Proof of The Projected Extension of an Affine-Controlled Transition-Rate Family

Throughout, $(\sigma,\gamma)$ ranges over the ordered pairs with $\sigma\neq\gamma$ in $\{1,\dots,l\}$, of which there are $l(l-1)$, and each fixed $\gamma$ has $l-1$ partners $\sigma$. We use two elementary facts about the Euclidean norm on $\mathbb{R}^k$: the coordinate bound $|z^i|\le|z|$ from its elementary properties, and $|z|\le\sqrt{k}\max_i|z^i|$, which holds because $|z|^2=\sum_i(z^i)^2\le k\max_i(z^i)^2$. We also use the Cauchy-Schwarz inequality $|z\cdot w|\le|z||w|$, applied to the positive semidefinite quadratic form given by the identity matrix, whose associated bilinear form is the dot product.

Claim 1. The set $\mathcal{A}$ is compact, hence closed and bounded by the Heine-Borel theorem; it is nonempty and convex by assumption, so the nearest-point projection $\pi_{\mathcal{A}}$ is defined, and $R=\sup_{\alpha\in\mathcal{A}}|\alpha|$ is a finite nonnegative real number.

The simplex $\Delta^l$ is nonempty, since it contains the standard basis vectors. It is convex: if $\Sigma,\Sigma'\in\Delta^l$ and $t\in[0,1]$, then $(1-t)\Sigma+t\Sigma'$ has nonnegative coordinates with sum $(1-t)+t=1$. It is bounded, since $|\Sigma|^2=\sum_\gamma(\Sigma^\gamma)^2\le\big(\sum_\gamma\Sigma^\gamma\big)^2=1$ for $\Sigma\in\Delta^l$, the coordinates being nonnegative. It is closed: if $\Sigma\notin\Delta^l$, then either some coordinate satisfies $\Sigma^\gamma<0$ or $|\sum_\gamma\Sigma^\gamma-1|>0$, and in either case, since the maps $\Sigma\mapsto\Sigma^\gamma$ and $\Sigma\mapsto\sum_\gamma\Sigma^\gamma$ change by at most $\sqrt{l}\,|\Sigma-\Sigma'|$ under a change of argument, some open ball around $\Sigma$ misses $\Delta^l$; hence the complement of $\Delta^l$ is open. So $\pi_{\Delta^l}$ is defined as well.

For finiteness of $K_1$ and $B$, fix $\Sigma_*\in\Delta^l$ and note that $|\Sigma-\Sigma_*|\le|\Sigma|+|\Sigma_*|\le2$ for $\Sigma\in\Delta^l$. By the Lipschitz clause,

and there are finitely many pairs $(\sigma,\gamma)$, so $K_1$ is a finite nonnegative real number. Since $\pi_{\mathcal{A}}(\alpha)\in\mathcal{A}$ for every $\alpha\in\mathbb{R}^m$, Cauchy-Schwarz gives $|\beta_1(\sigma,\gamma,\Sigma)\cdot\pi_{\mathcal{A}}(\alpha)|\le K_1R$, so $\beta$ is bounded above and $B$ is finite; $B\ge0$ because, as shown in Claim 2, $\beta\ge0$.

Claim 2. Let $\Sigma\in\Delta^l$ and $\alpha\in\mathbb{R}^m$. Then $\pi_{\mathcal{A}}(\alpha)\in\mathcal{A}$, so the nonnegativity clause of the definition gives $\beta(\sigma,\gamma,\Sigma,\alpha)\ge0$, and $\beta(\sigma,\gamma,\Sigma,\alpha)\le B$ by the definition of $B$. If $\alpha\in\mathcal{A}$ then $\pi_{\mathcal{A}}(\alpha)=\alpha$ by the fixed-point clause of the projection lemma, which gives the stated identity.

For joint continuity, let $\Sigma_k\to\Sigma$ in $\Delta^l$ and $\alpha_k\to\alpha$ in $\mathbb{R}^m$ in Euclidean distance. The Lipschitz clause gives $|\beta_0(\sigma,\gamma,\Sigma_k)-\beta_0(\sigma,\gamma,\Sigma)|\le\Lambda|\Sigma_k-\Sigma|\to0$ and $|\beta_1(\sigma,\gamma,\Sigma_k)-\beta_1(\sigma,\gamma,\Sigma)|\le\Lambda|\Sigma_k-\Sigma|\to0$, while nonexpansiveness of $\pi_{\mathcal{A}}$ gives $|\pi_{\mathcal{A}}(\alpha_k)-\pi_{\mathcal{A}}(\alpha)|\le|\alpha_k-\alpha|\to0$. Hence, by Cauchy-Schwarz,

after inserting the intermediate term $\beta_1(\sigma,\gamma,\Sigma)\cdot\pi_{\mathcal{A}}(\alpha_k)$ and using $|\pi_{\mathcal{A}}(\alpha_k)|\le R$. Therefore $\beta(\sigma,\gamma,\Sigma_k,\alpha_k)\to\beta(\sigma,\gamma,\Sigma,\alpha)$. Both requirements of a transition-rate family with rate bound $B$ are met.

Finally, for $\Sigma,\Sigma'\in\Delta^l$ and $\alpha\in\mathbb{R}^m$,

Claim 3. For $\alpha\in\mathcal{A}$ we may substitute $\beta(\sigma,\gamma,\Sigma,\alpha)=\beta_0(\sigma,\gamma,\Sigma)+\beta_1(\sigma,\gamma,\Sigma)\cdot\alpha$, from Claim 2, into the formula defining the aggregate state drift,

and collect the terms without $\alpha$ and the terms linear in $\alpha$; this gives exactly $b^\gamma_0(\Sigma)+b^\gamma_1(\Sigma)\cdot\alpha$.

Claim 4. Let $\Sigma,\Sigma'\in\Delta^l$ and $\alpha,\alpha'\in\mathbb{R}^m$. Since $0\le\Sigma^\sigma\le1$ and $0\le\beta\le B$,

and hence $|b(\Sigma,\alpha)|\le\sqrt{l}\cdot2(l-1)B$.

For the Lipschitz bound in the state, each of the $2(l-1)$ terms constituting $b^\gamma$ obeys

using the coordinate bound, $(\Sigma')^\sigma\le1$ and Claim 2, and similarly for the terms with $\Sigma^\gamma$. Summing, $|b^\gamma(\Sigma,\alpha)-b^\gamma(\Sigma',\alpha)|\le2(l-1)(B+\Lambda_\beta)|\Sigma-\Sigma'|$, whence $|b(\Sigma,\alpha)-b(\Sigma',\alpha)|\le\Lambda_b|\Sigma-\Sigma'|$.

For the Lipschitz bound in the control, Claim 2 and nonexpansiveness of $\pi_{\mathcal{A}}$ give

and summing the $2(l-1)$ terms, each weighted by a coordinate of $\Sigma$ which is at most $1$, gives $|b^\gamma(\Sigma,\alpha)-b^\gamma(\Sigma,\alpha')|\le2(l-1)K_1|\alpha-\alpha'|$ and then the stated bound.

Claim 5. Summing the drift over $\gamma$,

Both double sums run over all ordered pairs with distinct entries: the first is the sum of $\Sigma^\sigma\beta(\sigma,\gamma,\Sigma,\alpha)$ over such pairs $(\sigma,\gamma)$, and the second becomes the same sum after interchanging the names of the two summation indices. Hence the difference is $0$.

For the inflow bound, the first group of terms is nonnegative because $\Sigma^\sigma\ge0$ and $\beta\ge0$, so

Claim 6. For $x\in\mathbb{R}^l$ we have $\pi_{\Delta^l}(x)\in\Delta^l$, so $\hat{b}(x,\alpha)=b(\pi_{\Delta^l}(x),\alpha)$ is defined, and it satisfies the same bound as $b$ by Claim 4. For $x\in\Delta^l$, the fixed-point clause of the projection lemma gives $\pi_{\Delta^l}(x)=x$ and hence $\hat{b}(x,\alpha)=b(x,\alpha)$. Finally, by Claim 4 and nonexpansiveness of $\pi_{\Delta^l}$,