Stochastics and Dynamics

Let ϕ_t be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕ_t, and the stationary points of an associated vector field. As examples, SL(d,R)-flows and SO(1,d)-flows on SO(d) and S^{d - 1} are discussed in details.

Much effort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic differential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coefficients. Most extensively we study these problems for second order conservative systems with small random and periodic excitations. With respect to relations between the intrinsic period of the system and the period of perturbations we consider the incommensurable and commensurable cases. In the first case we obtain an asymptotic expansion of the moment Lyapunov exponent. In the second case we obtain a finite expansion except in situations of resonance. As an application we consider the Hill and Mathieu equations with random excitations.

We give algorithms for the asymptotic expansions of the almost sure and moment Lyapunov exponents associated with the two-dimensional stochastic differential equation obtained as a small perturbation of the deterministic rotation with rate ω. The matrices in the perturbation terms are all assumed to be periodic functions of time with period l. The form of the algorithms depends on whether or not the periods 2π/ω and l of the unperturbed system and the perturbation coefficients are commensurable (i.e. the ratio of the periods is rational). In the commensurable case certain resonances may cause jumps in the Lyapunov exponents. We give an example of a stochastically perturbed Hill's oscillator which is almost surely stable when 2ω is not an integer, but is almost surely unstable at resonant frequencies ω = m/2. This work extends recent results of Imkeller and Milstein.

We consider the two-step bifurcation scenario which has been studied by L. Arnold and his co-workers. We formulate a "continuous case" and a "measurable case" of the scenario, and present results and conjectures regarding sufficient conditions that it take place.

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φ_t = ξ_t°Ψ_t° Θ_t, where ξ_t and Ψ_t have the same properties of Liao's decomposition and (ξ_t°Ψ_t) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξ_t via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.

We consider the rotation of tangent vectors and the mutual rotation of two particles undergoing motion in a random two-dimensional velocity field. We look at random fields whose laws are invariant under translations and rotations, but not reflections. These polarized fields are intermediate between homogeneous and isotropic fields, and may possess a preferred sense of rotation. The covariance of such a field must have a certain form which we describe. In a Brownian flow based on a polarized random field, we show that tangent vectors can rotate at a constant asymptotic rate, and that under certain conditions, two particles will rotate about each other at the same asymptotic rate. For illustration we present simulations of a polarized Gaussian field and of particles moving in a polarized Brownian flow.