A note on the top Lyapunov exponent of linear cooperative systems

For all $T>0,$ let $\omega^{T}_{t}=\omega_{t/T}.$ Like $(\omega_{t})_{t\geq 0}$, $(\omega^{T}_{t})_{t\geq 0}$ is a Feller Markov process on $S,$ uniquely ergodic with invariant probability $\mu.$ The parameter $1/T$ can be understood as a velocity parameter. For instance, in the context of Example 1, $(\omega^{T}_{t})_{t\geq 0}$ is a $T$-periodic signal. In the context of Example 3, its mean sojourn time in each state $i\in S$ is proportional to $T.$

Consider the differential equation (1) with $(\omega_{t})_{t\geq 0}$ replaced by $(\omega^{T}_{t})_{t\geq 0}.$ We let $\pi^{T}$ and $\Lambda^{T}$ denote the corresponding invariant probabilities on $S\times\Delta$ and top Lyapunov exponent as defined in Proposition 1. This section considers the fast and slow regimes obtained as $T\rightarrow 0$ and $T\rightarrow\infty.$

For a $d\times d$ real matrix $M,$ we let $\lambda_{max}(M)$ denote the largest real part of its eigenvalues (sometimes called the spectral abscissa of $M$; see e.g [6]).

For $r>0$ sufficiently large, $\bar{A}+rI$ has nonnegative entries and is irreducible. Hence, by Perron-Frobenius theorem (applied to $\bar{A}+rI$), $\lambda_{max}(\bar{A})$ is an eigenvalue and there exists a unique vector, the Perron-Frobenius vector of $\bar{A},$ $\theta^{*}\in\Delta,$ such that

$$\bar{A}\theta^{*}=\lambda_{max}(\bar{A})\theta^{*}.$$

Note that Proposition 4 has been proven for Example 3 in ([5], Corollary 2.15). The next result generalizes [6] beyond Example 1. Let $\mathsf{supp}(\mu)$ be the topological support of $\mu.$ Assume that for all $s\in\mathsf{supp}(\mu),$ $A(s)$ is irreducible. Then, under this assumption, there exists for all $s\in\mathsf{supp}(\mu)$ a unique Perron-Frobenius vector for $A(s),\theta^{*}(s)\in\Delta$ characterized by

$$A(s)\theta^{*}(s)=\lambda_{max}(A(s))\theta^{*}(s).$$

If $X$ is a metric space (such as $S,\Delta,S\times\Delta$) we let $B(X)$ denote the space of real valued Borel bounded functions on $X$ and $C(X)\subset B(X)$ the subspace of bounded continuous functions. For all $f\in B(X)$ we let $\|f\|_{\infty}=\sup_{x\in X}|f(x)|.$ If $\nu$ is a probability on $X$ and $f\in B(X)$ we write $\nu(f)$ for $\int_{X}fd\nu.$

Our main assumption that $(\omega_{t})_{t\geq 0}$ is a Feller Markov process on $S,$ means, as usual, that $(\omega_{t})_{t\geq 0}$ is a Markov process whose transition semigroup $(P_{t})_{t\geq 0}$ is Feller. That is:

(a)

$P_{t}(C(S))\subset C(S);$

(b)

$\lim_{t\rightarrow 0}P_{t}f(s)=f(s)$ for all $f\in C(S)$ and $s\in S.$

It turns out (see e.g [10], Theorem 19.6) that $(a)$ and $(b)$ make $(P_{t})_{t\geq 0}$ strongly continuous in the sense that $\lim_{t\rightarrow 0}\|P_{t}f-f\|_{\infty}=0$ for all $f\in C(S).$

An invariant probability for $(\omega_{t})_{t\geq 0}$ (or $(P_{t})_{t\geq 0}$) is a probability $\mu$ on $S$ such that for all $t\geq 0,$ $\mu P_{t}=\mu$ (i.e $\mu(P_{t}f)=\mu(f)$ for all $f\in B(S)$). Feller continuity and compactness of $S$ imply that such a $\mu$ always exists (see e.g [2], Corollary 4.21). Our assumption that $(\omega_{t})_{t\geq 0}$ is uniquely ergodic means that $\mu$ is unique.

A useful consequence of Feller continuity is that we can assume without loss of generality that $(\omega_{t})_{t\geq 0}$ is defined on the space $\Omega$ consisting of càdlàg (right-continuous, left limit) paths $\omega:{\mathbb{R}}_{+}\rightarrow S$ equipped with the Skorohod topology and associated Borel sigma field (see e.g [10], Theorem 19.15). As usual, for all $s\in S$ we let $\mathbb{P}_{s}$ denote the law of $(\omega_{t})_{t\geq 0}$ starting from $\omega_{0}=s$ and $\mathbb{P}_{\mu}=\int_{S}\mathbb{P}_{s}\mu(ds).$ The associated expectations are denoted $\mathbb{E}_{s}$ and $\mathbb{E}_{\mu}.$

For all $\omega\in\Omega$ and $t\geq 0$ we let $\mathbf{\Theta}_{t}(\omega)$ denote the shifted path defined as $\mathbf{\Theta}_{t}(\omega)(s)=\omega(t+s).$ Ergodicity of $\mu$ for the Markov process $(\omega_{t})_{t\geq 0}$ makes $\mathbb{P}_{\mu}$ ergodic (but not uniquely ergodic) for the dynamical system $(\mathbf{\Theta}_{t})_{t\geq 0}$ on $\Omega$ (see e.g [2], Proposition 4.49).

For $\omega\in\Omega$ the solution to (1) writes $y(t)=\Phi(t,\omega)x$ where $(\Phi(t,\omega))_{t\geq 0}$ is solution to the matrix valued differential equation

$$\frac{dM}{dt}=A(\omega_{t})M,M(0)=Id.$$

Let $\cal{M}_{+}\subset{\cal M}$ denote the set of $d\times d$ Metzler matrices having positive diagonal entries and $\cal{M}_{++}\subset{\cal M}_{+}$ the set of matrices having positive entries. Observe that

$$\Phi(t,\omega)\in\cal{M}_{+}$$

for all $t\geq 0.$ Indeed, for $r$ large enough and all $s\in S,\,A(s)+2rId\geq rId$ so that $e^{2rt}\Phi(t,\omega)\geq e^{Rt}Id$ (componentwise).

For all $\theta\in\Delta$, the solution to (3) with initial condition $\theta(0)=\theta,$ writes

$$\theta(t)=\Psi(t,\omega)\theta:=\frac{\Phi(t,\omega)\theta}{\langle\Phi(t,\omega)\theta,\bf{1}\rangle}.$$

Proof:    $(i).$ First observe that $\Phi(t,\omega)\in{\cal M}_{++}\Leftrightarrow e^{rt}\Phi(t,\omega)\in{\cal M}_{++}$ for all $r>0.$ Therefore, replacing $A(s)$ by $A(s)+rId$ for $r>\|A\|_{\infty},$ we can assume without loss of generality that $A(s)\in{\cal M}_{+}$ for all $s\in S.$

Let $x(t)=\Phi(t,\omega)x$ with $x\in{\mathbb{R}}^{d}_{+}\setminus\{0\}.$ Suppose $x_{i}(0)>0.$ Then $x_{i}(t)>0$ because $\dot{x}_{i}(t)\geq A_{ii}(\omega_{t})x_{i}(t)\geq 0.$ By irreducibility of $\bar{A}$, for all $j\neq i$ there exists a sequence $i_{0}=i,i_{1},\ldots,i_{n}=j$ such that $\bar{A}_{i_{k}i_{k-1}}>0$ for $k=1,\ldots n.$ By ergodicity, there exists a Borel set $\tilde{\Omega}\subset\Omega$ with $\mathbb{P}_{\mu}(\tilde{\Omega})=1$ such that for all $\omega\in\tilde{\Omega}$

$$\frac{1}{t}\int_{0}^{t}A(\omega_{u})du\rightarrow\bar{A}.$$

Therefore, for all $\omega\in\tilde{\Omega},$ there exists a sequence $t_{1}>t_{2}>\ldots>t_{n}$ with

$$A_{i_{k}i_{k-1}}(\omega_{t_{k}})>0.$$

By right continuity of $(\omega_{t})$ we also have $A_{i_{k}i_{k-1}}(\omega_{t})>0$ for $t_{k}\leq t\leq t_{k}+\varepsilon$ for some $\varepsilon>0.$ It follows that $\dot{x}_{i_{1}}(t)\geq A_{i_{1},i}(\omega_{t})x_{1}(t)>0$ for all $t_{1}\leq t\leq t_{1}+\varepsilon.$ Hence $x_{i_{1}}(t)>0$ for all $t>t_{1}.$ Similarly $x_{i_{2}}(t)>0$ for all $t>t_{2}$ and, by recursion, $x_{j}(t)>0$ for all $t>t_{n}.$ In summary, we have shown that for all $i,j\in\{1,\ldots,d\}$ and $\omega\in\tilde{\Omega},$ there exists a time $t_{n}$ depending on $i,j,\omega$ such that for all $t\geq t_{n}$ $x_{j}(t)>0$ whenever $x_{i}(0)>0.$ This proves $(i).$

$(ii).$ Let ${\mathbb{R}}^{d}_{++}=\{x\in{\mathbb{R}}^{d}\>:x_{i}>0,\mbox{ for all }i=1\ldots d\}$ and $\dot{\Delta}=\Delta\cap{\mathbb{R}}^{d}_{++}$ be the relative interior of $\Delta.$ The projective or Hilbert metric $d_{H}$ on ${\mathbb{R}}^{d}_{++}$ (see Seneta [18]) is defined by

$$d_{H}(x,y)=\log\frac{\max_{1\leq i\leq d}x_{i}/y_{i}}{\min_{1\leq i\leq d}x_{i}/y_{i}}.$$

Note that for all $\alpha,\beta>0,$ $d_{H}(\alpha x,\beta y)=d_{H}(x,y)$ so that $d_{H}$ is not a distance on ${\mathbb{R}}^{d}_{++}.$ However its restriction to $\dot{\Delta}$ is. Furthermore, for all $\theta$, $\theta^{\prime}\in\dot{\Delta}$,

$$\max_{1\leq i\leq d}|\theta_{i}-\theta^{\prime}_{i}|\leq e^{d_{H}(\theta,\theta^{\prime})}-1.$$

(5)

By a theorem of Birkhoff (see e.g [18], Section 3.4), for all $M\in{\cal M}_{+},$

$$\sup_{\{x,y\in{\mathbb{R}}^{d}_{++}\>d_{H}(x,y)>0\}}\frac{d_{H}(Mx,My)}{d_{H}(x,y)}=\tau[M]$$

(6)

where $0\leq\tau(M)\leq 1$ is the number defined as $\tau(M)=\frac{1-\sqrt{r(M)}}{1+\sqrt{r(M)}}$ with $r(M)=\min_{i,j,k,l}\min\frac{M_{ik}M_{jl}}{M_{jk}M_{il}}$ if $M\in{\cal M}_{++}$ and $r(M)=0$ if $M\in{\cal M}_{+}\setminus{\cal M}_{++}.$ In particular, for $M\in{\cal M}_{+}$, $\tau(M)<1$ if and only if $M\in{\cal M}_{++}$.

For all $0\leq s\leq t,$ let

$$F_{s,t}(\omega)=\max\{\log(\tau[\Phi(t-s,\mathbf{\Theta}_{s}(\omega))]),s-t\}\in[s-t,0].$$

We claim that $(F_{s,t})_{0\leq s\leq t}$ is a sub-additive process. That is:

(i)

$F_{s,t}\circ\mathbf{\Theta}_{v}=F_{s+v,t+v},$ and

(ii)

$F_{s,u}\leq F_{s,t}+F_{t,u},$

for all $s\leq t\leq u$ and $v\geq 0.$

The first assertion is immediate because $\mathbf{\Theta}_{s}\circ\mathbf{\Theta}_{v}=\mathbf{\Theta}_{s+v}.$ For the second, by the cocycle property

$$\Phi(u-s,\mathbf{\Theta}_{s}(\omega))=\Phi(u-t,\mathbf{\Theta}_{t}(\omega))\circ\Phi(t-s,\mathbf{\Theta}_{s}(\omega)).$$

Thus,

$$\log(\tau[\Phi(u-s,\mathbf{\Theta}_{s}(\omega))]\leq\log(\tau[\Phi(u-t,\mathbf{\Theta}_{t}(\omega))]+\log(\tau[\Phi(t-s,\mathbf{\Theta}_{s}(\omega))].$$

This proves $(ii)$.

Note also that $t,s\rightarrow F_{s,t}(\omega)$ is continuous and that $\displaystyle\sup_{0\leq s\leq t\leq 1}|F_{s,t}|\leq 1,$ so that the integrability conditions required for the continuous time version of Kingman’s subadditive ergodic theorem (as stated in [11], Theorem 5.6) are satisfied. Therefore, by this theorem,

$$\limsup_{t\rightarrow\infty}\frac{\log(\tau[\Phi(t,\omega)])}{t}\leq\lim_{t\rightarrow\infty}\frac{F_{0,t}(\omega)}{t}=\gamma,$$

$\mathbb{P}_{\mu}$ almost surely, where

$$\gamma=\inf_{t>0}\mathbb{E}_{\mu}\frac{F_{0,t}}{t}.$$

Clearly $\gamma<0.$ For otherwise we would have that $\tau[\Phi(n,\omega)]=1\Leftrightarrow\Phi(n,\omega)\in{\cal M}_{+}\setminus{\cal M}_{++}$ for all $n\in{\mathbb{N}},\,\mathbb{P}_{\mu}$ almost surely, in contradiction with $(i).$

Let $N$ be like in assertion $(i)$ of the Lemma. Then, by what precedes, $\mathbb{P}_{\mu}$ almost surely,

$$\limsup_{t\rightarrow\infty}\frac{\log(d_{H}(\Psi(t+N,\omega)\theta,\Psi(t+N,\omega)\theta^{\prime}))}{t}$$

$$\leq\limsup_{t\rightarrow\infty}\frac{\log(\tau[\Phi(t,\mathbf{\Theta}_{N}(\omega))])}{t}+\limsup_{t\rightarrow\infty}\frac{\log(d_{H}(\Psi(N,\omega)\theta,\Psi(N,\omega)\theta^{\prime}))}{t}=\gamma.$$

By inequality (5), this concludes the proof. $\Box$
Let $(Q_{t})_{t\geq 0}$ denote the semigroup of the process $(\omega_{t},\theta_{t})_{t\geq 0}.$ Then, for all $f\in B(S\times\Delta)$ $(s,\theta)\in S\times\Delta,$

$$Q_{t}f(s,\theta)=\mathbb{E}_{s}[f(\omega_{t},\Psi(t,\omega)(\theta))].$$

Proof:    We need to show that $(a)$ $Q_{t}(C(S\times\Delta)\subset C(S\times\Delta)$ and $(b)$ $\lim_{t\rightarrow 0}Q_{t}f(s,\theta)=f(s,\theta)$ for all $f\in C(S\times\Delta).$

$(a).$ It is easy to verify that there exist constants $c_{1},c_{2}\geq 0$ such that for all $s,s^{\prime}\in S,\theta,\theta^{\prime}\in\Delta$

	$\displaystyle\|F(s,\theta)-F(s,\theta^{\prime})\|$	$\displaystyle\leq$	$\displaystyle c_{1}\|\theta-\theta^{\prime}\|$		(7)
	$\displaystyle\|F(s,\theta)-F(s^{\prime},\theta)\|$	$\displaystyle\leq$	$\displaystyle c_{2}\|A(s)-A(s^{\prime})\|$

where $F$ is defined by (4). Fix $\varepsilon>0$ and let $\tilde{\omega}$ be the path defined as $\tilde{\omega}_{u}=\omega_{k\varepsilon}$ for all $k\varepsilon\leq u<(k+1)\varepsilon.$ Then, by Gronwall’s lemma,

$$\|\Psi(t,\omega)(\theta)-\Psi(t,\tilde{\omega})(\theta)\|\leq c_{t}\int_{0}^{t}\|A(\omega(u))-A(\tilde{\omega}(u))\|du$$

(8)

where $c_{t}=e^{c_{1}t}c_{2}.$ Thus, by Jensen inequality,

$$\mathbb{E}_{s}(\|\Psi(t,\omega)\theta-\Psi(t,\tilde{\omega}))\theta)\|)^{2}\leq\mathbb{E}_{s}(\|\Psi(t,\omega)(\theta)-\Psi(t,\tilde{\omega})(\theta)\|^{2})$$

$$\leq c_{t}^{2}t\int_{0}^{t}\mathbb{E}_{s}(\|A(\omega_{u}))-A(\tilde{\omega}_{u})\|^{2})du$$

The choice of the norm being arbitrary we can assume that the norm on the right hand side of the preceding inequality is the Euclidean on ${\mathbb{R}}^{d^{2}}.$ Then, for all $k\varepsilon\leq u<(k+1)\varepsilon,$

$$\mathbb{E}_{s}(\|A(\omega_{u}))-A(\tilde{\omega}_{u})\|^{2})=\mathbb{E}_{s}\left(\mathbb{E}(\|A(\omega_{u}))-A(\tilde{\omega}_{u})\|^{2})|{\cal F}_{k\varepsilon})\right)$$

$$=\mathbb{E}_{s}(P_{u-k\varepsilon}(\|A\|^{2})(\omega_{k\varepsilon})-2\langle A(\omega_{k\varepsilon}),P_{u-k\varepsilon}(A)(\omega_{k\varepsilon})\rangle+\|A(\omega_{k\varepsilon})\|^{2})$$

$$\leq\sup_{0\leq h\leq\varepsilon}\|P_{h}(\|A\|^{2})-\|A\|^{2})\|_{\infty}+2\|A\|_{\infty}\|P_{h}A-A\|_{\infty}:=\delta(\varepsilon).$$

Observe that $\delta(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$ by strong continuity of $(P_{t})_{t\geq 0}.$ Combining the two last inequalities, we get

$$\mathbb{E}_{s}(\|\Psi(t,\omega)(\theta)-\Psi(t,\tilde{\omega})(\theta)\|^{2})\leq c_{t}^{2}t^{2}\delta(\varepsilon)$$

(9)

Let now $f\in C(S\times\Delta).$ Then, for every $\delta>0$ there exists $\alpha>0,$ such that

$$|f(s,\theta)-f(s,\theta^{\prime})|\leq\delta+2\|f\|{\bf 1}_{\|\theta-\theta^{\prime}\|\geq\alpha}$$

Thus

$$|Q_{t}f(s,\theta)-\mathbb{E}_{s}(f(\omega_{t},\Psi(t,\tilde{\omega})(\theta))|$$

$$\leq\mathbb{E}_{s}\left(|f(\omega_{t},\Psi(t,\omega)(\theta))-f(\omega_{t},\Psi(t,\tilde{\omega})(\theta))|\right)\leq\delta+2\frac{\|f\|}{\alpha^{2}}c_{t}^{2}t^{2}\delta(\varepsilon).$$

This shows that the left hand term goes to $0$ uniformly in $(s,\theta)\in S\times\Delta$ as $\varepsilon\rightarrow 0.$