aa r X i v : . [ m a t h . P R ] F e b PLANAR BROWNIAN MOTION WINDS EVENLYALONG ITS TRAJECTORY
ISAO SAUZEDDE
Abstract.
Let D N be the set of points around which a planar Brownian motion winds atleast N times. We prove that the random measure on the plane with density πN D N withrespect to the Lebesgue measure converges almost surely weakly, as N tends to infinity, towardsthe occupation measure of the Brownian motion. Introduction
Let X : [0 , → R be a planar Brownian motion started from . Let ¯ X be the oriented loopobtained by concatenating X with the straight line segment joining X to X .For each point z in R outside the range of ¯ X , let θ ( z ) be the number of times ¯ X windsaround z . For z on the range of ¯ X , we set θ ( z ) = 0 . Define D N = { z ∈ R : θ ( z ) ≥ N } . The Lebesgue measure |D N | of this set is known to be of the order of πN . More precisely,Werner proved in [8] that the following convergence holds: πN |D N | L −→ N →∞ . (1)For all N ≥ , we denote by µ N the random measure on the plane with density πN D N withrespect to the Lebesgue measure: d µ N ( z ) = 2 πN D N ( z ) d z. Let ν be the occupation measure of X , defined as the push-forward of the Lebesgue measure on [0 , by X . In other words, ν is the random Borel probability measure on the plane characterisedby the fact that for every continuous test function f : R → R , Z R f d ν = Z f ( X t ) d t. The main result of this paper is the following.
Theorem 1.
Almost surely, µ N = ⇒ N →∞ ν . To be clear, we mean that almost surely, for all bounded continuous function f : R → R , thefollowing convergence holds: lim N →∞ πN Z R f ( z ) [ N, + ∞ ) ( θ ( z )) d z = Z f ( X u ) d u. Mathematics Subject Classification.
The assumption that the test function is bounded is not essential, because almost surely, thesupports of the measures µ N , N ≥ and ν are contained in the convex hull of the range of X ,which is compact.In the course of the proof, we will obtain an estimation of the rate of convergence in terms ofthe modulus of continuity of the test function f (see Lemma 5).The study of the windings of the planar Brownian motion has a long history. The first in-vestigations were mostly concerned with the winding around a fixed point, the most prominentexample being the celebrated Spitzer theorem [7]. There followed among other works a compu-tation by Yor of the exact law of the winding [4, 10], as well as many fine asymptotic resultsconcerning related functionals (see for example [6] and references therein).In [8, 9], Werner shifted the attention from the winding around a point to the winding as a function , as well as to the set of points with a given winding number. He established, for instance,in [8], the convergence (1). His results suggest in particular that when N is large, the set D N ,which is located near the trajectory X , has a very balanced distribution along this trajectory.Our main result gives a rigorous statement of this idea.Our proof uses some results that we obtained in our previous work [5] on this subject, andwhich we recall briefly in the next section for the convenience of the reader.2. Prior results
The Brownian motion X is defined under a probability that we denote by P .Let T be a positive integer. For all i ∈ { , . . . , T } , let X i be the restriction of X to theinterval [ i − T , iT ] . As we did for X , let us denote by ¯ X i the concatenation of X i with a straightline segment from X iT to X i − T , and by θ i the winding function of the loop ¯ X i , taken to be onthe trajectory. We then set, for all N ≥ , D iN = { z ∈ R : θ i ( z ) ≥ N } and D i,jN = { z ∈ R : | θ i ( z ) | ≥ N, | θ j ( z ) | ≥ N } , with absolute values intended in the second definition.Our proof of Theorem 1 relies on the following lemmas, which are mild reformulations ofresults that we proved in [5] (see Equation (28), Theorem 1.5 and Lemma 2.4 there). Lemma 2.
Let µ be a Borel measure on R , absolutely continuous with respect to the Lebesguemeasure. For all positive integers N, T, M such that T ( M + 1) < N , T X i =1 µ (cid:0) D iN + T + M ( T − (cid:1) − X ≤ i For all δ < and p > , there exists C > such that for all N ≥ and all R > , P (cid:16) N δ (cid:12)(cid:12) πN |D N | − (cid:12)(cid:12) ≥ R (cid:17) ≤ CR − p . Lemma 4. For all ε > , there exists C > such that for all positive integers T, M , E h(cid:16) X ≤ i Let f : R → R be a bounded continuous function. Let ω f be the modulus of continuity of f :for all t ≥ , ω f ( t ) = sup {| f ( z ) − f ( w ) | : z, w ∈ R , k z − w k ≤ t } ∈ [0 , + ∞ ] . For all Borel subset E of R , we also set f ( E ) = R E f ( z ) d z .For α ∈ (0 , ) , let k X k C α denote the α -Hölder norm of the Brownian motion: k X k C α = sup ≤ s Lemma 5. For all t ∈ (0 , ) and α ∈ (0 , ) , there exists η > such that P -almost surely, thereexists a constant C such that for all bounded continuous function f : R → R and all N ≥ , (cid:12)(cid:12)(cid:12)(cid:12) πN f ( D N ) − Z f ( X u ) d u (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) ω f (2 k X k C α N − αt ) + k f k ∞ N − η (cid:1) . Let us explain why this lemma directly implies Theorem 1. Proof of Theorem 1 assuming Lemma 5. Thanks to the Portmanteau theorem, is suffices to showthat P -almost surely, for any bounded Lipschitz continuous function f , (cid:12)(cid:12)(cid:12)(cid:12) πN f ( D N ) − Z f ( X u ) d u (cid:12)(cid:12)(cid:12)(cid:12) −→ N → + ∞ . For such a function f , one has ω f ( t ) ≤ k f k Lip t and the result follows from Lemma 5 applied forinstance to t = and α = . (cid:3) In order to prove Lemma 5, we introduce the following subset of N , which depends on a positivereal parameter γ > : N γ = {⌊ K γ ⌋ : K ∈ N } \ { } . Let us fix two positive real parameters t and m with m + t < and set, for all N ≥ , T = ⌊ N t ⌋ and M = ⌊ N m ⌋ . We advise the reader to think of m as being larger than , and of t as a smallnumber. Precise conditions can be found in the statement of Lemma 7.We also set N ′ = max { n ∈ N γ : n ≤ N − T − M ( T − } , which is well defined when N islarge enough. The difference between N and N ′ is O ( N − /γ + N m + t ) .We also define the following events, which depend on t and m , and also on other positive realparameters s, ζ, δ : E N = (cid:8) ∀ i ∈ { , . . . , T } , N ′ δ (cid:12)(cid:12) πN ′ |D iN ′ | − T (cid:12)(cid:12) ≤ T − + st (cid:9) ,F N = n X ≤ i The event [ N ≥ \ N ∈ N γ N ≥ N ( E N ∩ F N ∩ G N ) has probability .Proof. The scaling properties of the Brownian motion imply that |D iN ′ | is equal in distributionto T − |D N ′ | . Thus, − P ( E N ) ≤ T P ( N ′ δ (cid:12)(cid:12) πN ′ |D N ′ | − (cid:12)(cid:12) ≥ T + st ) . Using Lemma 3 with p = 2 gives − P ( E N ) ≤ CT − st , and for N large enough, this quantity is smaller than CN − s . In particular, X N ∈ N γ (cid:0) − P ( E N ) (cid:1) ≤ C + ∞ X K =1 K − sγ . Besides, by Markov inequality, − P ( F N ) ≤ N ζ E h(cid:16) X ≤ i Using Borel–Cantelli lemma, we conclude the proof, but for the presence of G N . However,using the fact that N ′ is not larger than N and equivalent to N as N tends to infinity, andthe inequality T ≤ N t , one verifies that if t + 2 s < δ , then for N large enough, the inclusion E N ⊂ G N holds. Hence, the proof is complete. (cid:3) We now turn to the second step of the proof. Lemma 7. Almost surely, there exists a constant C such that for all N ∈ N γ and all boundedcontinuous function f : R → R , (cid:12)(cid:12)(cid:12)(cid:12) πN f ( D N ) − Z f ( X u ) d u (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) ω f (cid:0) k X k C α T − α (cid:1) + k f k ∞ ( N − m + t + N − γ +1 + N − δ + t + s + N − ζ ) (cid:17) . Proof. We first assume that f is non-negative. Replacing C if necessary by a larger constant,it suffices to show the inequality for N ≥ N , for a possibly random N which does not dependon f . Using Lemma 6, we can thus assume that the event E N ∩ F N ∩ G N holds.Using Lemma 2, the assumption that f is non-negative and the fact that the sequence ( D iN ) N ≥ is non-increasing, we have N f ( D N ) ≤ T X i =1 N f ( D iN − T − M ( T − ) + X ≤ i Comparing the Riemann sum with the integral and f to its upper bound, we turn this inequalityinto π T X i =1 N f ( D iN ′ ) ≤ Z f ( X u ) d u + ω f ( k X k C α T − α ) + k f k ∞ N − N ′ N ′ + k f k ∞ N T X i =1 (cid:0) π |D iN ′ | − T N ′ (cid:1) + 2 πω f ( k X k C α T − α ) N T X i =1 |D iN ′ | . Our next goal is to bound the last three terms of the right-hand side. Let us discuss the first,then the third and finally the second.For the first term, it follows from the definition of N ′ and by elementary arguments that for N large enough, indeed larger than a certain N that does not depend on f , N − N ′ N ′ < N m + t − + γN − γ +1 ) . For the third term, since the event G N holds, we have T X i =1 |D iN ′ | ≤ T max i ∈{ ,...,T } |D iN ′ | ≤ πN . Finally, since the event E N holds, and for N large enough, T X i =1 (cid:0) π |D iN ′ | − T N ′ (cid:1) ≤ N ′− − δ T + st ≤ N − − δ + t + s . Here the second inequality holds for N larger than a certain N which does not depend on f .We end up with π T X i =1 N f ( D iN ′ ) − Z f ( X u ) d u ≤ ω f ( k X k C α T − α ) + 2 k f k ∞ ( N m + t − + γN − γ +1 + N − δ + t + s ) . (3)We now turn to the second term of the right-hand side of (2). Since F N holds, N X ≤ i This concludes the proof when f is non-negative. To remove this assumption, it suffices todecompose f into the sum of its positive and negative parts. (cid:3) We now extend Lemma 7 from N ∈ N γ to N ∈ N ∗ , in order to obtain Lemma 5. Proof of Lemma 5. The reals t and α being given, choose positive real numbers s, ζ, m, δ, γ whichsatisfy the assumptions (A). Set η = min(1 − m − t, γ − , δ − t − s, ζ ) > .Let us first assume f is non-negative. Set ˜ N = max { n ∈ N γ : n ≤ N } , the largest integersmaller than N in N γ .Since the sequence ( f ( D N )) N ≥ is non-increasing, we have πN f ( D N ) − Z f ( X u ) d u ≤ πN f ( D ˜ N ) − Z f ( X u ) d u = N ˜ N (cid:16) π ˜ N f ( D ˜ N ) − Z f ( X u ) d u (cid:17) + (cid:16) N ˜ N − (cid:17) Z f ( X u ) d u. The first term is taken care of by Lemma 7 and the fact that N ≤ N for N large enough. Thesecond term is bounded above, for N sufficiently large, by γ k f k ∞ N − γ +1 . Altogether, we findthe upper bound πN f ( D N ) − Z f ( X u ) d u ≤ C (cid:0) ω f ( k X k C α T − α ) + k f k ∞ N − η (cid:1) for some constant C . The corresponding lower bound is obtained by the same argument with ˜ N defined as min { n ∈ N γ : n ≥ N } . This concludes the proof when f is non-negative. For thegeneral case, we simply decompose f into its positive and negative parts. This concludes theproof of Lemma 5, and also the proof of Theorem 1. (cid:3) Further perspectives It is possible that a similar result also holds when we consider the joint windings of independentBrownian motions. To be more specific, for two independent planar Brownian motions X, X ′ ,we can define their intersection measure ℓ , which is carried by the plane (see [1]).One possible way to approximate the mass of this measure is to look at the Lebesgue measureof the intersection of Wiener sausages with small radius ε around X and X ′ . In [2] (and also in[3]), it is shown that ℓ ( R ) can be obtained as the properly normalized limit of these measuresas ε → .For two independent planar Brownian motions X, X ′ , define D (2) N = { z ∈ R : θ X ( z ) ≥ N, θ X ′ ( z ) ≥ N } . Conjecture 8. There exists a constant C which depends only k X − X ′ k and such that CN |D (2) N | converges, as N → ∞ , towards ℓ ( R ) . The converges holds both in L p for any p ∈ [1 , + ∞ ) andalmost surely.Besides, almost surely, the measure CN D (2) N d z converges weakly towards ℓ . For such a result to hold, it is necessary that the exponent of N is equal to . Nonetheless,we cannot exclude that some logarithmic corrections should be added. ISAO SAUZEDDE References [1] Donald Geman, Joseph Horowitz, and Jay Rosen. A local time analysis of intersections of Brownian pathsin the plane. Ann. Probab. , 12(1):86–107, 1984.[2] Jean-François Le Gall. Sur la saucisse de Wiener et les points multiples du mouvement brownien. 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Z. Wahrsch. Verw. Gebiete ,53(1):71–95, 1980. Isao Sauzedde – LPSM, Sorbonne Université, Paris Email address ::