On zeros of Martin-Löf random Brownian motion

Abstract

We investigate the sample path properties of Martin-Löf random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-Löf random Brownian path, (2) that the effective dimension of zeroes of a Martin-Löf random Brownian path must be at least 1/2, and conversely that every real with effective dimension greater than 1/2 must be a zero of some Martin-Löf random Brownian path, and (3) we will demonstrate a new proof that the solution to the Dirichlet problem in the plane is computable.

Expanding the Reals by Continuous Functions Adds No Computational Power

“…set W A . A settling time function for A 1 is a function s : Ñ such that n P A 1 if and only if n P W Aaes(n) s(n) , i.e., n is the stage s(n) approximation to the set W A and references the oracle A only on numbers less than n. If a jump ideal S contains A, then the least (with respect to majorizing) settling time function s A 1 is computable from A 1 , and hence it is in F. To define J i , we make an initial guess towards a P-index for a settling time function of (E 0 ' ¨¨¨' E i ) 1 : we guess that P 0 is such a function. We then try to compute (E 0 ' ¨¨¨' E i ) 1 using our current guess of that function and the columns E 0 , ... , E i .…”

“…In the present paper, we investigate structures of the form R f = (R, f), the ordered field of reals expanded by a function f. Igusa, Knight, and Schweber [7] showed that if f is analytic, then R f " ẘ R. They asked whether this remains true for arbitrary continuous functions f. They believed that the answer should be negative, witnessed possibly by something like Brownian motion, with complicated level sets as studied in [1]. Here, we show that the answer to the question is actually positive.…”

“…Let S be a countable jump ideal, and let E be an enumeration of S. We may think of E as a function taking each index n to the set E n . We consider the companion function J taking n to the set (E 0 ' ¨¨¨' E n ) 1 . More formally, J will be an enumeration of a subfamily of S, with specified indices.…”

“…Definition 1.5 (Running jump). Let E be an enumeration of a jump ideal S. The running jump for E is the relation J Ď 2 such that (n, x) P J if and only if x P (E 0 ' ¨¨¨' E n ) 1 .…”

Computable Measure Theory and Algorithmic Randomness

“…Also the Wiener measure is a computable probability measure on this space. The algorithmically random Brownian motion paths have been thoroughly studied by Fouché and others [4,36,37,77,38,39,28,13,40,42,41].…”