Random Dynamical Systems

Introduction

A random dynamical system is a dynamical system whose law of motion itself varies randomly with time: one is given a probability space $(\Omega,\mathcal F,\mathbb P)$ carrying an ergodic measure-preserving transformation $\theta$, called the noise or the driving system, and over each point $\omega$ of $\Omega$ a map $\varphi(\cdot,\omega)$ of the phase space, with the cocycle property that the map at time $t+s$ over $\omega$ is the composition of the map at time $s$ over $\omega$ followed by the map at time $t$ over the shifted noise $\theta_s\omega$. The pair $(\theta,\varphi)$ is then a genuine dynamical system — the skew product $(\omega,x)\mapsto(\theta\omega,\varphi(\omega)x)$ — on the product space, and the whole machinery of the preceding articles applies to it; what is new is that the invariant objects are not single sets or measures but measurable families $A(\omega)$, $\mu_\omega$ depending on the noise, and that the asymptotic quantities are almost-sure limits which are constant only by ergodicity of $\theta$. The central theorem is the multiplicative ergodic theorem of Oseledets, the random analogue of the Jordan decomposition: for a linear cocycle whose logarithm is integrable, the product of the matrices along the noise has almost-sure exponential growth rates and an equivariant splitting of the phase space, the Lyapunov exponents, and every stability, entropy and invariant-manifold statement of the smooth theory has a random version in which the constants become random variables and the uniformity is almost sure.

The article begins with the definition of a random dynamical system, the cocycle and skew-product formulations and the basic examples: the products of random matrices, the iterated function systems, the random maps of a compact space, and the stochastic flows generated by stochastic differential equations. It states the multiplicative ergodic theorem with the Oseledets filtration and the Lyapunov spectrum, together with the subadditive ergodic theorem of Kingman that is its engine and the Furstenberg–Kesten theorem that computes the top exponent as an integral over the noise, and it gives the Furstenberg theorem on the positivity of the top exponent for a non-degenerate product of matrices with the induced dynamics on the projective space. The random attractors are treated next, in the pullback sense of Crauel and Flandoli, with the existence theorem for dissipative systems and the random invariant measures and their disintegration $\mu_\omega$ over the noise; the random invariant manifolds of the hyperbolic theory follow, as the almost-sure version of the Hadamard–Perron theorem, and the entropy formula of Ledrappier and Young. The article closes with the stochastic differential equations and the stochastic flows they generate, the Furstenberg–Khasminskii formula for the top exponent of a linear stochastic flow, and the applications to the stability of the random products.

The measure theory, the probability spaces, the conditional expectations, the martingales, the Markov chains and the Brownian motion are those of Measure-Theoretic Probability, Independence and Conditional Expectation, Martingales, Markov Chains and Processes and Brownian Motion and Stochastic Calculus; the stochastic differential equations and their flows are those of Stochastic Differential Equations. The measure-preserving transformations, the ergodicity, the mixing, the entropy and the Birkhoff ergodic theorem are those of Ergodic Theory; the Lyapunov exponents, the Pesin theory, the entropy formula and the SRB measures of the deterministic systems are those of Hyperbolic Dynamics and Anosov Systems, and the flows, the linearisation and the Poincaré return maps are those of Smooth Dynamical Systems and Topological Dynamics. The random dynamical systems of arithmetic origin lie outside this article.

No physics is invoked.

Random Dynamical Systems and Cocycles

The Definition

Definition. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, let $\theta:\mathbb{R}\times\Omega\to\Omega$ (or $\theta:\mathbb{Z}\times\Omega\to\Omega$) be a measurable measure-preserving action, and let $X$ be a measurable space. A random dynamical system on $X$ over $\theta$ is a measurable map

$$ \varphi:\mathbb{R}\times\Omega\times X\to X, \qquad (t,\omega,x)\mapsto\varphi(t,\omega,x), $$

such that (i) $\varphi(0,\omega)=\mathrm{id}_X$ for all $\omega$ and (ii) the cocycle property holds:

$$ \varphi(t+s,\omega)=\varphi(t,\theta_s\omega)\circ\varphi(s,\omega) \qquad \text{for all } t,s \text{ and } \omega . $$

The skew product is the map $\Theta:\mathbb{R}\times\Omega\times X\to\Omega\times X$, $\Theta_t(\omega,x)=(\theta_t\omega,\varphi(t,\omega)x)$; it is a measurable action of $\mathbb{R}$ on $\Omega\times X$ and it carries the whole random system into a single deterministic system. A random map is the case $t=1$ of a discrete system, and the iteration over the noise is the composition $\varphi(n,\omega)=\varphi(\theta_{n-1}\omega)\circ\cdots\circ\varphi(\omega)$; when each $\varphi(\omega)$ is invertible the system is invertible and $\varphi(n,\omega)^{-1}$ is defined for negative $n$.

Definition. A random set is a measurable family $A=\{A(\omega)\}_{\omega \in\Omega}$ of subsets of $X$, in the sense that $\{(\omega,x):x \in A(\omega)\}$ is measurable in $\Omega\times X$; a random compact set has $A(\omega)$ compact for almost every $\omega$, and a random set is invariant if $\varphi(t,\omega)A(\omega)=A(\theta_t\omega)$ for all $t$ almost surely. A random measure is a measurable family $\mu=\{\mu_\omega\}$ of measures on $X$, and it is invariant if $\varphi(t,\omega)_*\mu_\omega=\mu_{\theta_t\omega}$ for all $t$.

Example. (i) Products of random matrices. Let $A:\Omega\to\mathrm{GL}(d,\mathbb{R})$ be measurable with $\log^+\|A(\cdot)\| \in L^1(\mathbb{P})$, and put $X=\mathbb{R}^d$, $\varphi(n,\omega)x=A(\theta_{n-1}\omega)\cdots A(\omega)x$. The cocycle property is immediate from the definition, and the system is the linear random dynamical system generated by the sequence of matrices.

(ii) Iterated function systems. Let $f_1,\dots,f_k$ be contractions of a complete metric space $X$ and let $\theta$ be the one-sided Bernoulli shift on $\Omega=\{1,\dots,k\}^{\mathbb{N}}$ with the product measure. Then $\varphi(n,\omega)=f_{\omega_0}\circ\cdots\circ f_{\omega_{n-1}}$ (in the appropriate order) is a random dynamical system, the iterated function system, whose attractor is the unique random compact set invariant under the system; the deterministic self-similar sets are the special case in which the maps are similarities and the measure is chosen so that the attractor is the unique invariant set.

(iii) Stochastic flows. Let $W$ be a Brownian motion and let $\theta$ be the Wiener shift on the space of paths; the solution of a stochastic differential equation generates a cocycle of diffeomorphisms, which is the principal example, treated in the last section.

Linear Cocycles and the Skew Product

Definition. A linear cocycle over $\theta$ is a measurable map $A:\mathbb{Z}\times\Omega\to\mathrm{GL}(d,\mathbb{K})$ with $A(m+n,\omega)=A(m,\theta_n\omega)A(n,\omega)$ and $A(1,\omega)=A(\omega)$; equivalently it is a cocycle acting on $X=\mathbb{K}^d$ by linear maps. A subadditive sequence over $\theta$ is a family $a_n:\Omega\to\mathbb{R}\cup\{-\infty\}$ with $a_{m+n}\le a_m+a_n\circ\theta_m$ for all $m,n$.

Theorem (Kingman's subadditive ergodic theorem). Let $a_n$ be a subadditive sequence over a measure-preserving $\theta$ with $a_1^+ \in L^1(\mathbb{P})$ and $\inf_n\frac1n\int a_n\,d\mathbb{P}>-\infty$. Then there is an integrable function $a$ with $a_n/n\to a$ almost surely and in $L^1$, and $a=\inf_n\frac1n\int a_n\,d\mathbb{P}=\lim_n\frac1n\int a_n\,d\mathbb{P}$.

Proof. The theorem is the subadditive analogue of the Birkhoff ergodic theorem, proved by reducing to the case $a_1\le0$ and applying the maximal ergodic inequality to the sequence $a_n$; the details are the standard proof, cited in the references.

The Multiplicative Ergodic Theorem

The Oseledets Theorem

Theorem (Oseledets multiplicative ergodic theorem). Let $\theta$ be an ergodic measure-preserving transformation of $(\Omega,\mathcal F,\mathbb{P})$, let $A:\Omega\to\mathrm{GL}(d,\mathbb{R})$ be measurable with

$$ \int\log^+\|A(\omega)\|\,d\mathbb{P}(\omega)<\infty, \qquad \int\log^+\|A(\omega)^{-1}\|\,d\mathbb{P}(\omega)<\infty , $$

and let $A(n,\omega)$ be the associated linear cocycle. Then there are a set $\Omega_0$ of full measure, an integer $r$ with $1\le r\le d$, and real numbers

$$ \lambda_1>\lambda_2>\cdots>\lambda_r , $$

the Lyapunov exponents of the cocycle, which do not depend on $\omega$ by ergodicity, such that for every $\omega \in\Omega_0$ there is a filtration

$$ \{0\}=V_0(\omega)\subset V_1(\omega)\subset\cdots\subset V_r(\omega)=\mathbb{R}^d $$

of subspaces with $A(\omega)V_i(\omega)=V_i(\theta\omega)$ and

$$ \lim_{n\to\pm\infty}\frac1n\log\|A(n,\omega)v\|=\lambda_i \qquad \text{for } v \in V_i(\omega)\setminus V_{i-1}(\omega). $$

The filtration determines measurable subspaces $E_i(\omega)$ with $V_i(\omega)=E_1(\omega)\oplus\cdots\oplus E_i(\omega)$ and $A(\omega)E_i(\omega)=E_i(\theta\omega)$, the Oseledets splitting, so that $\mathbb{R}^d=\bigoplus_iE_i(\omega)$, and $\lim_{n\to\pm\infty}\frac1n\log\|A(n,\omega)v\|=\lambda_i$ for $v \in E_i(\omega)\setminus\{0\}$, the limit being two-sided; each $E_i$ is the orthogonal complement of $V_{i-1}$ inside $V_i$ for the Lyapunov inner product $\langle u,v\rangle'_\omega=\sum_{n\in\mathbb{Z}}e^{-2\varepsilon|n|}\langle A(n,\omega)u,A(n,\omega)v\rangle$ with $\varepsilon>0$ small, with respect to which the splitting is orthogonal, and the Lyapunov product rather than the Euclidean one is essential: for a single measure-preserving matrix with non-orthogonal eigenvectors the Euclidean complement $V_i\cap V_{i-1}^\perp$ is not equivariant; the multiplicity $m_i=\dim E_i(\omega)$ is independent of $\omega$, and the sum of the exponents counted with multiplicity is the integral

$$ \sum_im_i\lambda_i=\int\log|\det A(\omega)|\,d\mathbb{P}(\omega). $$

Proof (sketch). The argument is an application of Kingman's theorem to the subadditive sequences $a_n(\omega)=\log\|A(n,\omega)\|$ and $\log\|\wedge^kA(n,\omega)\|$ for the exterior powers, which give the sums of the $k$ largest exponents; the limits $\beta_k$ are the sums of the $k$ largest exponents, the exponents are the successive differences, and the subspaces are obtained by intersecting the images of the sets $\{v:\limsup_n\frac1n\log\|A(n,\omega)v\|\le\beta_k\}$. The determinantal identity is the statement that the exterior power of the top degree computes the determinant.

Theorem (Furstenberg–Kesten). Let $A_0,A_1,\dots$ be independent and identically distributed random matrices in $\mathrm{GL}(d,\mathbb{R})$ with $\mathbb{E}\log^+\|A_0\|<\infty$. Then the top Lyapunov exponent of the product is

$$ \lambda_1=\lim_{n\to\infty}\frac1n\log\|A_{n-1}\cdots A_0\|=\lim_{n\to\infty}\frac1n\mathbb{E}\log\|A_{n-1}\cdots A_0\| $$

almost surely and in $L^1$; the limit on the right is the Furstenberg–Kesten integral and it equals the top exponent of the cocycle over the Bernoulli shift generated by the distribution.

Example (commuting matrices). Suppose the matrices of the cocycle commute almost surely with a fixed diagonalisation, so that there are measurable functions $a_1,\dots,a_d$ with $A(\omega)=\operatorname{diag}(a_1(\omega),\dots,a_d(\omega))$. Then the product is the diagonal matrix with entries $\prod_{k

$$ \lambda_i=\int\log|a_i(\omega)|\,d\mathbb{P}(\omega), $$

and the Oseledets splitting is the coordinate splitting whenever the integrals are distinct. In the special case in which $A$ takes the value $\operatorname{diag}(2,1)$ with probability $\frac12$ and $\operatorname{diag}(1,2)$ with probability $\frac12$, the exponents are both $\int\log|a_i|=\frac12\log2=0.346573\ldots$, since the diagonal entries are $2^{N}$ with $N$ a binomial count of the choices, and the ergodic average $\frac1n\log|a_i|=\frac{N}{n}\log2$ fluctuates about $\frac{\log2}{2}$ with the standard deviation $\frac{\log2}{2\sqrt n}$, of order $3\cdot10^{-4}$ for $n=10^6$. In the special case $A=\operatorname{diag}(2,\frac12)$ or $\operatorname{diag}(\frac12,2)$ with equal probabilities, the exponents are both $\int\log|a_i|=\frac12(\log2-\log2)=0$; the product is nevertheless unbounded, since $|a_i|=2^{N_1-N_2}$ with $N_1-N_2$ a simple random walk, so its norm grows like $2^{c\sqrt n}$ subexponentially, and here the fluctuation of $\frac1n\log|a_i|$ about $0$ has the standard deviation $\frac{\log2}{\sqrt n}$, of order $7\cdot10^{-4}$ for $n=10^6$.

The Furstenberg Theory on the Projective Space

Theorem (Furstenberg). Let $\mu$ be a probability measure on $\mathrm{GL}(d,\mathbb{R})$ with $\int\log^+\|A\|\,d\mu<\infty$, and let $\nu$ be a $\mu$-invariant probability measure on the projective space $\mathbb{P}^{d-1}$, that is, $\nu=\int(A_*\nu)\,d\mu(A)$; such a measure exists by the compactness of $\mathbb{P}^{d-1}$ and the continuity of the action. Then the top exponent of the associated random product is

$$ \lambda_1=\int_{\mathbb{P}^{d-1}}\int_{\mathrm{GL}(d,\mathbb{R})}\log\frac{\|Av\|}{\|v\|}\,d\mu(A)\,d\nu([v]), $$

(the Furstenberg formula), and if $\mu$ is strongly irreducible and contracting — the closed semigroup generated by the support of $\mu$ contains no finite union of proper subspaces and contains a rank-one element — then the invariant measure $\nu$ is unique, the top exponent is simple, and $\lambda_1>0$ unless the semigroup generated by the support of $\mu$ is relatively compact modulo the scalars, that is, unless its image in the projective linear group has compact closure.

Proof (sketch). The formula is the Birkhoff ergodic theorem applied to the skew product on $\Omega\times\mathbb{P}^{d-1}$ and the function $\log(\|Av\|/\|v\|)$, together with the additivity of the telescoping products and the identification of the top exponent with the average growth of a nonzero vector. The uniqueness of $\nu$ and the positivity of the exponent are the content of Furstenberg's theorem: under the irreducibility and contraction hypotheses $\lambda_1>0$ unless the semigroup generated by the support of $\mu$ is contained, modulo scalars, in a compact subgroup of $\mathrm{GL}(d,\mathbb{R})$. The proof analyses the action of the semigroup on the space of probabilities on $\mathbb{P}^{d-1}$ and uses the contraction of the projective distance by the rank-one element.

Random Attractors and Invariant Measures

Pullback Attractors

Definition. Let $\varphi$ be a random dynamical system on a metric space $X$ and let $\mathcal B$ be a family of bounded subsets of $X$ containing the bounded sets. A random compact set $A$ is a pullback attractor for $\mathcal B$ if it is invariant and if for every $B \in\mathcal B$ and almost every $\omega$,

$$ \operatorname{dist}(\varphi(n,\theta_{-n}\omega)B,A(\omega))\to0 \qquad (n\to+\infty), $$

$\operatorname{dist}$ being the Hausdorff semidistance. The convergence is pullback because the orbit is started at the time $-n$ so that it arrives at the fixed time $0$: it is the natural notion for a nonautonomous system, in which the initial condition cannot be prescribed at time $0$.

Theorem (existence of the pullback attractor; Crauel–Flandoli). Suppose the random dynamical system on the complete separable metric space $X$ has a random compact set $K$ that is absorbing in the pullback sense: for every bounded $B$ there is a random time $\tau_B$ with $\varphi(n,\theta_{-n}\omega)B \subseteq K(\omega)$ for all $n \ge\tau_B(\omega)$ and almost every $\omega$. Then the system has a unique pullback attractor $A$, and

$$ A(\omega)=\bigcap_{n\ge1}\overline{\bigcup_{m\ge n}\varphi(m,\theta_{-m}\omega)K(\theta_{-m}\omega)}, $$

the set of the values at $\omega$ of all complete bounded orbits, that is, of the sequences $x(\omega)$ with $\varphi(n,\omega)x(\omega)=x(\theta_n\omega)$ for all $n$ and bounded values.

Proof. The sets $\overline{\bigcup_{m\ge n}\varphi(m,\theta_{-m}\omega)K(\theta_{-m}\omega)}$ form a decreasing sequence of compact sets, so their intersection is nonempty and compact; the pullback convergence follows from the absorbing property, the invariance from the shift in the definition, and the uniqueness from the invariance and the convergence of every bounded set.

Example (iterated function systems). For an iterated function system of contractions of a complete metric space, the pullback attractor is the random compact set $\{x(\omega)\}$ given by the unique backward composition, and it is a single point for each $\omega$ when the contractions have contracting ratios bounded by $\rho<1$; the distribution of that point is the invariant measure of the system, and the support of the invariant measure of the Markov chain is the deterministic attractor of the system of maps. The Hausdorff dimension of the random attractor and the multifractal analysis of the invariant measure belong to the fractal geometry of the iterated function systems and are treated in the literature cited below.

Invariant Measures and Their Disintegration

Theorem (invariant measures). Let $\varphi$ be a random dynamical system on a Polish space $X$ with a compact pullback attractor. Then there is an invariant random measure $\mu=\{\mu_\omega\}$ with $\mu_\omega$ a probability measure on $X$ for almost every $\omega$, which is the disintegration of a $\Theta$-invariant probability measure $m$ on $\Omega\times X$ with marginal $\mathbb{P}$ on $\Omega$,

$$ m(d\omega,dx)=\mu_\omega(dx)\,\mathbb{P}(d\omega), \qquad \varphi(t,\omega)_*\mu_\omega=\mu_{\theta_t\omega}; $$

every invariant measure arises in this way, the set of invariant measures is convex and compact in the weak topology, its extreme points are the ergodic ones, and every invariant measure is ergodic with respect to the skew product if and only if it is indecomposable. If the system has a compact pullback attractor, every invariant measure is supported on it. The ergodic decomposition of an invariant measure is the disintegration over the ergodic measures, and the ergodicity of the skew product is equivalent to the almost-sure ergodicity of the random system with respect to $\mu_\omega$.

Proof (sketch). One applies the Krylov–Bogoliubov procedure to the skew product: the averages of the iterates of a measure on $\Omega\times X$ have weak limit points by the compactness, and every limit point is invariant; the disintegrations exist by the standard theorem on the disintegration of a measure on a product, and the equivariance of the disintegrations is the invariance of $m$. The ergodic decomposition is the Choquet decomposition of the compact convex set of invariant measures.

Random Invariant Manifolds and Entropy

The Random Stable Manifold Theorem

Theorem (random Hadamard–Perron; Ruelle, Pesin). Let $\varphi$ be a $C^1$ invertible random dynamical system on a smooth manifold $X$ over an ergodic $\theta$, and let $\mu$ be an invariant measure. Suppose that at $\mu$-almost every point the Lyapunov exponents of the cocycle split into a negative and a positive part separated by the gap $[\alpha,\beta]$ with $\alpha<0<\beta$; that is, on a measurable equivariant splitting $T_xX=E^s(\omega,x)\oplus E^u(\omega,x)$ the growth rates are bounded by $\alpha$ respectively $\beta$. Then for $\mu$-almost every $(\omega,x)$ there are $C^1$ random stable and unstable manifolds $W^s(\omega,x)$, $W^u(\omega,x)$ tangent to $E^s$ and $E^u$, invariant in the sense that $\varphi(t,\omega)W^s(\omega,x)=W^s(\theta_t\omega,\varphi(t,\omega)x)$, and characterised as the sets of points whose orbits converge to that of $x$ in the corresponding time direction; their sizes vary measurably with $(\omega,x)$ and are positive $\mu$-almost surely.

Proof (sketch). One applies the Hadamard–Perron graph transform, as in the deterministic stable manifold theorem of Hyperbolic Dynamics and Anosov Systems, to the cocycle restricted to the hyperbolic block; the uniformity that the deterministic proof obtains from the compactness of $\Lambda$ is replaced by the almost-sure uniformity of the exponents given by the multiplicative ergodic theorem, and the measurability of the manifolds is obtained by a measurable selection of the graph transforms.

The Entropy Formula

Theorem (Ledrappier–Young). Let $\varphi$ be a $C^{1+\alpha}$ random dynamical system with an invariant measure $\mu$ and Lyapunov exponents $\lambda_i$, and let $h_\mu$ be the measure-theoretic entropy of the skew product. Then Ruelle's inequality holds,

$$ h_\mu\le\int\sum_i\lambda_i^+\,d\mu , $$

and equality holds if and only if the measure has absolutely continuous conditional measures on the unstable manifolds, the u-Gibbs property; in that case the invariant measure is the random SRB measure and the entropy is the sum of the positive exponents integrated over the noise. The deterministic statement is Pesin's formula of Hyperbolic Dynamics and Anosov Systems, and the random statement is its almost-sure version.

Stochastic Flows and Differential Equations

Stochastic Flows as Random Dynamical Systems

Definition. Let $W$ be a $k$-dimensional Brownian motion on a filtered probability space and let $F,\sigma_1,\dots,\sigma_k$ be smooth vector fields on a manifold $M$. The stochastic differential equation

$$ dX_t=F(X_t)\,dt+\sum_{j=1}^k\sigma_j(X_t)\circ dW^j_t $$

in the Stratonovich sense, with the initial condition $X_0=x$, generates a stochastic flow $\varphi(t,\omega,x)$: a family of diffeomorphisms with the cocycle property over the Wiener shift $\theta_t\omega(\cdot)=\omega(t+\cdot)-\omega(t)$ on the space of paths, which preserves the Wiener measure and is ergodic. The stochastic flow is thus a random dynamical system, invertible and $C^\infty$ in $x$ when the vector fields are; the differential equation, the Stratonovich integral and the smoothness of the flow are treated in Stochastic Differential Equations and Brownian Motion and Stochastic Calculus.

Theorem (Khasminskii; the linear case). Let $X_t$ solve the linear stochastic differential equation, written here in the Itô sense,

$$ dX_t=AX_t\,dt+\sum_{j=1}^kB_jX_t\,dW^j_t , $$

with $A,B_j \in\mathrm{Mat}(d,\mathbb{R})$. Then the flow is the linear cocycle $X_t(\omega)=\Phi(t,\omega)x$ and the multiplicative ergodic theorem applies with the integrability automatic by the Gaussian tails; the Lyapunov exponents $\lambda_1>\cdots>\lambda_r$ are the almost-sure exponential rates of the flow, they are nonrandom, and the top exponent is given by the Furstenberg–Khasminskii formula,

$$ \lambda_1=\int_{\mathbb{P}^{d-1}}Q(x)\,d\nu(x), \qquad Q(x)=\langle Ax,x\rangle+\frac12\sum_{j=1}^k\bigl(\langle B_jx,B_jx\rangle-\langle B_jx,x\rangle^2\bigr), $$

where $\nu$ is the invariant measure of the induced projective flow, $Q$ is the drift of $\log\|\Phi_tx\|$ at $t=0$ computed by the Itô formula, and the formula is the specialisation of the Furstenberg formula to the diffusive case.

Proof (sketch). The flow is linear by the linearity of the equation; the integrability of $\log^+\|\Phi(1,\omega)\|$ follows from the Gaussian moment bounds of the solution; the projective flow $\pi(X_t)$ is a diffusion on $\mathbb{P}^{d-1}$ whose generator is computed by the Itô formula applied to $u=X/\|X\|$, and the Furstenberg formula is the ergodic average of $\log(\|\Phi(1,\omega)x\|/\|x\|)$ against the invariant measure of that diffusion; the computation of the drift gives $Q(x)$ above, the term $-\frac12\sum\langle B_jx,x\rangle^2$ arising from the Itô correction in the change to the unit vector.

Corollary (stability and the random spectrum). The trivial solution of the linear stochastic equation is almost surely exponentially stable if and only if $\lambda_1<0$, the decay being bounded by $e^{(\lambda_1+\epsilon)t}$ for large $t$ when $\epsilon>0$ is small; the exponents control the random version of the Lyapunov stability of Smooth Dynamical Systems, in which the deterministic Lyapunov function is replaced by the linearisation along the random orbit and the exponential rates are those of the multiplicative ergodic theorem. The random invariant manifolds of the previous section then give the nonlinear stability near a random equilibrium.

Summary

A random dynamical system is a cocycle $\varphi$ over a measure-preserving noise $(\Omega,\mathcal F,\mathbb{P},\theta)$ with $\varphi(t+s,\omega)=\varphi(t,\theta_s\omega)\circ\varphi(s,\omega)$; the skew product $\Theta_t(\omega,x)=(\theta_t\omega,\varphi(t,\omega)x)$ turns it into a deterministic system, and the random sets, random compact sets and random measures are the invariant objects, with the pullback convergence $\varphi(n,\theta_{-n}\omega)B\to A(\omega)$ as the appropriate asymptotic notion. Products of random matrices, iterated function systems and stochastic flows are the basic examples; for a linear cocycle with $\log^+\|A^{\pm1}\|\in L^1$ the Oseledets multiplicative ergodic theorem produces the Lyapunov exponents $\lambda_1>\cdots>\lambda_r$, the equivariant filtration $V_i$ and the splitting $E_i$ with $\frac1n\log\|A(n,\omega)v\|\to\lambda_i$ on $E_i$, and $\sum_im_i\lambda_i=\int\log|\det A|\,d\mathbb{P}$; the subadditive ergodic theorem of Kingman is the engine, and the Furstenberg–Kesten theorem computes the top exponent of an i.i.d. product as the almost-sure and $L^1$ limit $\lim_n\frac1n\log\|A_{n-1}\cdots A_0\|$, while the Furstenberg formula computes it as the integral of $\log(\|Av\|/\|v\|)$ against the invariant measure on the projective space, with positivity under strong irreducibility and contraction. For commuting matrices the exponents are the integrals $\int\log|a_i|$, and the two diagonal examples $\operatorname{diag}(2,1)/\operatorname{diag}(1,2)$ and $\operatorname{diag}(2,\frac12)/\operatorname{diag}(\frac12,2)$ have exponents $\frac12\log2\approx0.3466$ and $0$.

A pullback attractor is a random compact set attracting every bounded set in the pullback sense; it exists under a pullback absorbing condition and consists of the bounded complete orbits, by the theorem of Crauel and Flandoli. The invariant random measures are the disintegrations $m(d\omega,dx)=\mu_\omega(dx)\mathbb{P}(d\omega)$ of the $\Theta$-invariant measures, equivariant under the flow, supported on the attractor, with the ergodic decomposition and the Choquet theory; and the random stable and unstable manifolds exist almost surely whenever the Lyapunov spectrum has a gap around zero, by the random Hadamard–Perron theorem. The entropy satisfies Ruelle's inequality $h_\mu\le\int\sum\lambda_i^+d\mu$ with equality exactly for the u-Gibbs measures, by the theorem of Ledrappier and Young, which is the random form of Pesin's formula. A stochastic flow, generated by a Stratonovich stochastic differential equation, is a random dynamical system over the Wiener shift, and for the linear equation $dX=AX\,dt+\sum_jB_jX\,dW^j$ in the Itô sense the exponents are given by the Furstenberg–Khasminskii formula, with $\lambda_1<0$ as the criterion for almost-sure exponential stability.

Summary of Notation

Symbol	Meaning
$(\Omega,\mathcal F,\mathbb{P},\theta)$	noise: measure-preserving driving system
$\varphi(t,\omega,x)$	random dynamical system (cocycle)
$\Theta_t$	skew product $(\omega,x)\mapsto(\theta_t\omega,\varphi(t,\omega)x)$
$A(\omega)$, $A(n,\omega)$	linear cocycle and its $n$-fold product
$\lambda_1>\cdots>\lambda_r$, $m_i$	Lyapunov exponents and their multiplicities
$V_i(\omega)$, $E_i(\omega)$	Oseledets filtration and splitting
$\langle\cdot,\cdot\rangle'_\omega$	Lyapunov inner product making the splitting orthogonal
$A(\omega)$, $\mu_\omega$	random (pullback) attractor, random invariant measure
$m$	invariant measure on $\Omega\times X$
$h_\mu$	entropy of the skew product
$\mathbb{P}^{d-1}$, $\nu$	projective space and its invariant measure
$W^s(\omega,x)$, $W^u(\omega,x)$	random stable and unstable manifolds
$\Phi(t,\omega)$	fundamental matrix of a linear stochastic flow
$\theta_t$	Wiener shift