for each $\gamma$, so $|M_t(\omega)|^2=\sum_{\gamma}(M^{\gamma}_t(\omega))^2\le\sum_{\gamma}(\overline{M}^{\gamma}(\omega))^2=\overline{M}(\omega)^2$, and $|M_t(\omega)|\le\overline{M}(\omega)$ because the nonnegative square root is nondecreasing. The path $r\mapsto\mathcal{M}_r(\omega)$ is nondecreasing by claim 1 of the tracking lemma, and $I(\omega)=\mathcal{M}_T(\omega)$ by part (b) of the restricted-moments lemma, so $\mathcal{M}_t(\omega)\le I(\omega)$.

Combining the three estimates,

$$\bigl|\Sigma_t(\omega)-\Phi_t(\omega)\bigr|\ \le\ \overline{M}(\omega)+\Lambda_be^{\Lambda_bT}I(\omega)+e^{\Lambda_bT}N^{-1/2}|\mathfrak{s}_0(\omega)|\ =\ \overline{\mathfrak{b}}(\omega)-\varepsilon_Y\ =\ Q(\omega),$$

which is the claim.