Characteristic exponents for stochastic flows

Löwner evolution driven by a stochastic boundary point

“…Suppose again that the family φ t (z) is a family of disk automorphisms,p(z) = Rep(0) 1+z 1−z + Imp(0) i. Since the unit circle T is always mapped by φ t (z) bijectively onto itself, we can study the stochastic family of diffeomorphisms of the unit circle described by the equation The functionp(z) = 1+z 1−z corresponds to the noisy North-South flow (28) dΘ t = −2 sin Θ t dt − kdB t , studied in [7,6] (see also [21,25]).…”

Ergodic Theory of Random Transformations

“…Since II(J! (II)) = 1 it follows from (2.30) that p(p,of (II)) = 0: (11). x E 11(,,1) then in the same way as above we can apply argl:ments of the previous section to obtain that p x p-a.s. the limit (3.25) exists and it is non-random.…”

On certain almost Brownian filtrations

“…A proof of this remark is given by Gangbo [7]. 3 Another, more direct, proof is the following. We shall work with ]0, 1] instead of [0, 1]; by 'interval', we shall always mean an interval of the form ]a, b].…”

“…The drift term − d 2 X t dt compensates for the extrinsic curvature of the sphere, so X remains on S and is a Brownian motion on S. This SDE generates a stochastic flow of diffeomorphisms Φ st on S. A. Carverhill, M. Chappell and D. Elworthy have shown in[3] that Φ has only one characteristic exponent, namely −d/2.…”