# Renyi's conditional probability fields and turbulence

I've come to the conclusion that what is universal, in the statistics of high Reynolds number turbulence of viscous incompressible fluids, could be modelled exactly only with Alfred Renyi's concept of a "conditional probability field" defined from an unbounded measure (on a set of velocity or vorticity fields on $$\mathbb R^3$$). While the (physical) reasoning towards this view is by no means far-fetched, I wonder if people like Kolmogorov, Onsager, von Weizsäcker, Obukhov etc, who made the first breakthrough in turbulence theory in the years 1941--1962, were aware of it.

On the other hand, Renyi's motivation for his construction was clearly from physics. And Kolmogorov himself seems to have agreed that it could be necessary in some cases:

So my question is: was Renyi's theory of conditional probability fields applied to turbulence ? Either back then in the 1950s, or recently?

EDIT: The application I have in mind is to the infinite integral scale limit: $$\log\varepsilon(r)$$, the $$\log$$ of the energy transfer rate at scale $$r$$, behaves as a Gaussian with variance $$\mu\log(R/r)$$, which becomes uniformly distributed on any fixed interval as $$R\to\infty$$. A proper description of what is universal in this limit is then with a group-invariant conditional probability field on the set of incompressible velocity fields $$\mathbb u:\mathbb R^3\to\mathbb R^3$$ satisfying $$\mathbb u(0)=0$$, for the group $$(0,\infty)$$ acting through $$S_\lambda\mathbb u(x)=\lambda\mathbb u(\lambda x)$$. The corresponding unbounded measure should be invariant (differentially) for the dynamics (local Navier-Stokes), i.e. an eigenvector of some evolution operator.

Weren't such simple ideas discussed then?

## 1 Answer

An early contribution along these lines is by Mandelbrot, Sporadic random functions and conditional spectral analysis: self-similar examples and limits (1967):

There is, however, a simple way of generalizing the concept of a stationary random function, so as to accommodate the sporadically varying series $X(t)$ [encountered in turbulence]. It suffices to amend in two ways the classical Kolmogorov's probability space triplet $(\Omega,{\cal A}, \mu)$: (a) the measure $\mu$ is assumed unbounded though sigma-finite; and (b) a family ${\cal B}$ of conditioning events $B$ is added, where $0 < \mu(B) < \infty$. It is further agreed that, henceforth, the only well-posed questions will be those relative to conditional probabilities $\text{P}r \{A| B\}$ where $B \in {\cal B}$.

Mandelbrot then goes on to acknowledge Renyi's general axiomatic theory of conditional probability, but argues that "we shall not strive for generality for its own sake, nor for irreducible axiomatics. Our sole object is to select new mathematical objects adapted to the needs of physics."

• That's a start, thanks. I think "sporadic" is too radical for turbulence, which is "intermittent" instead, as in KO62 (Kolmogorov-Obukhov). But a description using a special kind of conditional probability field should be relevant anyway. I would expect something else had been discussed in those years. Don't really know how to search... – Jean Duchon Sep 28 '17 at 15:27