Caustic Frequency in 2D Stochastic Flows Modeling Turbulence

Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such flows can collide and we study the collision (caustic) frequency. Caustics occur when the Jacobian of a flow vanishes. First, a system of nonlinear stochastic differential equations involving the Jacobian is derived and reduced to a smaller number of unknowns. Then, for special cases of the stochastic forcing, upper and lower bounds are found for the mean number of caustics as a function of Stokes number. The bounds yield an exact asymptotic for small Stokes numbers. The efficiency of the bounds is verified numerically. As auxiliary results we give rigorous proofs of the well known expressions for the caustic frequency and Lyapunov exponent in the one-dimensional model. Our findings may also be used for estimating the mean time when a 2D Riemann type partial differential equation with a stochastic forcing loses uniqueness of solutions.

On a Brownian motion conditioned to stay in an open set

UDC 519.21 Distribution of a Brownian motion conditioned to start from the boundary of an open set and to stay in for a finite period of time is studied. Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained. Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on