aa r X i v : . [ m a t h . P R ] A ug BROWNIAN BRICKLAYER: A RANDOM SPACE-FILLING CURVE
NOAH FORMAN
Abstract.
Let ( B ( t ) , t ≥
0) denote the standard, one-dimensional Wiener process and ( ℓ ( y, t ); y ∈ R , t ≥
0) its local time at level y up to time t . Then (cid:0) ( B ( t ) , ℓ ( B ( t ) , t )) , t ≥ (cid:1) is a random paththat fills the upper half-plane, covering one unit of area per unit time. The existence of space-filling curves was first proved by Peano, who gave a deterministic, recursiveconstruction, which was later simplified by Hilbert [15]. In this note we examine a stochastic process,based on Brownian local times, that gives rise to a random space-filling curve. More broadly, thisnote belongs to the genre of so-called “pathological” examples that arise naturally in the study ofBrownian motion. Before presenting the main result, we describe its simpler discrete analogue.
The discrete random bricklayer.
Imagine a worker building a wall by stacking blocks vertically,in side-by-side columns. The worker starts by setting a single block in front of themself. They thenflip a coin, stepping to the right if they get heads or to the left if tails before placing the next block.They repeat this process ad infinitum, flipping, moving, and stacking blocks.We label sites by integers, with the worker’s initial position labeled ‘0’. The worker’s left-to-rightmotion is the simple random walk on Z . The number of blocks stacked at site j after the first n steps is called the occupation time of the walk at j up to time n .The simple random walk on Z is transitive and recurrent [9], meaning that it visits every siteinfinitely many times. Thus, the wall ultimately grows infinitely high at every site. Now supposethe blocks are labeled by the time at which they were placed and we run a string across the frontof the infinite wall, connecting consecutively numbered blocks. It is possible that adjacent columnscan be at significantly different heights, so the string may run nearly vertical. But this will beuncommon, with the string usually taking smaller vertical steps, since the worker will have visitedadjacent sites similar numbers of times [8, Theorem 1]. The Brownian bricklayer.
For each n ≥ j ∈ Z , let S ( n ) denote the worker’s position after n steps and L ( j, n ) the height of the column at site j at this time. Donsker’s theorem statesthat simple random walk converges in distribution, as a process, in a scaling limit to standard,one-dimensional Brownian motion: (cid:16) n − / S ⌈ nt ⌉ , t ≥ (cid:17) d → ( B ( t ) , t ≥ . (1)Knight [12] strengthened this result, showing the joint convergence (cid:16)(cid:16) n − / S (cid:0) ⌈ nt ⌉ (cid:1) , t ≥ (cid:17) , (cid:0) n − L (cid:0) ⌈ ny ⌉ , ⌈ nt ⌉ (cid:1) ; t ≥ , y ∈ R (cid:1)(cid:17) d → (cid:0) ( B ( t ) , t ≥ , ( ℓ ( y, t ); t ≥ , y ∈ R ) (cid:1) , (2)where ℓ ( y, t ) is the (occupation density) local time of Brownian motion at level y , up to time t .This is defined as ℓ ( y, t ) := lim ǫ → (2 ǫ ) − Leb { u ∈ [0 , t ] : | B ( u ) − y | < ǫ } for y ∈ R , t ≥ . (3) Date : August 25, 2017.2010
Mathematics Subject Classification.
Key words and phrases.
Local times, space-filling curve, Brownian motion.
Informally, the local time ℓ ( y, t ) quantifies the amount of time spent by the Brownian path at (ormore accurately, near) level y , prior to time t . See [16, Chapter 6] for more background.We define the Brownian bricklayer to be the process K ( t ) := (cid:0) B ( t ) , ℓ ( B ( t ) , t ) (cid:1) , t ≥ . (4)In analogy with our discrete-time description, the first coordinate of this process describes thecurrent location of a bricklayer as they move up and down the line, and the second describes theheight of the wall in front of the bricklayer’s current location. Extending the analogy, the wall itselfat time t is described by the local time profile ( ℓ ( y, t ) , y ∈ R ). Theorem 1.
The process (cid:0) K ( t ) , t ≥ (cid:1) is almost surely (a.s.) path continuous, and its path a.s.fills the upper half plane. I.e. { K ( t ) : t ≥ } = R × [0 , ∞ ) almost surely. To prove this result, we appeal to two well-known properties of Brownian local times.(1) Trotter’s theorem [16, Theorem 6.19]: the random field ( ℓ ( y, t ); y ∈ R , t ≥
0) admits ana.s. continuous version. I.e. there is an a.s. event on which the limit ℓ ( y, t ) is defined andcontinuous at all points ( y, t ) ∈ R × [0 , ∞ ).(2) It is a.s. the case that, for every y ∈ R , the process ( ℓ ( y, t ) , t ≥
0) increases withoutbound. This can be deduced from Ray’s Theorem [16, Theorem 6.38] and the strongMarkov property.Let Ω ′ denote an a.s. event on which both of the above properties hold. Proof of Theorem 1.
The continuity of ( K ( t ) , t ≥
0) follows from Trotter’s theorem and the conti-nuity of Brownian motion itself. It remains to show that the path is a.s. space-filling. To that end,fix ( y, s ) ∈ R × [0 , ∞ ). It suffices to show that for every outcome ω ∈ Ω ′ there is some T = T ( ω, y, s )for which (cid:0) B ( T ) , ℓ ( B ( T ) , T ) (cid:1) = ( y, s ).If s = 0 then, by the continuity of ℓ , taking T = inf { t ≥ B ( t ) = y } gives the desired result.Now, assume s >
0. Consider the set { t ≥ ℓ ( y, t ) < s } . This set is non-empty and, by property(ii) above, it is bounded, so it has a supremum T . By the continuity of ℓ we get ℓ ( y, T ) = s , asdesired. Moreover, for all t < T we have ℓ ( y, t ) < ℓ ( y, T ). By the definition in (3), this implies thatfor every ǫ > t < T , there is some t ′ ∈ ( t, T ) for which | B ( t ′ ) − y | < ǫ . In particular,by the continuity of B we get B ( T ) = y , also as desired. (cid:3) Properties of the Brownian bricklayer.
The following results are consequences of well knownproperties of Brownian local times and excursions. See [6, 10, 16] for general references on thesetopics.(a)
The area of the brick wall grows determinstically, at unit rate . For each y , the local time( ℓ ( y, u ) , u ≥
0) increases only at times u when B ( u ) = y . Thus, at time t , at each position y , thewall reaches height ℓ ( y, t ). I.e. (cid:8) K ( u ) , u ∈ [0 , t ] (cid:9) = (cid:8) ( y, s ) : y ∈ { B ( u ) , u ∈ [0 , t ] } , s ∈ [0 , ℓ ( y, t )] (cid:9) . The area of the wall at time t is thus R ∞−∞ ℓ ( y, t ) dy = t , by the definition in (3).For c = 0 and d >
0, the process (cid:0)(cid:0) cB ( t ) , dℓ ( B ( t ) , t ) (cid:1) , t ≥ (cid:1) is a space-filling curve covering anarea that grows deterministically, at rate | c | d .(b) The process ( K ( t ) , t ≥ is non-Markovian . Its future evolution depends on the local timeprofile, ( ℓ ( y, t ) , y ∈ R ), which we think of as the state of the growing wall. However, the augmentedprocess (cid:0)(cid:0) K ( t ) , (cid:0) ℓ ( y, t ) , y ∈ R (cid:1)(cid:1) , t ≥ (cid:1) is a strong Markov process.(c) Barlow [4] proved that (cid:0) ℓ ( B ( t ) , t ) , t ≥ (cid:1) is not a semimartingale by showing that it fails tobe H¨older-continuous with index . ROWNIAN BRICKLAYER: A RANDOM SPACE-FILLING CURVE 3 (d) The law of K ( t ), at fixed times t , isPr { B ( t ) ∈ dy, ℓ ( B ( t ) , t ) ∈ ds } = | y | + s √ πt exp (cid:18) − ( | y | + s ) t (cid:19) dyds. (5)We deduce this by a sequence of transformation identities. First, the identity ( B ( u ) , u ∈ [0 , t ]) d =( B ( t ) − B ( t − u ) , u ∈ [0 , t ]) implies that( B ( t ) , ℓ ( B ( t ) , t )) d = ( B ( t ) , ℓ (0 , t )) . (6)L´evy’s theorem [16, Theorem 7.38] states that( | B ( t ) | , ℓ (0 , t )) d = ( S ( t ) − B ( t ) , S ( t )) , where S ( t ) := max u ∈ [0 ,t ] B ( u ) . (7)The identity ( B ( u ) , u ∈ [0 , t ]) d = ( − B ( u ) , u ∈ [0 , t ]) implies that the sign of B ( t ) is independent of( | B ( t ) | , ℓ (0 , t )). Thus, ( B ( t ) , ℓ (0 , t )) d = (( S ( t ) − B ( t )) · I, S ( t )) , (8)where I = ± each, independent of the Brownian motion. The reflectionprinciple (see [16]) implies that, for x ∈ R and s > x < s ,Pr { B ( t ) ∈ dx, S ( t ) ≥ s } = Pr { s − B ( t ) ∈ dx } = 1 √ πt exp (cid:18) − (2 s − x ) t (cid:19) . (9)Putting these pieces together gives (5).(e) For any fixed y , the set of points on the line { y } × [0 , ∞ ) that the process K visits multipletimes is a.s. countable, and no point on this line gets visited more than twice. In particular, { t : B ( t ) = y } is known to be homeomorphic to the Cantor middle-third set, with the removedmiddle thirds corresponding to excursions of B away from y . If ( G, D ) denotes the time interval ofone such excursion, then B ( G ) = B ( D ) = y and ℓ ( y, G ) = ℓ ( y, D ). Thus K ( t ) revisits a previousvalue in { y } × [0 , ∞ ) at the end of each excursion. But there are a.s. no two excursions about level y arising at the same local time, so these are a.s. the only repetitions. Related literature.
As noted above, Barlow [4] proved that (cid:0) ℓ ( B ( t ) , t ) , t ≥ (cid:1) is not a semi-martingale. Subsequent papers [1, 5] have encountered this process as well. It is closely related to (cid:0) sup y ℓ ( y, t ) , t ≥ y ℓ ( y, t ) = sup u ∈ [0 ,t ] ℓ ( B ( u ) , u ). See [19], for example, for discussionof this supremum.The local time profile ( ℓ ( y, T ) , y ∈ R ) at a deterministic or random time T has been studiedextensively, most notably with the Ray-Knight theorems [16, Chapter 6]. These theorems statethat, at certain natural random times, the local time as a process in level behaves like a certaincontinuous state branching process, which is a natural model for random population growth anddecay. Jeulin’s theorem [11] gives an alternative descrition of the local time profile of a Brownianexcursion, as a certain time-change of another Brownian excursion. Generalizations of the Ray-Knight and Jeulin descriptions to fixed times appear in [13] and [3, 14], respectively.The name “Brownian bricklayer” is an homage to Warren and Yor’s “Brownian burglar” [22]:Brownian motion conditioned on its local time profile. Aldous [2] conducted a similar study. Thelocal time profile, as a process evolving in time, arises in the study of self-repelling processes, suchas true self-repelling motion (TSRM) [21]. See [17] for a survey of foundational results in this area.On the discrete side, many authors have studied the occupation times, or discrete local times, ofsimple random walks. Such counts were the basis for Knight’s proof of the Ray-Knight theorems[12]. See also subsequent work by Cs¨org˝o and R´ev´esz, such as [7, 18].Schramm-Loewner evolution with parameter κ > κ )) is another random path that fillsthe upper half-plane [20], but it does not equal the Brownian bricklayer. For example, the bricklayer NOAH FORMAN fills each vertical column from bottom to top – it cannot visit the point (0 ,
1) before visiting (0 , / Acknowledgements.
The author thanks Soumik Pal, Jim Pitman, and Matthias Winkel for theirhelpful comments.
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