## The Annals of Probability

### Isotropic Stochastic Flows

#### Abstract

We consider isotropic stochastic flows in a Euclidean space of $d$ dimensions, $d \geq 2$. The tendency of two-point distances and of tangent vectors to shrink or expand is related to the dimension and the proportion of the flow that is solenoidal or potential. Tangent vectors from the same point tend to become aligned in the same or opposite directions. The purely potential flows are characterized by an analogue of the curl-free property. Liapounov exponents are treated briefly. The rate of increase or decrease of the length of an arc of small diameter is related to the shape of the arc. In the case $d = 2$ a sufficient condition is given under which the length of a short arc has a high probability of approaching 0.

#### Article information

Source
Ann. Probab., Volume 14, Number 4 (1986), 1155-1179.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176992360

Digital Object Identifier
doi:10.1214/aop/1176992360

Mathematical Reviews number (MathSciNet)
MR866340

Zentralblatt MATH identifier
0606.60014

JSTOR