Kolmogorov complexity and the geometry of Brownian motion
aa r X i v : . [ c s . CC ] S e p Kolmogorov complexity and the geometry of Brownian motion
Willem L. Fouch´e
Department of Decision Sciences,School of Economic SciencesUniversity of South Africa, PO Box 392, 0003 Pretoria, South Africa [email protected]
Abstract
In this paper, we continue the study of the geometry of Brownian motions which are encoded byKolmogorov-Chaitin random reals (complex oscillations). We unfold Kolmogorov-Chaitin complexityin the context of Brownian motion and specifically to phenomena emerging from the random geometricpatterns generated by a Brownian motion.
Key words:
Kolmogorov complexity, Martin-L¨of randomness, Brownian motion, countable denserandom sets, descriptive set theory.
Finally, we would like to comment on the hidden role of Kolmogorov complexity in the reallife of classical computing ....The inherent tension, incompatability of shortest descriptions with most-economical algorith-mical processing, is the central issue of any computability theory.The place-value notation of numbers that played such a great role in the development of humancivilizations is the ultimate system of short descriptions that bridges the abyss. Kolmogorovcomplexity goes far beyond this point. ([Manin 2010] p 327.)It is well-known that the notion of randomness, suitably refined, goes a a long way in dealing with thistension. (See, for example, [Chaitin 1987, Martin-L¨of 1966, Nies 2008].) In this paper, we continue toexplore this interplay between short descriptions and randomness in the context of Brownian motionand its associated geometry. In this way one sees how random phenomena associated with the geom-etry of Brownian motion, are implicitly enfolded in each real number which is complex in the sense ofKolmogorov. These random phenomena range from fractal geometry, Fourier analysis and non-classicalnoises in quantum physics.We study in this paper algorithmically random Brownian motion, the representations of which werealso called complex oscillations in [Fouch´e(1) 2000, Fouch´e(2) 2000] for example. This terminology wassuggested to the author by the following Kolmolgorov theoretic interpretation of this notion by Asarinand Prokovskii [Asarin and Prokovskii 1986], who are the pioneers of this theme. One can characterise aBrownian motion which is algorithmically random (or, equivalently Martin-L¨of random) as an effectiveand uniform limit of a sequence ( x n ) of “finite random walks”, where, moreover, each x n can be encodedby a finite binary string s n of length n , such that the (prefix-free) Kolmogorov complexity, K ( s n ), of s n satisfies, for some constant d >
0, the inequality K ( s n ) > n − d for all values of n .Two other characterisations of the class of complex oscillations were developed by the author in[Fouch´e(1) 2000] and [Fouch´e(2) 2000]. In [Fouch´e(1) 2000] the class of complex oscillations were de-scribed in terms of effective subalgebras of the Borel σ -algebra on C [0 , Kolmogorov complexity and Brownian motion exactly those real-valued continuous functions on the unit interval that can be computed from Martin-L¨of-random reals (relative to the Lebesgue measure) by means of an associated Franklin-Wiener series.Here, the guiding motif was the fact that one can, as is well-known, also think of Brownian motion asa linear superposition of deterministic oscillations with normally distributed random amplitudes. Fur-ther applications and developments of this idea can be found in the papers [Fouch´e 2008, Fouch´e 2009,Hoyrup and Rojas 2009, Potgieter 2012, Kjos-Hanssen and nerode 2009]. A sharper version, from a com-putational point of view, of the main result in [Fouch´e(2) 2000] has recently been developed by GeorgeDavie and the author [Davie and Fouch´e 2012].Countable dense random sets arise naturally in the theory of Brownian motion [Tsirelson 2006], in non-classical noises [Tsirelson 2004] and the understanding of percolation phenomena in statistical physics(see [Tsirelson 2004], [Camia, Fontes and Newman 2005], for example). It is an interesting fact thatthe study of countable dense random sets quite naturally brings one in contact with studying randomprocesses over spaces which are not even Polish. One has to do probability theory over orbit spaces underthe action of the group S ∞ , which is the symmetry group of a countable set, on the space of all injectionsof N into the unit interval. These are examples of what Kechris [Kechris 1999] referred to as singularspaces of Borel cardinality F .In [Tsirelson 2006] Tsirelson develops a very powerful approach to random processes over these sin-gular spaces and his results imply that the Kechris-singularity manifests in very concrete and interestingstatistical properties of countable dense random sets and new aspects of Brownian motion.Tsirelson [Tsirelson 2006] shows that the minimizers of a Browian motion are, in the language of[Tsirelson 2004], instances of so-called stationary local random dense countable sets over the white noiseand that they play a pivotal role in the understanding of non-classical noises.This work suggested to the author the problem of constructing the minimizers of Brownian motiondirectly from an unbiased coin-tossing experiment. This can be seen as an extension of [Fouch´e(2) 2000]where a generic Brownian motion was constructed from a generic point in the unit interval. We shallagain adopt the viewpoint of Kolmogorov complexity to define what we mean by the word generic. In thisway, we shall be able to find Σ definitions, within the arithmetical hierarchy, for countable dense randomsets, which can be considered to be “generic” countable dense sets of reals and moreover symmetricallyrandom over white noise. We provide an explicit computable enumeration of the elements of such setsrelative to Kolmogorov-Chaitin-Martin-L¨of random real numbers. This opens the way to relate certainnon-classical noises to Kolmogorov complexity. For example, the work of the present paper enables one torepresent Warren’s splitting noise (see [Tsirelson 2004]) directly in terms of infinite binary strings whichare Kolmogorov-Chaitin-Martin-L¨of random. This line of thought will also be pursued in a sequel to thispaper.In this sequel to this paper, we shall study the images of certain Π perfect sets of Hausdorff dimensionzero under a complex oscillation. We have given a sketch in the extended abstract [Fouch´e 2009] of aproof that there are instances of such sets where such images under complex oscillations have elementsall of which are linearly independent over the field of rational numbers. In Fourier analysis, these setsare called sets of independence. (See, for example, Chapter 5 of Rudin’s book [Rudin 1960], pp 97-130.)We shall provide a generalisation of this result and show in fact that one can obtain sets via complexoscillations which are linearly independent over the field of recursive real numbers. Moreover, all theelements in these images are non-computable.The author is grateful to the Department of Mathematics at the Corvinus University, Budapest, forhosting my frequent visits to the department and for sharing with me so much of the subtleties of measuretheory and stochastic processes.The research in this paper has been supported by the National Research Foundation (NRF) of SouthAfrica and by the European Union grant agreement PIRSES-GA-2011-2011-294962 in Computable Anal-ysis (COMPUTAL).Many thanks are due to the referee whose remarks led to a significant strengtening of Theorem 5. illem L. Fouch´e A Brownian motion on the unit interval is a real-valued function ( ω, t ) X ω ( t ) on Ω × [0 , X ω (0) = 0 a.s. and for t < . . . < t n inthe unit interval, the random variables X ω ( t ) , X ω ( t ) − X ω ( t ) , · · · , X ω ( t n ) − X ω ( t n − ) are statisticallyindependent and normally distributed with means all 0 and variances t , t − t , · · · , t n − t n − , respectively.We say in this case that the Brownian motion is parametrised by Ω. Alternatively, the map X definesa Brownian motion iff for t < . . . < t n in the unit interval, the random vector ( X ω ( t ) , · · · , X ω ( t n )) isGaussian with correlation matrix (min( t i , t j ) : 1 ≤ i, j ≤ n ).It is a fundamental fact that any Brownian motion has a “continuous version”. This means thefollowing: Write Σ for the σ -algebra of Borel sets of C [0 ,
1] where the latter is topologised by the uniformnorm topology. There is a probability measure W on Σ such that for 0 ≤ t < . . . < t n ≤ B of R n , we have P ( { ω ∈ Ω : ( X ω ( t ) , · · · , X ω ( t n )) ∈ B } ) = W ( A ) , where A = { x ∈ C [0 ,
1] : ( x ( t ) , · · · , x ( t n )) ∈ B } . The measure W is known as the Wiener measure . We shall usually write X ( t ) instead of X ω ( t ).In the sequel, we shall denote by (0 , ∞ the Borel space consisting of the product of countably manycopies of the unit interval and with Borel structure being given by the natural product structure whichis induced by the standard Borel structure on the unit interval. We write (0 , ∞6 = for the Borel subspaceconsisting of the infinite sequences in the unit interval which are pairwise distinct.We write S ∞ for the symmetric group of a countable set (which we can take to be N ). We place on S ∞ the pointwise topology. We thus give S ∞ the subspace topology under the embedding of S ∞ into theBaire space N N . The group S ∞ acts naturally (and continuously) on (0 , ∞6 = as follows: σ. ( u j : j ≥
1) := ( u σ − ( j ) : j ≥ , for all ( u j ) ∈ (0 , ∞6 = and σ ∈ S ∞ (the logical action). The orbit space under this action is denoted by(0 , ∞6 = /S ∞ . We place a Borel structure on this space via the topology induced by the canonical mapping π : (0 , ∞6 = −→ (0 , ∞6 = /S ∞ . Let Ω be standard Borel space. A strongly random countable set in the unit interval is a measurablemapping X : Ω → (0 , ∞6 = /S ∞ that factors through some (traditional) random sequence Y as shown:Ω (0 , ∞6 = /S ∞ X (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ Ω (0 , ∞6 = . Y / / (0 , ∞6 = . (0 , ∞6 = /S ∞ π (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ One can think of X as a random countable set induced via S ∞ -equivalence, by a random sequence Y , bothin the unit interval. In the sequel, we shall sometimes denote the Borel space (0 , ∞6 = /S ∞ by CS (0 , X : Ω → (0 , ∞6 = /S ∞ factors through some Y as above, is an open problem.The following fundamental theorem of Tsirelson’s explains exactly what it means for two stronglyrandom countable sets to be “statistically similar”. Theorem 1 ([Tsirelson 2006]). For standard measure spaces (Ω , P ) and (Ω , P ) , let, for i = 1 , ,there be, some P i -measurable strongly random set X i : Ω i → CS (0 , such that the induced probabilitydistributions on CS (0 , are the same, i.e., for every Borel subset Σ of CS (0 , it is the case that P ( X − (Σ)) = P ( X − (Σ)) . Kolmogorov complexity and Brownian motion
Then there is a probability distribution P on Ω × Ω such that the marginal of P to Ω i is P i , and moreover,for P almost all ( ω , ω ) ∈ Ω × Ω it is the case that X ( ω ) = X ( ω ) . The statement “the marginal of P to Ω i is P i ”, means that for measurable Σ i ⊂ Ω i i = 1 , P (Σ × Ω ) = P (Σ );and P (Ω × Σ ) = P (Σ ) . We say in this case that the strongly random sets X and X are statistically similar relative to theprobabilities P , P and we simply write X ∼ X .A strongly random countable set X : Ω → CS (0 ,
1) is said to be generic relative to a probabilitymeasure P on Ω if the following is true:If B is a Borel subset of the unit interval such that λ ( B ) >
0, then P - almost surely, B ∩ X = ∅ .On the other hand, if λ ( B ) = 0, then P -almost surely, B ∩ X = ∅ . Equivalently, if C is aBorel set such that λ ( C ) = 1, then, almost surely, X ⊂ C .Here we have written λ for the Lebesgue measure on the unit interval. Note that if X is generic and if Y ∼ X , then Y too is generic.A partial converse of this statement can be found in [Tsirelson 2006]: If X , X are both strongly ran-dom, each satisfying what Tsirelson calls the “independence condition” relative to a probability measure P i , and each being almost surely dense in the unit interval, then they are statistically similar providedthey are both generic!! (Tsirelson 2006).Write λ ∞ for the product measure on (0 , ∞ which is the countable product of the Lebesgue measure λ on the unit interval and write Λ for the measure on CS (0 ,
1) which is the pushout of λ ∞ under π . Inother words, for a Borel subset Σ of CS (0 , λ ∞ ( π − Σ) . Write U : (0 , ∞ → CS (0 ,
1) for the strongly random set as defined by the following commutativediagram: (0 , ∞ CS (0 ,
1) = (0 , ∞6 = /S ∞ U (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ (0 , ∞ (0 , ∞ . Id / / (0 , ∞ .CS (0 ,
1) = (0 , ∞6 = /S ∞ π (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Then U is almost surely dense and generic. In statistics U is a model of an unordered uniform infinitesample. Moreover, it follows from the Hewitt-Savage theorem, that for every Borel subset Σ of CS (0 , ∈ { , } . (1)Note that Λ is non-atomic. Consequently, CS (0 ,
1) is not a Polish space! We shall refer to the stronglyrandom set U as the uniform random set .If X is a continuous function on the unit interval, then a local minimizer of X is a point t such thatthere is some closed interval I ⊂ [0 ,
1] containing t such that the function X assumes its minimum valueon I at the point t . We denote by MIN ( X ) the set of local minimizers of X . It is well-known that if X is a continuous version of Brownian motion on the unit interval, then MIN ( X ) is almost surely a denseand countable set and that all the local minimizers of X are strict . This means that, for each closedsubinterval I of the closed unit interval, there is a unique ν ∈ I where the minimum of X on I is assumed.This, as will be explained in this paper, has the implication that there is a subset Ω of C [0 ,
1] of full illem L. Fouch´e min : C [0 , ⊃ Ω −→ (0 , ∞6 = in such away that the composition of min with the projection π will define a measurable mapping X MIN ( X ).In the sequel this strongly random set will be denoted by MIN . To summarise, we have the followingcommutative diagram: C [0 , ⊃ Ω (0 , ∞6 = /S ∞ MIN (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄ C [0 , ⊃ Ω (0 , ∞6 = . min / / (0 , ∞6 = . (0 , ∞6 = /S ∞ π (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ The next theorem of Tsirelson (2006) says essentially that the local minimizers of a Brownian motionis a generic countable dense random set. (It is quite trivial to show that it satisfies the independenceproperty.)
Theorem 2 [Tsirelson 2006]. If X is a continuous version of Brownian motion on the unit interval and B is a Borel subset of the unit interval such that λ ( B ) > , then almost surely, B ∩ MIN ( X ) = ∅ . On theother hand, if λ ( B ) = 0 , then almost surely, B ∩ MIN ( X ) = ∅ . In particular, if λ ( C ) = 1 , then, almostsurely, MIN ( X ) ⊂ C . It follows that any generic countable dense random set with the independence property will be statisticallysimilar to the random set of minimizers of a Brownian motion. In particular
MIN ∼ U. (2)The set of words over the alphabet { , } is denoted by { , } ∗ . If a ∈ { , } ∗ , we write | a | forthe length of a . If α = α α . . . is an infinite word over the alphabet { , } , we write α ( n ) for theword Q j 0. (See, for example, [Hinman 1978]). We again write λ for the Lebesgueprobability measure on { , } . For a binary word s of length n , say, we write [ s ] for the “interval” { α ∈ { , } N : α ( n ) = s } . A sequence ( a n ) of real numbers converges effectively to 0 as n → ∞ if forsome total recursive f : N → N , it is the case that | a n | ≤ ( m + 1) − whenever n ≥ f ( m ).For any finite binary word a we denote its (prefix-free) Kolmogorov complexity by K ( a ). Recall thatan infinite binary string α is Kolmogorov-Chaitin complex if ∃ d ∀ n K ( α ( n )) ≥ n − d. (3)In the sequel, we shall denote this set by KC and refer to its elements as KC -strings. (See, e.g.,[Chaitin 1987], [Martin-L¨of 1966] or [Nies 2008] for more background.)For n ≥ 1, we write C n for the class of continuous functions on the unit interval that vanish at 0 andare linear with slopes ±√ n on the intervals [( i − /n, i/n ] , i = 1 , . . . , n . With every x ∈ C n , one canassociate a binary string a = a · · · a n by setting a i = 1 or a i = 0 according to whether x increases ordecreases on the interval [( i − /n, i/n ]. We call the sequence a the code of x and denote it by c ( x ). Thefollowing notion was introduced by Asarin and Prokovskii in [Asarin and Prokovskii 1986]. Definition 1 A sequence ( x n ) in C [0 , is complex if x n ∈ C n for each n and there is a constant d > such that K ( c ( x n )) ≥ n − d for all n . A function x ∈ C [0 , is a complex oscillation if there is a complexsequence ( x n ) such that k x − x n k converges effectively to as n → ∞ . The class of complex oscillations is denoted by C . It was shown by Asarin and Prokovskii [Asarin and Prokovskii 1986]that the class C has Wiener measure 1. In fact, they implicitly showed that the class corresponds exactly,in the broad context and modern language of Hoyrup and Rojas [Hoyrup and Rojas 2009], to the Martin-L¨of random elements of the computable measure space ([Weihrauch 1999, Weihrauch 2000, G´acs 2005]) R = ( C [0 , , d, B, W ) , (4) Kolmogorov complexity and Brownian motion where C [0 , 1] is the set of continuous functions on the unit interval that vanish at the origin, d is themetric induced by the uniform norm, B is the countable set of piecewise linear functions f vanishing atthe origin with slopes and points of non-differentiability all rational numbers and where W is the Wienermeasure.For recent refinements of this result, the reader is referred to the work of Kjos-Hanssen and Szabados[Klos-Hanssen and Szabados 2011]. They note that Brownian motion and scaled, interpolated simplerandom walks can be jointly embedded in a probability space in such a way that almost surely, the n -stepwalk is, with respect to the uniform norm, within a distance O ( n − log n ) of the Brownian path, for allbut finitely many positive integers n . In the same paper, Kjos-Hanssen and Szabados show that, almostsurely, their constructed sequence ( x n ) of n -step walks is complex in the sense of Definition 1 and allMartin-L¨of random paths have such an incompressible close approximant. This strengthens a result ofAsarin [Asarin 1988], who obtained instead the bound O ( n − log n ).The following theorem can be extracted from [Fouch´e(2) 2000]: Theorem 3 There is a bijection Φ : KC → C and a uniform algorithm that, relative to any KC -string α , with input a dyadic rational number t in the unit interval and a natural number n , will output the first n bits of the the value of the complex oscillation Φ( α ) at t . The construction in [Fouch´e(2) 2000] of the complex oscillation Φ( α ) from a given α ∈ KC is asfollows. Beginning with α ∈ KC we can construct a sequence of reals ξ , ξ , ξ jn , j ≥ , ≤ n < j ; thesequence is computable in α, j and n . Thereafter, we recursively find x ( n/ j ) for n, j ∈ N with n ≤ j from the ξ -sequence by solving the equations x (1) = ξ , x ( 12 ) = ξ + ξ , and 2 x (cid:18) n + 12 j +1 (cid:19) = 2 − j/ ξ jn + x (cid:18) n + 12 j (cid:19) + x (cid:16) n j (cid:17) . By the arguments in [Fouch´e(2) 2000], the complex oscillation Φ( α ) associated with a given α ∈ KC turnsout to be the unique continuous function which assumes, for every dyadic rational d , the value x ( d ). Inthis way, one can effectively compute any finite initial segment of the value of Φ( α ) at a given dyadicrational number from some initial segment of α . It also follows from the construction in [Fouch´e(2) 2000]that Φ( α ) = − Φ(ˆ α ) . (5)Here ˆ α denotes the binary string obtained from α by replacing each bit α i of α by 1 − α i .The mapping Φ is also measure-preserving in the following sense: Let B be a Borel subset of C [0 , λ ( α ∈ KC : Φ( α ) ∈ B ) = W ( B ) . Let N be the function that associates with every x ∈ C , the set of local minimizers of x . We shalldiscuss the measurability and computability of N in Section 4 of this paper. Thus N is the restrictionof MIN to C . We then define the function MIN : KC −→ (0 , ∞6 = /S ∞ by α MIN (Φ( α ));this means that the diagram KC C Φ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ KC (0 , ∞6 = /S ∞MIN / / (0 , ∞6 = /S ∞ C ? ? N ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ illem L. Fouch´e , ∞6 = /S ∞ : λ ( α ∈ KC : MIN ( α ) ∈ Σ) ∈ { , } . (The zero-one law for the minimizers of complex oscillations.)What is essentially at stake here is the Hewitt-Savage theorem together with the statistical similarityof three strongly random sets: U ∼ N ∼ MIN . Remark. It would be interesting to better understand the Borel subsets Σ of (0 , ∞6 = /S ∞ having Λmeasure one such that MIN ( α ) ∈ Σ for all α ∈ KC In this paper we shall prove Theorem 4 There is a uniform procedure that, relative to a given α ∈ KC , will yield, for any closeddyadic subinterval I of the unit interval, a sequence t , t , . . . of rationals in I that converges to the(unique) local minimizer of the complex oscillation, Φ( α ) , in I . Moreover all the local minimizers of acomplex oscillation are non-computable real numbers. We shall also prove Theorem 5 There is a Σ predicate C ( α, ν ) over { , } N × { , } N such that for α, ν ∈ { , } N C ( α, ν ) ⇐⇒ ν ∈ MIN (Φ( α )) ∧ α ∈ KC. This is a Σ -representation, in effective descriptive set theory, of countably random dense sets, indepen-dent and generic as explained above and given by the minimizers of Brownian motions which are encodedby KC -strings. Remark . By specialising to a ∆ -element Ω in KC (a Chaitin real), we thus find a Σ -predicatedescribing the local minimizers of the complex oscillation Φ(Ω ).The proofs of these theorems appear in Section 4 of this paper. It is a daunting task to reflect sample path properties of Brownian motion (nowhere differentiability,law of the iterated algorithm, fractal geometry) into complex oscillations by defining these phenomenain terms of the basic events in B (see (4)) which is an effective basis for the uniform norm topology in C [0 , C by using basic descriptions relative to effective Boolean subalgebras of the Borel algebra on C [0 , subset of C [0 , 1] which is of constructive measure 0. If F is a subset of C [0 , F itstopological closure in C [0 , 1] with the uniform norm topology. For ǫ > 0, we let O ǫ ( F ) be the ǫ -ball { f ∈ C [0 , 1] : ∃ g ∈ F k f − g k < ǫ } of f . (Here k . k denotes the supremum norm.) We write F for thecomplement of F and F for F . Definition 2 A sequence F = ( F i : i < ω ) in Σ is an effective generating sequence if1. for F ∈ F , for ǫ > and δ ∈ { , } , we have, for G = O ǫ ( F δ ) or for G = F δ , that W ( G ) = W ( G ) ,2. there is an effective procedure that yields, for each sequence ≤ i < . . . < i n < ω and k < ω abinary rational number β k such that | W ( F i ∩ . . . ∩ F i n ) − β k | < − k , 3. for n, i < ω , a strictly positive rational number ǫ and for x ∈ C n , both the relations x ∈ O ǫ ( F i ) and x ∈ O ǫ ( F i ) are recursive in x, ǫ, i and n , relative to an effective representation of the rationals. Kolmogorov complexity and Brownian motion Remark . This definition was motivated by the desire to have a class of basic statistical events, which,firstly, are relevant to the practice of Brownian motion, and, secondly, is such that one can prove aneffective version of the Donsker invariance principle and therefore, thirdly, to capture the entire class ofcomplex oscillations.If F = ( F i : i < ω ) is an effective generating sequence and F is the Boolean algebra generated by F ,then there is an enumeration ( T i : i < ω ) of the elements of F (with possible repetition) in such a way,for a given i , one can effectively describe T i as a finite union of sets of the form F = F δ i ∩ . . . ∩ F δ n i n where 0 ≤ i < . . . < i n and δ i ∈ { , } for each i ≤ n . We call any such sequence ( T i : i < ω ) a recursiveenumeration of F . We say in this case that F is effectively generated by F and refer to F as an effectivelygenerated algebra of sets.Let ( T i : i < ω ) be a recursive enumeration of the algebra F which is effectively generated by thesequence F = ( F i : i < ω ) in Σ. It is shown in [Fouch´e(1) 2000] that there is an effective procedure thatyields, for i, k < ω , a binary rational β k such that | W ( T i ) − β k | < − k , in other words, the function i W ( T i ) is computable.A sequence ( A n ) of sets in F is said to be F - semirecursive if it is of the form ( T φ ( n ) ) for some totalrecursive function φ : ω → ω and some effective enumeration ( T i ) of F . (Note that the sequence ( A cn ),where A cn is the complement of A n , is also an F -semirecursive sequence.) In this case, we call the union ∪ n A n a Σ ( F ) set. A set is a Π ( F )-set if it is the complement of a Σ ( F )-set. It is of the form ∩ n A n for some F -semirecursive sequence ( A n ). A sequence ( B n ) in F is a uniform sequence of Σ ( F )- sets if,for some total recursive function φ : ω → ω and some effective enumeration ( T i ) of F , each B n is of theform B n = [ m T φ ( n,m ) . In this case, we call the intersection ∩ n B n a Π ( F )-set. If, moreover, the Wiener-measure of B n converges effectively to 0 as n → ∞ , we say that the set given by ∩ n B n is a Π ( F )-set of constructive measure 0.The proof of the following theorem appears in [Fouch´e(1) 2000]. Theorem 6 Let F be an effectively generated algebra of sets. If x is a complex oscillation, then x is inthe complement of every Π ( F ) -set of constructive measure . This means, that every complex oscillation is, in an obvious sense, F -Martin-L¨of random. The converseis also true. Definition 3 An effectively generated algebra of sets F is universal if the class C of complex oscillationsis definable by a single Σ ( F ) -set, the complement of which is a set of constructive measure . In otherwords, F is universal iff a continuous function x on the unit interval is a complex oscillation iff x is F -Martin-L¨of random. We introduce two classes of effectively generated algebras G and M which are very useful for reflectingproperties of one-dimensional Brownian motion into complex oscillations.Let G be a family of sets in Σ each having a description of the form: a X ( t ) + · · · + a n X ( t n ) ≤ L (6)or of the form (6) with ≤ replaced by < , where all the a j , t j (0 ≤ t j ≤ 1) are non-zero computable realnumbers, L is a recursive real number and X is one-dimensional Brownian motion.We require that it be possible to find an enumeration ( G i : i < ω ) of G such that, for given i , if G i is gi ven by (6), we can effectively compute the sign, and for every n , a rational approximation to eachof a j , t j with error at most 1 /n . illem L. Fouch´e G = ( G i : i < ω ) is an effective generating sequencein the sense of Definition 2. The argument on p 325 of [Fouch´e(2) 2000] holds verbatim for this slightgeneralisation. The associated effectively generated algebra of sets G will be referred to as a gaussianalgebra .It is shown in [Fouch´e(1) 2000] that if G is defined by events of the form (6) with n = 1 and a = 1,then the associated G is in fact universal in the sense of Definition 3.For a closed subinterval I of the unit interval and a real number b , we write [ M ( I ) ≥ b ] for the event[sup { X ( t ) : t ∈ I } ≥ b ] and [ m ( I ) ≤ b ] for the event [inf { X ( t ) : t ∈ I } ≤ b ], where X is one-dimensionalBrownian motion on the unit interval. We let M be the set of the events of the form [ M ( I ) ≤ b ] or[ m ( I ) ≤ b ] where b is an arbitrary rational number and where I is a subinterval of the unit interval withrational endpoints. It follows from the arguments on pp 434 - 438 in [Fouch´e(1) 2000] that the elementsof M can be effectively enumerated rendering M an effective generating sequence. We denote by M the Boolean algebra generated by M . It is shown in [Fouch´e(1) 2000] that M too is in fact universal.We shall also make frequent use of the following result from [Fouch´e(1) 2000] which is an easy con-sequence of Theorem 6. It is the analogue, for continuous functions, of the well-known fact that Kurtz-random reals contain the class of Martin-L¨of random reals. Theorem 7 If B is a Σ ( F ) set and W ( B ) = 1 , then C , the set of complex oscillations, is contained in B . In this section we shall prove Theorems 4 and 5. A crucial remark is that the the local minimizers insubintervals of the unit interval of complex oscillations are uniquely determined: Proposition 1 If x ∈ C , and I , I are closed disjoint subintervals of the unit interval having rationalendpoints, then inf t ∈ I x ( t ) = inf t ∈ I x ( t ) . The proof is a constructive version of the argument on p 20 of [Peres]. We shall need the following Lemma 1 Let Z , Z be independent real-valued random variables on some probability space with Z having a non-atomic distribution. Then, almost surely, Z + Z = 0 . Proof . For i = 1 , 2, write µ i for the distribution measure of Z i . Then Z + Z has the convolutionproduct µ ∗ µ as its distribution measure, which will be non-atomic when µ is. Indeed, for a Borel set A of real numbers, ( µ ∗ µ )( A ) = Z R µ ( A − t ) dµ ( t ) , and for A = { } , it is the case that µ ( A − t ) = 0 for all t (the measure µ being non-atomic). The resultfollows since µ ∗ µ ( { } ) is the probability of the event [ Z + Z = 0]. Proof of Proposition 1 . It is well known (see p20 of [Peres]) that, under the hypotheses on I , I ,almost surely, m ( I ) = m ( I ). Indeed, let I = [ a , b ] and I = [ a , b ] denote the lower and higherinterval, respectively. Then the event [ m ( I ) = m ( I )] is the same as X ( a ) − X ( b ) = ( m ( I ) − X ( b )) − ( m ( I ) − X ( a )) . Since successive increments of Brownian motion are statistically independent, the random variablegiven by the expression on the right-hand side of this equation is independent from the the randomvariable on the left-hand side while the latter is non-atomic, being absolutely continuous with respect toLebesgue measure. It follows from the preceding lemma that m ( I ) = m ( I ) almost surely.The event [ m ( I ) = m ( I )] is described by the following Σ ( M ) event of Wiener measure one: ∃ r ∈ Q (cid:0) m ( I ) < r < m ( I ) (cid:1) ∨ (cid:0) m ( I ) < r < m ( I ) (cid:1) . Kolmogorov complexity and Brownian motion The proposition follows from Theorem 7 with F = M .The proposition has the following Corollary 1 For every complex oscillation, for every dyadic interval I in the unit interval, there is aunique point in I where the minimum of x is assumed. We shall refer to this point as the minimizer of x in I . We shall also need Lemma 2 If x ∈ C and d , d are distinct rational numbers in the unit interval, then x ( d ) = x ( d ) . Proof . Indeed, for a one-dimensional Brownian motion X , the random variable X ( d ) − X ( d ) is normalwith variance | d − d | and is therefore non-atomic being absolutely continuous with respect to Lebesguemeasure. Consequently, almost surely, X ( d ) = X ( d ).Moreover, the almost sure event [ X ( d ) = X ( d )] has a Σ ( G ) description with respect to a suitablegaussian algebra G . The description is given by the predicate ∃ r ∈ Q + | x ( d ) − x ( d ) | > r over C [0 , F = G . Remark . The preceding argument can very easily be adapted to show that each complex oscillation is injective when restricted to the computable reals in the unit interval.A consequence of Proposition 1 is that for x ∈ C , one can associate, with every dyadic subinterval I of the unit interval, the unique real number τ xI ∈ I which is the local minimizer of x in the interval I . By using this fact we shall now show that one can find a measurable mapping N : KC → (0 , ∞6 = which upon composition with the projection π : (0 , ∞ → (0 , ∞6 = /S ∞ yields the strongly random set MIN : KC → (0 , ∞6 = /S ∞ . To define N we firstly note that it follows from L´evy’s arcsine law that the distribution of the variables τ xI are all absolutely continuous with respect to Lebesgue measure. (See, for example [Peres].)In fact, it follows from L´evy’s arcsine law that, if I = [ a, b ] and a < α < β < b then W ( α < τ xI < β ) = 1 π Z β − ab − aα − ab − a dt p t (1 − t ) . In particular, for a recursive real number r in and a dyadic subinterval I of the unit interval, the event A given by x ∈ A ⇔ τ xI = r is such that W ( A ) = 0. Moreover, writing D for the set of dyadic rationals in the unit interval: x ∈ A ⇔ ∀ q ∈ D x ( r ) ≤ x ( q ) , which means that A has a Π ( G ) description relative to a suitable gaussian algebra G . It again followsfrom Theorem 7 that A contains no complex oscillations.Therefore, writing R r for the field of recursive real numbers, we have: Theorem 8 If α ∈ KC then MIN (Φ( α )) ∩ R r = ∅ . We now define the mapping N : KC → (0 , ∞6 = in stages. At stage 0 we select the unique minimizerof Φ( α ) in the unit interval. At stage n , we first select (cid:0) τ Φ( α )[ k n , k +12 n ] : 0 ≤ k < n (cid:1) illem L. Fouch´e Remark. It would be interesting to know, whether, for α ∈ KC it is the case that MIN (Φ( α )) ⊂ KC. Proof of Theorem 4 . In the sequel we shall, for α ∈ KC , denote the function Φ( α ) also by x α .Let I be a fixed dyadic interval. For n ≥ 1, set D n = { k n : 0 ≤ k < n ∧ [ k n , k + 12 n ] ⊂ I } . For d ∈ D n , set I d = [ k n , k +12 n ), when d = k n . Note that, for each β ∈ I , there is some N such that β ∈ ∪ d ∈ D n I d for all n ≥ N .It follows from Lemma 2 and Theorem 3 that for α ∈ KC and dyadic rationals d , d , the relation x α ( d ) < x α ( d ) is decidable in α, d , and d . It is because we know that x α ( d ) = x α ( d ) . (Lemma 2.)Fix α ∈ KC and write x α for the associated complex oscillation. The sequence T = ( t n ) is computedby the prescription that for all n with D n nonempty, we let t n be the unique element of D n such that x α ( t n ) < x α ( d ) for all d ∈ D n with d = t n . In view of Theorem 3, the sequence T is computable from α .Let ( t n k ) be any convergent subsequence of T with limit ν , say. For given η > 0, we have for all k sufficiently large ( ≥ L , say) that ν ∈ [ t n k − η, t n k + η ]. Fix β ∈ I . Next choose L ≥ L such for k ≥ L we can find some d k in D n k such that β ∈ I d k .For k ≥ L x α ( ν ) − x α ( β ) = x α ( ν ) − x α ( t n k ) + x α ( t n k ) − x α ( d k ) + x α ( d k ) − x α ( β ) , and, since, by construction, x α ( t n k ) ≤ x α ( d k ) , the difference x α ( ν ) − x α ( β ) can be made to be arbitrarily small by first choosing η sufficiently small andthen k sufficiently large. We conclude that x α ( ν ) ≤ x α ( β ). In particular, x α ( ν ) = m ( I ) . Recall that x α has a unique minimizer in I . Hence all the convergent subsequences of T have thesame limit. We can therefore conclude that T is a convergent sequence converging to the unique point in I where the minimum of x α on I is assumed. This concludes the proof of the theorem. Remark. Even though the construction of the sequence ( t i : i ≥ 0) in the theorem is effective relativeto α ∈ KC , the proof renders no information on the rate of convergence to the local minimizer in thedyadic interval. This problem will be addressed in a sequel of this paper (in collaboration with GeorgeDavie where it will be shown how it can be uniformly computed from the incompressibility coefficient of α ∈ KC ).We again write D for the dyadic rationals in the unit interval. For the proof of the Theorem 5, weshall need the following Proposition 2 The relations x α ( µ ) < x α ( t ) and x α ( µ ) > x α ( t ) are each Σ in α ∈ KC, µ ∈ { , } N and t ∈ D . To prove this Proposition, we first discuss the following Lemma: Lemma 3 There is a uniform algorithm that, having access to an oracle for α ∈ KC , will decide whether Φ( α )( t ) < q, for t ∈ D and q ∈ R r . Kolmogorov complexity and Brownian motion Proof: It follows from [Fouch´e(1) 2000] that, under the above hypotheses on α, t and q Φ( α )( t ) = q. Since Φ( α )( t ) > q ⇔ ∃ n Φ( α )( t )( n ) > q, the inequality Φ( α )( t ) > q can be algorithmically affirmed if true. (This a direct consequence of Theorem3.) To affirm the inequality Φ( α )( t ) < q we need only apply (5) and note thatΦ( α )( t ) < q ⇔ Φ(ˆ α )( t ) > − q, to conclude the proof of the lemma. It is shown in [Fouch´e(2) 2000] that every complex oscillation iseverywhere β -H¨older continuous for any 0 < β < . The proof of of the Σ -definability of the relation x α ( µ ) < x α ( t ) now follows from this observation together with Lemma 3 which allows one to infer that: x α ( µ ) < x α ( t ) ⇔ ∃ q ∈ Q ∃ k ∀ L ≥ k x α ( µ ( L )) < q + 12 L/ ∧ q < x α ( t ) . The Σ -definability of x α ( µ ) > x α ( t ) now follows from symmetry. (Replace α by ˆ α .) Proof of Theorem 5 . Using the notation in the proof of the preceding theorem, define, for µ in theCantor space { , } N , and for α ∈ KC , the predicate I d ( µ, α ) to mean that the real in the unit intervalcorresponding to µ is the unique minimum on I d of x α . Here again, we have set I d = [ k n , k +12 n ), when d = k n . Note that for α ∈ KC . I d ( µ, α ) ⇔ µ ∈ I d ∧ ∀ t ∈ I d ∩ D [ x α ( µ ) < x α ( t )] . Since every local minimizer of a complex oscillation is a non-dyadic number, we can replace [ x α ( µ ) < x α ( t )]by [ x α ( µ ) ≤ x α ( t )] in the definition of I d ( µ, α ). It therefore follows from Proposition 2 that the predicate I d ( µ, α ) is a Π -formula in µ and α . Writing again N ( α ) for the set of local minimizers of x α , we find, µ ∈ N ( α ) ↔ ∃ d ∈ D I d ( µ, α ) . This is a Σ -formula in µ and α . Finally note that the set KC is Σ -definable. Remark: In view of Tsirelson’s Theorem [Tsirelson 2006] (see Theorem 2 above), it is an interestingproblem, to characterise, for x ∈ C , the Borel sets B of Lebesgue measure 0 that are disjoint from MIN ( x ).This is not always the case. For instance, if B = { z } where z is the minimum of x on the unit intervalthen of course B will intersect the local minimisers of x . On the other hand, if B = Z x , the zero set of x , then B does have Lebesgue measure 0 and will be disjoint from MIN ( x ). To see this, note that if X is a continuous version of one-dimensional Brownian motion, then, almost surely, no local minimum of X will be a zero of X . For otherwise, there will be a neighbourhood of some zero of X containing noother zeroes of X , which contradicts the well-known fact that the zero set of X is almost surely perfect(the zero set being, for instance, almost surely, a set of non-zero Hausdorff dimension). It follows that,we have, for each interval I in the unit interval, almost surely ∃ r ∈ Q + m ( I ) < − r ∨ m ( I ) > r. This is, for each closed interval I with rational endpoints, a Σ ( M ) event of full Lebesgue measure and isconsequently reflected in every complex oscillation. We conclude that if x is a complex oscillation, then Z x ∩ MIN ( x ) = ∅ . illem L. 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