Random Dynamical Systems V´ıtor Araujo´ Centro de Matematica´ da Universidade do Porto Rua do Campo Alegre, 687, 4169-007 Porto, Portugal E-mail: [email protected] 1 Introduction randomness be embedded within the model to begin with. We present the most basic classes of models in what The concept of random dynamical system is a com- follows, then define the general concept and present paratively recent development combining ideas and some developments and examples of applications. methods from the well developed areas of probabil- ity theory and dynamical systems. Let us consider a mathematical model of some physi- ¥ 2 Dynamics with noise ¡ ¡ k k ¢£¢£¢ cal process given by the iterates T0 T0 T0 ¤ k 1, of a smooth transformation T : M ¦ of a man- 0 To model random perturbations of a transformation ifold into itself. A realization of the process with § T0 we may consider a transition from the image T0 x ¨ initial condition x0 is modelled by the sequence § § k to some point according to a given probability law, © T0 x0 ¨£¨ k 1, the orbit of x0. obtaining a Markov Chain, or, if T0 depends on a pa- Due to our inaccurate knowledge of the particular rameter p, we may choose p at random at each iter- physical system or due to computational or theo- ation, which also can be seen as a Markov Chain but retical limitations (lack of sufficient computational whose transitions are strongly correlated. power, inefficient algorithms or insufficiently devel- oped mathematical or physical theory, for example), the mathematical models never correspond exactly to 2.1 Random noise the phenomenon they are meant to model. Moreover § ¢ ¦ when considering practical systems we cannot avoid Given T0 : M and a family p x ¨ : x M either external noise or measurement or inaccuracy of probability measures on M such that the sup- § § ¢ ¨ errors, so every realistic mathematical model should port of p x ¨ is close to T0 x , the random or- § ¨ allow for small errors along orbits not to disturb too bits are sequences xk k © 1 where each xk 1 is a ran- § ¢ much the long term behavior. To be able to cope dom variable with law p xk ¨ . This is a Markov with unavoidable uncertainty about the “correct” pa- Chain with state space M and transition probabil- § ¢ rameter values, observed initial states and even the ities p x ¨ x M . To extend the concept of in- specific mathematical formulation involved, we let variant measure of a transformation to this setting, 1 we say that a probability measure µ is stationary if M M, let us define § § § § § § ¨ ¨ µ A ¨ p A x dµ x for every measurable (Borel) § $ ¨ ¨£¨ m A B T x ¤ ε e 0 § § § subset A. This can be conveniently translated by say- p A x ¨ ¨ ¨£¨ m B T0 x ¤ ε ing that the skew-product measure µ p on M M given by where m denotes some choice of Riemannian volume § e ¢% ¨ § § form on M. Then p x is the normalized volume § ¨ ¨ ¤£££¤ ¤£££ d µ p x0 ¤ x1 xn restricted to the ε-neighborhood of T0 x ¨ . This de- § § § ¢£¢£¢ ¢£¢£¢ ¨ ¨ dµ x0 ¨ p dx1 x0 p dxn 1 xn fines a family of transition probabilities allowing the § points to “jump” from T0 x ¨ to any point in the ε- § ¦ is invariant by the shift map S : M M on the neighborhood of T0 x ¨ following a uniform distribu- space of orbits. Hence we may use the Ergodic The- tion law. orem and get that time averages of every continuous § © observable ϕ : M , i.e. writing x xk ¨ k 0 and 2.2 Random maps n 1 § § 1 ¨ ϕ˜ x ¨ lim ϕ x ¤£££&¤ k Alternatively we may choose maps T1 ¤ T2 Tk in- n ¥ n ∑ k 0 dependently at random near T0 according to a proba- § n 1 § § 1 § k bility law ν on the space T M ¨ of maps, whose sup- lim ϕ π0 S x ¨£¨£¨ n ¥ n ∑ port is close to T in some topology, and consider 0 k 0 § ¡ ¡ ¢£¢£¢ sequences xk Tk T1 x0 ¨ obtained through ran- ¥ exist for µ p -almost all sequences x, where π0 : dom iteration, k 1 ¤ x0 M. M M M is the natural projection on the first This is again a Markov Chain whose transition prob- coordinate. It is well known that stationary measures § abilities are given for any x M by ¢! always exist if the transition probabilities p x ¨ de- § § § )( pend continuously on x. '£ ¨ ¨ ¨ p A x ν T T M : T x A ¤ § A function ϕ : M is invariant if ϕ x ¨ § § so this model may be reduced to the first one. How- ¨ ϕ z ¨ p dz x for µ-almost every x. We then say ever in the random maps setting we may associate that µ is ergodic if every invariant function is con- to each random orbit a sequence of maps which are stant µ-almost everywhere. Using the Ergodic Theo- iterated, enabling us to use robust properties of the rem again, if µ is ergodic, then ϕ˜ " ϕdµ, µ-almost transformation T0 (i.e. properties which are known everywhere. to hold for T0 and for every nearby map T) to derive Stationary measures are the building blocks for more properties of the random orbits. sophisticated analysis involving e.g. asymptotic so- Under some regularity conditions on the map x * § journ times, Lyapunov exponents, decay of correla- p A x ¨ for every Borel subset A, it is possible to tions, entropy and/or dimensions, exit/entrance times represent random noise by random maps on suit- from/to subsets of M, to name just a few frequent ably chosen spaces of transformations. In fact the notions of dynamical and probabilistic/statistical na- transition probability measures obtained in the ran- ture. dom maps setting exhibit strong spatial correlation: § § ¢+ ¢% # ¨ Example 1 (Random jumps). Given ε 0 and T0 : p x ¨ is close to p y is x is near y. 2 If we have a parameterized family T : U M M case suggests the name additive random perturba- of maps we can specify the law ν by giving a proba- tions for random perturbations defined using families ¤£££ bility θ on U. Then to every sequence T1 ¤£££&¤ Tk of maps of this type. of maps of the given family we associate a sequence For the probability measure on U we may take ¤£££ ω1 ¤£££&¤ ωk of parameters in U since θe any probability measure supported in the ε- ¡ ¡ ¡ ¡ ¢£¢£¢ ¢£¢£¢ k Tk T1 T T T neighborhood of e and absolutely continuous with - - -., wk w1 w1 , wk # § § ¥ respect to the Riemannian metric on M, for any ε 0 ¨ ¨ for all k 1, where we write Tw x T ω ¤ x . In small enough. this setting the shift map S becomes a skew-product Example 4 (Local additive perturbations). If M transformation d and U0 is a bounded open subset of M § § § */' ( ¦ ¨ ¨ ¨ ¤ ¤ S : M U x ¤ ω Tw1 x σ ω strictly invariant under a diffeomorphism T0, i.e., § § closure T U ¨£¨65 U , then we can define an isomet- to which many of the standard methods of dynami- 0 0 0 ric random perturbation setting cal systems and ergodic theory can be applied, yield- § § 7 ing stronger results that can be interpreted in random ¨ V T0 U0 ¨ (so that closure V § terms. § closure T0 U0 ¨£¨5 U0); Example 2 (Parametrical noise). Let T : P M 7 d d G 89 the group of translations of ; M be a smooth map where P¤ M are finite dimen- sional Riemannian manifolds. We fix p P, de- 7 0 V a small enough neighborhood of 0 in G. note by m some choice of Riemannian volume form § § § # ¨ ¨ ¤ on P, set Tw x T w x and for every ε 0 write § § § § § ¨ 3 Then for v and x V we set T x x v, with 1 ¢ V v ¨£¨ ¨£¨ ¤ θ m B p ¤ ε m B p ε , the normalized e 0 0 the standard notation for vector addition, and clearly restriction of m to the ε-neighborhood of p0. Then § Tv is an isometry. For θe we may take any probability ¨ T together with θ defines a random perturba- w w P e measure on the ε-neighborhood of 0, supported in V tion of T , for every small enough ε # 0. p0 and absolutely continuous with respect to the volume Example 3 (Global additive perturbations). Let M d # in , for every small enough ε 0. be a homogeneous space, i.e., a compact connected Lie Group admitting an invariant Riemannian metric. Fixing a neighborhood U of the identity e M we 2.3 Random perturbations of flows § § § * ¨ ¨£¨ ¤ can define a map T : U M M ¤ u x Lu T0 x , § ¢ In the continuous time case the basic model to where Lu x ¨ u x is the left translation associated to u M. The invariance of the metric means that start with is an ordinary differential equation dXt § § § : 3 ¨ ¨ ¨ ¨ ¤ left (an also right) translations are isometries, hence f t ¤ Xt dt, where f : 0 ∞ X M and X M is § M ¨ the family of vector fields in . We embed random- fixing u U and taking any x ¤ v TM we get § 0 § 0 0 § § § 0 ness in the differential equation basically through ¢ ¢ ¨£¨ ¨ ¨ DTu x ¨ v DLu T0 x DT0 x v diffusion, the perturbation is given by white noise or 0 § 0 ¢ ¨ DT0 x v Brownian motion “added” to the ordinary solution. 21 d n In the particular case of M , the d-dimensional In this setting, assuming for simplicity that M , § § ¨43 torus, we have Tu x ¨ T0 x u and this simplest the random orbits are solutions of stochastic differ- 3 ential equations 2.4 The abstract framework § § ¢ ; ; ¨ 3 ¨ ¤ ¤ ¤ ¤ dXt f t ¤ Xt dt ε σ t Xt dWt 0 t T X0 Z The illustrative particular cases presented can all be written in skew-product form as follows.