Flows, coalescence and noise

Abstract

We are interested in stationary ``fluid'' random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.Comment: Published at http://dx.doi.org/10.1214/009117904000000207 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models

“…42,43 In the kinematic dynamo model of Kazantsev and Kraichnan 44,45 these statements have been proved rigorously to hold. [46][47][48][49][50] It has furthermore been proved that the Lagrangian trajectories in the Kazantsev-Kraichnan model remain stochastic as , → 0, a result that has been termed "spontaneous stochasticity." 51 The limiting probability distributions of trajectories are known to be very robust and universal for the case of an incompressible fluid velocity, with the same result being obtained for limits of a wide class of regularizations.…”

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

“…This phenomenon of "spontaneous stochasticity" was first noted for Lagrangian trajectories in the Kraichnan model of random advection [82,83]. It has since been rigorously proved in the Kraichnan model that the solutions for the Lagrangian trajectories correspond to a random process, with a fixed initial condition x 0 for the fluid particle and a fixed advecting velocity u [84,85]. These considerations carry over plausibly also to the equation (33) for the magnetic field-lines.…”

Turbulent diffusion of lines and circulations

“…A novel phenomenon has been discovered there called spontaneous stochasticity: Lagrangian particle trajectories for a non-Lipschitz advecting velocity are non-unique and split to form a random process in pathspace for a fixed velocity realization [23,24,25,26,27,28,29]. This phenomenon raises many fundamental questions, including whether material objects such as lines and surfaces can even exist in the limit of infinite Reynolds number.…”

“…Equation (9) can be solved iteratively to generate a representation Pu[C, t] = S * u (t, t0)P [C, t0] as a Wiener chaos expansion in white-noise e u; cf. [28,29].…”

“…This stochastic equation is solved by the method of LeJan and Raimond [28,29] (cf. footnote #1), writing it as a (backward) Ito integral and iterating to obtain Φ E (C, t) = S u (t, t ′ )Φ E (C, t ′ ), where the Markov operator semi-group S u (t, t ′ ) is defined by a Wiener chaos expansion.…”