Weak synchronization for isotropic flows
Abstract.
We study Brownian flows on manifolds for which the associated Markov process is strongly mixing with respect to an invariant probability measure and for which the distance process for each pair of trajectories is a diffusion . We provide a sufficient condition on the boundary behavior of at which guarantees that the statistical equilibrium of the flow is almost surely a singleton and its support is a weak point attractor. The condition is fulfilled in the case of negative top Lyapunov exponent, but it is also fulfilled in some cases when the top Lyapunov exponent is zero. Particular examples are isotropic Brownian flows on as well as isotropic OrnsteinUhlenbeck flows on .
Key words and phrases:
synchronization, random dynamical system, random attractor, Lyapunov exponent, stochastic differential equation, statistical equilibrium, isotropic Brownian flow, isotropic OrnsteinUhlenbeck flow2010 Mathematics Subject Classification:
37B25; 37G35, 37H151. Introduction
We study the asymptotic behavior of whitenoise random dynamical systems^{1}^{1}1For notation and some background on random dynamical systems and random attractors see Section 1.1 below. (RDS) on complete, seperable metric spaces . We present necessary and sufficient conditions for weak synchronization, which means that there is a weak point attractor for consisting of a single random point. In particular, in this case
in probability, for all .
In [6] (weak) synchronization for RDS generated by SDE driven by additive noise
(1.1) 
has been analyzed and general necessary and sufficient conditions for (weak) synchronization have been derived, based on a local asymptotic stability condition. More precisely, the existence of a nonempty open set and a sequence such that is contracted along the flow, that is,
was assumed in [6]. This condition was shown to be satisfied in the case that the top Lyapunov exponent is negative using the stable manifold theorem. In addition, in [6] weak synchronization has been shown for (1.1) in the gradient case, i.e. if for some among further assumptions.
These results are complemented by the present work which concentrates on the case of vanishing top Lyapunov exponent . In this case, local asymptotic stability as used in [6] is not satisfied in general. As a first main result we present necessary and sufficient conditions for weak synchronization. More precisely, in Theorem 2.4 we show that weak synchronization holds if and only if is strongly mixing and satisfies a global weak pointwise stability condition (cf. Definition 2.1). In a sense, this condition replaces the local asymptotic stability condition required in [6]. As it turns out, this global weak pointwise stability condition is particularly easy to check in the case of the distance being a diffusion. In this case it follows from the speed measure of being infinite.
This general result is then used in order to prove weak synchronization for isotropic Brownian flows on the sphere satisfying , as well as for isotropic OrnsteinUhlenbeck flows with . As detailed above, the cases of vanishing top Lyapunov exponent could not be treated by previous methods.
1.1. Preliminaries and notation
We start by fixing our setup which is the same as in [6]. Let be a complete separable metric space with Borel algebra and be an ergodic metric dynamical system, that is, is a probability space (not necessarily complete) and is a group of jointly measurable maps on with ergodic invariant measure .
Further, let be a perfect cocycle, that is, is measurable with and for all , , . We will assume continuity of the map for each and . The collection is then called a random dynamical system (in short: RDS), see [1] for a comprehensive treatment.
Since our main applications are RDS generated by SDE driven by Brownian motion, we will assume that the RDS is suitably adapted to a filtration and is of white noise type in the following sense: We will assume that there exists a family of sub algebras of such that whenever , for all and are independent for all . For each , denotes the smallest algebra containing all , and denotes the smallest algebra containing all , . Note that for each , the algebras and are independent. We will assume that is measurable for each . The collection (or just ) is then called a white noise (filtered) RDS.
If is a white noise RDS then we define the associated Markovian (Feller) semigroup by
for measurable and bounded. If there exists an invariant probability measure for , then for every sequence the weak limit, the socalled statistical equilibrium of ,
(1.2) 
exists a.s. and does not depend on the sequence , a.s. The measure can be chosen to be measurable and it satisfies, for each , a.s. and .
A Markovian semigroup with invariant measure is said to be strongly mixing if
for each continuous, bounded and all . Similarly, an RDS is said to be strongly mixing if the law of converges to for for all .
Definition 1.1.
Let be a whitenoise RDS. We say that weak synchronization occurs if there exists an measurable random variable such that

for all , almost surely, and

for every , we have
We say that synchronization occurs if (1) holds and (2) is replaced by

for every compact set we have
Note that (weak) synchronization means that there exists a weak (point) attractor which is a singleton (see [6] for further details).
2. Main result
2.1. RDS on metric spaces
Let be a complete, separable metric space and assume that is an valued white noise random dynamical system for which the associated Markov process has an ergodic invariant probability measure .
Definition 2.1.
The RDS is said to satisfy

weak local pointwise stability iff there is a set with and a sequence such that for all there is a such that for all :
(2.1) 
weak global pointwise stability iff in (1) can be chosen such that .
We first present a slight generalization of [6, Lemma 2.19 (ii)] (which in turn is based on the main result in [10]).
Lemma 2.2.
Assume that satisfies weak local pointwise stability. Then the statistical equilibrium is discrete, that is consists of finitely many atoms of the same mass a.s., i.e. there is an and measurable random variables such that
Proof.
Due to [6, Lemma 2.19 (i)], the statistical equilibrium is either discrete or diffuse, which means that does not have point masses a.s..
Hence, we only have to show that has a point mass with positive probability. Denote the diagonal in by . Then there exists a measurable function such that for and
By invariance of we get
(2.2) 
By assumption there is a sequence such that, for all ,
(2.3) 
We can and will assume that depends measurably upon (e.g. by defining as the of the left hand side of (2.3)). We define and observe that
for all and all . From (2.2) we obtain that
Using that is measurable, is measurable and that , are independent, we conclude that
Using this above, taking and using Fatou’s lemma yields
If has no point masses, then and thus
Assume that the right hand side is . Then, for a.a. and a.a. we have
which implies . Since this implies , a contradiction. Thus,
Since is arbitrary we obtain a contradiction. This concludes the proof. ∎
Lemma 2.3.
Assume that satisfies weak global pointwise stability. Then the statistical equilibrium is supported by a single point.
Proof.
By Lemma 2.2 the statistical equilibrium is discrete, with
(2.4) 
Let
Step 1: We show that for all there is a such that for every , we have
(2.5) 
By assumption, there is a set with such that (2.1) holds for all . Due to (2.4) we have for a.a. and all .
Since is a white noise RDS we have that
Thus,
Theorem 2.4.
Let be a white noise valued RDS which is strongly mixing with respect to an invariant probability measure and satisfies weak global pointwise stability. Then weak synchronization holds for .
Proof.
Lemma 2.5.
Let be a white noise valued RDS with invariant probability measure . Assume that the distance of each pair of points converges to in probability as . Then is strongly mixing.
Proof.
Denote the law of a random variable by . For and we have
Since in probability for all , by dominated convergence, we have . Again by dominated convergence we conclude
for . ∎
Corollary 2.6.
Let be a white noise valued RDS with invariant probability measure , such that for all
in probability. Then weak synchronization holds for .
Remark 2.7.
The following converse statement holds true. Assume that satisfies weak synchronization. Then is strongly mixing and for all
in probability. In particular, satisfies weak global pointwise stability.
Proof.
Let a.s. be the weak point attractor of . Then, is an invariant probability measure for . By Lemma 2.5, is strongly mixing. Moreover,
in probability. ∎
2.2. Case of distance being a diffusion
In this section we consider the special case in which the distance process of any pair of trajectories is a diffusion. The setup is the same as in the previous subsection. In addition, we assume that for each pair of initial points the distance process satisfies a scalar stochastic differential equation
(2.8) 
on for some , where is standard Brownian motion. We assume that and are continuous on , on , and . Recall the following definition of the scale function and the speed measure on (see e.g. [7] or [8]):
for and an arbitrary point . Assume that none of the boundary points of are accessible (which can be expressed in terms of and via Feller’s test for explosions, see [8, p.342ff]) and that the speed measure is finite away from 0, i.e. for some (and hence for all) .
Lemma 2.8.
Under the assumptions above we have the following equivalences

in probability

is infinite.
3. Examples
Let be a white noise RDS on a complete, dimensional smooth Riemannian manifold with respect to an ergodic metric dynamical system and let be the associated Markovian semigroup. Further assume that for some and all and that has an ergodic invariant measure such that
Following [3] for the case of being compact and [6] for the case this implies the existence of constants , the socalled Lyapunov spectrum, such that
for a.a. and all . We define the top Lyapunov exponent by .
3.1. Isotropic Brownian flows on the sphere
We study isotropic Brownian flows on for , where denotes the Euclidean norm. On , we define the metric , where denotes the standard inner product on . Note that the metric takes values in . The reason for excluding the case (i.e. ) is that in that case the boundary of the twopoint distance is accessible and therefore requires a slightly different treatment. Given smooth vector fields and on the sphere and standard Brownian motions , a stochastic flow on is defined by the solution of the Stratonovich equation
(3.1) 
where the vector field valued semimartingale is defined as
(3.2) 
The flow is completely determined by its characteristics and which are given by
In terms of the vector fields used in (3.2) one has
where denotes the Riemannian connection on compatible with the standard metric defined above on the sphere. The flow is called Brownian if the one point motion is a Brownian motion on and this corresponds to the requirement where is the LaplaceBeltrami operator on The definition of isotropy for a flow on a sphere relies on the fact that the sphere is a homogeneous space. In the present context, the space where is the orthogonal group. The group acts transitively on and the flow is called isotropic if for all
(3.3) 
In terms of the characteristics, the flow is isotropic if and only if for every
Here means the differential of the mapping at the point It was shown in [11] that the solution of (3.1) defines a flow of diffeomorphisms of into itself and satisfies the definition of an RDS.
Theorem 3.1.
Let be an isotropic Brownian flow on satisfying . Then weak synchronization holds.
Proof.
We recall from [12, Theorem 4.1] that is a diffusion and
where
with
and
where are the Gegenbauer polynomials. The sequences and are summable with nonnegative terms. Moreover, from the proof of [12, Theorem 4.1] we recall that, since ,
In [12, Theorem 3.1] the Lyapunov spectrum is shown to be given by
In particular, .
Moreover, if then a.s. and if then .
Let be the scale function and the speed measure associated to the diffusion From [12, proof of Theorem 4.1] we recall, close to ,
with , where is a positive constant and means the ratio of the two sides tends to as tends to On the other hand, close to ,
We note that iff which is seen by noting is the same as Thus, if
and the other equivalence is proven in a similar fashion. Hence, iff . Furthermore, iff .
Remark 3.2.
Note that synchronization can never occur for an IBF on since is a homeomorphism almost surely and is compact.
3.2. Isotropic OrnsteinUhlenbeck flows
Next, we consider isotropic OrnsteinUhlenbeck flows (IOUF) on . Consider a stochastic flow generated by the following Kunitatype SDE on
(3.4) 
where and is a (normalized) isotropic Brownian field, i.e.