%0 Journal Article %1 MR1284989 %A Arnold, Ludwig %A Gundlach, Volker Matthias %A Demetrius, Lloyd %D 1994 %J Ann. Appl. Probab. %K Perron_Frobenius mean_growth_rate random_matrices %N 3 %P 859--901 %T Evolutionary formalism for products of positive random matrices %U http://www.jstor.org/stable/pdfplus/2245067.pdf %V 4 %X Let $(Ømega,F,P)$ be a probability space, and let $þeta$ be an invertible map of $Ømega$ onto itself such that $ømega$ and $ømega^-1$ are $F$-measurable, $P$ is $þeta$-invariant, and $þeta$ is $P$-ergodic. Furthermore, let $ømegaA(ømega)$ be an $F$-measurable map from $Ømega$ into $M_+$, the space of $dd$ matrices with positive entries. Write $A=(a_ij)$ and $m=\min_ija_ij$, $M=\max_ija_ij$. One of the main results is the following random Perron-Frobenius theorem: If $łog^+(1/m)$ and $łog^+M$ are in $L^1(P)$, then (i) there exist a unique positive random unit vector $u$ and a positive random scalar $q$ with $logqL^1(P)$, such that $A(ømega)u(ømega)=q(ømega)u(þetaømega)$ $P$-a.e. Moreover, there is a unique invariant splitting $R^d=W(ømega)øplusR u(ømega)$, i.e., $A(ømega)W(ømega)W(þetaømega)$ $P$-a.e. (ii) If $xW(ømega)$, then $łim_n\toınfty(1/n)log\|A(þeta^n-1ømega)A(ømega)x\|=łambda=Eq$, where $łambda$ is the top Lyapunov exponent. If $xW(ømega)$, then $łimsup_n\toınfty(1/n)log\|A(þeta^n-1ømega)A(ømega)x\|<łambda$. A formalism to study directionality principles in evolution described by a positive matrix cycle such as $A(þeta^n-1ømega)A(ømega)$ above is presented. The Perron-Frobenius theorem is used to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. A thermodynamic formalism for random dynamical systems is developed, and the analyticity of the top Lyapunov exponent is proved under mild assumptions. A fluctuation theory for $A(þeta^n-1ømega)A(ømega)$ is worked out, by which directionality of mutation and selection in evolutionary dynamics is proved, using the notion of entropy. The paper considers only linear models of competitive interaction between ancestral and mutant types in an asexual population.

@article{MR1284989, abstract = {Let $(\Omega,{\bf F},{\bf P})$ be a probability space, and let $\theta$ be an invertible map of $\Omega$ onto itself such that $\omega$ and $\omega^{-1}$ are ${\bf F}$-measurable, ${\bf P}$ is $\theta$-invariant, and $\theta$ is ${\bf P}$-ergodic. Furthermore, let $\omega\mapsto A(\omega)$ be an $F$-measurable map from $\Omega$ into $M_+$, the space of $d\times d$ matrices with positive entries. Write $A=(a_{ij})$ and $m=\min_{ij}a_{ij}$, $M=\max_{ij}a_{ij}$. One of the main results is the following random Perron-Frobenius theorem: If $\log^+(1/m)$ and $\log^+M$ are in $L^1({\bf P})$, then (i) there exist a unique positive random unit vector $u$ and a positive random scalar $q$ with $logq\in L^1(P)$, such that $A(\omega)u(\omega)=q(\omega)u(\theta\omega)$ ${\bf P}$-a.e. Moreover, there is a unique invariant splitting $R^d=W(\omega)\oplusR u(\omega)$, i.e., $A(\omega)W(\omega)\subseteq W(\theta\omega)$ ${\bf P}$-a.e. (ii) If $x\notin W(\omega)$, then $\lim_{n\to\infty}(1/n)log\|A(\theta^{n-1}\omega)\cdots A(\omega)x\|=\lambda=E\log q$, where $\lambda$ is the top Lyapunov exponent. If $x\in W(\omega)$, then $\limsup_{n\to\infty}(1/n)log\|A(\theta^{n-1}\omega)\cdots A(\omega)x\|<\lambda$. A formalism to study directionality principles in evolution described by a positive matrix cycle such as $A(\theta^{n-1}\omega)\cdots A(\omega)$ above is presented. The Perron-Frobenius theorem is used to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. A thermodynamic formalism for random dynamical systems is developed, and the analyticity of the top Lyapunov exponent is proved under mild assumptions. A fluctuation theory for $A(\theta^{n-1}\omega)\cdots A(\omega)$ is worked out, by which directionality of mutation and selection in evolutionary dynamics is proved, using the notion of entropy. The paper considers only linear models of competitive interaction between ancestral and mutant types in an asexual population. }, added-at = {2009-08-16T22:31:36.000+0200}, author = {Arnold, Ludwig and Gundlach, Volker Matthias and Demetrius, Lloyd}, biburl = {https://www.bibsonomy.org/bibtex/2ba4181f122bc559fcf630cc2bdd872e6/peter.ralph}, description = {Evolutionary formalism for products of positive random matrices}, fjournal = {The Annals of Applied Probability}, interhash = {f2ec97b7ff63713871888b5bbc59b39a}, intrahash = {ba4181f122bc559fcf630cc2bdd872e6}, issn = {1050-5164}, journal = {Ann. Appl. Probab.}, keywords = {Perron_Frobenius mean_growth_rate random_matrices}, mrclass = {28D99 (28D20 60J10 90A14 92D25)}, mrnumber = {MR1284989 (95h:28028)}, mrreviewer = {J{\'o}zsef Sz{\H{u}}cs}, number = 3, pages = {859--901}, timestamp = {2009-08-16T22:31:36.000+0200}, title = {Evolutionary formalism for products of positive random matrices}, url = {http://www.jstor.org/stable/pdfplus/2245067.pdf}, volume = 4, year = 1994 }