Multiple points of the Brownian sheet in critical dimensions

Transcription

1 Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne Based on joint work with: Carl Mueller Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 1 / 26

2 Overview Introduction to the Brownian sheet Hitting points, multiple points Intersection equivalence Main result: absence of multiple points in critical dimensions Method of proof Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 2 / 26

3 The Brownian sheet Fix two positive integers d and N. An N-parameter Brownian sheet with values in R d is a Gaussian random field B = (B 1,..., B d ), defined on a probability space (Ω, F, P), with parameter set R N +, continuous sample paths and covariances Cov(B i (s), B j (t)) = δ i,j N l=1 (s l t l ), where δ i,j = 1 if i = j and δ i,j = 0 otherwise, s, t R N +, s = (s 1,..., s N ) and t = (t 1,..., t N ). The case N = 1: Brownian motion B = (B(t), t R +). The case N > 1: multi-parameter extension of Brownian motion. A few references: Orey & Pruitt (1973), R. Adler (1978), W. Kendall (1980), J.B. Walsh (1986), D. & Walsh (1992), Khoshnevisan & Shi (1999) D. Khoshnevisan, Multiparameter processes, Springer (2002). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 3 / 26

4 A sample path of the Brownian sheet N = 2, d = 1 Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 4 / 26

5 Hitting probabilities Basic questions, N = 1. (1) Hitting points. For which d is P{ t R + : B(t) = x} > 0 (x R d )? (2) Polar sets. For I R +, let B(I ) = {B(t) : t I } be the range of B over I. Which sets A R d are polar (P{B(I ) A } = 0)? (3) Hitting probabilities. For A R d, what are bounds on P{B(I ) A }? (4) What is the Hausdorff dimension of the range of B? dim B(R +) = 2 d a.s. (5) What is the Hausdorff dimension of level sets of B? If d = 1, then dim L(x) = 1 2 a.s. L(x) = {t R + : B(t) = x} Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 5 / 26

6 Intersections of Brownian motions Given k 2 independent Brownian motions B 1,..., B k with values in R d, what is the probability that their sample paths have a non-empty intersection? Is P (y1,...,y k ){ (t 1,..., t k ) (R +) k : B 1 (t 1) = = B k (t k )} > 0? (y l distinct) Equivalently, is P (y1,...,y k ){B 1 (R +) B k (R +) } > 0? d 5: 2 independent BM s do not intersect (Kakutani 1944). d 4: 2 independent BM s do not intersect. d = 3: P (x1,x 2 ){B 1 (R +) B 2 (R +) } > 0 (Dvoretzky, Erdős & Kakutani 1950), but 3 independent BM s do not intersect (& Taylor 1957). d = 2: For all k 2, sample paths of k independent BM s intersect (D-E-K 1954). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 6 / 26

7 Self-intersections, multiple points x R d is a k-multiple point of B if there exist distinct t 1,..., t k R + such that B(t 1) = = B(t k ) = x. We let M k denote the (random, possibly empty) set of all k-multiple points of B. Existence of multiple points. Given a positive integer k, is P{M k } > 0? Existence of multiple points within a given set. Given a positive integer k and a subset A R d, is P{M k A } > 0? Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 7 / 26

8 Relationship between self-intersections and intersections of independent motions Case of Brownian motion (or of Markov processes). Given a Brownian motion B, a double point (or 2-multiple point) occurs if there are two disjoint intervals I 1 and I 2 in R + such that B(I 1) B(I 2). (1) Suppose I 1 precedes I 2. By the Markov property of B, given the position of B at the right endpoint of I 1, B I2 is conditionally independent of B I1, so the study of (1) is essentially the study of intersections of two independent Brownian motions. For random fields, this type of Markov property is absent, so other methods are needed. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 8 / 26

9 Case N > 1: the Brownian sheet (1) (Orey & Pruitt, 1973). Points are polar for the (N, d)-brownian sheet if and only if d 2N. (2) (J. Rosen, 1984) If d(k 1) < 2kN, then the Hausdorff dimension of the set of k-multiple points is kn d(k 1)/2. (3) (Khoshnevisan, 1997). The (N, d)-brownian sheet does not have k-multiple points if d(k 1) > 2kN. Remark. The critical case d(k 1) = 2kN remained open. (4) Double points (D. et al, Annals Probab. 2012). In the case k = 2 and for critical dimensions d = 4N, the (N, d)-brownian sheet does not have double points. Theorem 1 (D. & Mueller, today s talk) Absence of k-multiple points, k 2: If N, d and k are such that (k 1)d = 2kN, then an (N, d)-brownian sheet has no k-multiple points. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 9 / 26

10 Non-critical dimensions Case N = 1, d 3: standard Brownian motion with values in R d does not hit points. Explanation. Let t k = 1 + k2 2n. Fix x R d. Then 2 2n P{ t [1, 2] : B(t) = x} = P { t [t k 1, t k ] : B(t) = x} k=1 P{ t [t k 1, t k ] : B(t) = x} 2 2n k=1 P{ B tk x n2 n } 2 2n k=1 c(n2 n ) d 2 2n k=1 = c2 2n c(n2 n ) d = cn d 2 (2 d)n 0 as n + (because d 3). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 10 / 26

11 Anisotropic Gaussian fields, generic results (Xiao, 2008) Let (V (x), x R k ) be a centered continuous Gaussian random field with values in R d with i.i.d. components: V (x) = (V 1(x),..., V d (x)). Set σ 2 (x, y) = E[(V 1(x) V 1(y)) 2 ]. Let I be a rectangle. Assume the two conditions: (C1) There exists 0 < c < and H 1,..., H k ]0, 1[ such that for all x I, and for all x, y I, c 1 c 1 σ 2 (0, x) c, k x j y j 2H j σ 2 (x, y) c j=1 (H j is the Hölder exponent for coordinate j). (C2) There is c > 0 such that for all x, y I, Var(V 1(y) V 1(x)) c k x j y j 2H j j=1 k x j y j 2H j. j=1 Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 11 / 26

12 Anisotropic Gaussian fields Theorem 1 (Biermé, Lacaux & Xiao, 2007) Fix M > 0. Set Q = k j=1 1 H j. Assume d > Q. Then there is 0 < C < such that for every compact set A B(0, M), C 1 Cap d Q (A) P{V (I ) A } CH d Q (A). Special case obtained by D., Khoshnevisan and E. Nualart (2007); see also D. & Sanz-Solé (2010). This results tells us what sort of inequality to aim for when we have information about Hölder exponents. Notice the Hausdorff measure appearing on the right-hand side. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 12 / 26

13 Measuring the size of sets: capacity Capacity. Cap β (A) denotes the Bessel-Riesz capacity of A: and Cap β (A) = E β (µ) = Examples. If A = {z}, then: R d 1 inf µ P(A) E β (µ), R d k β (x y)µ(dx)µ(dy) x β if 0 < β < d, k β (x) = ln( 1 ) if β = 0, x 1 if β < 0. Cap β ({z}) = { 1 if β < 0, 0 if β 0. If A is a subspace of R d with dimension l {1,..., d 1}, then: { > 0 if β < l, Cap β (A) = 0 if β l. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 13 / 26

14 Another measure of the size of sets: Hausdorff measure For β 0, the β-dimensional Hausdorff measure of A is defined by { } H β (A) = lim inf (2r i ) β : A B(x i, r i ), sup r i ɛ. ɛ 0 + i 1 i=1 When β < 0, we define H β (A) to be infinite. Note. For β 1 > β 2 > 0, i=1 Cap β1 (A) > 0 H β1 (A) > 0 Cap β2 (A) > 0. Example. A = {z} if β < 0 H β ({z}) = 1 if β = 0 0 if β > 0. Remark. For β = 0, Cap 0 ({z}) = 0 < H 0({z}) = 1. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 14 / 26

15 Situation in the critical dimension d = Q When d = Q and A = {x}, the inequality C 1 Cap d Q (A) P{V (I ) A } CH d Q (A) is equivalent to which is uninformative! 0 P{x is polar} 1, Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 15 / 26

16 Reducing multiple points to hitting points Let X (t 1,..., t k ) = (B(t 1 ) B(t 2 ), B(t 2 ) B(t 3 ),..., B(t k 1 ) B(t k )). Then B(t 1 ) = = B(t k ) X (t 1,..., t k ) = 0 (R d ) k 1, Existence of k-multiple points for B is equivalent to hitting 0 for X. Remark. The probability density function of X (t 1,..., t k ) is related to the joint probability density function of (B(t 1 ),..., B(t k )), so the difficulty lies in getting properties of the joint density. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 16 / 26

17 Intersection-equivalence Let TN k denote the set of parameters (t 1,..., t k ) with t i ]0, [ N such that no two t i = (t1, i..., tn) i and t j = (t j 1,..., tj N ) (i j) share a common coordinate: TN k = {(t 1,..., t k ) (]0, [ N ) k : tl i t j l, for all 1 i < j k and l = 1,..., N}. Theorem 2 Let A R d be a Borel set. For all k {2, 3,...}, we have if and only if P{ (t 1,..., t k ) T k N : B(t 1 ) = = B(t k ) A} > 0 P{ (t 1,..., t k ) T k N : W 1(t 1 ) = = W k (t k ) A} > 0, where W 1,..., W k are independent N-parameter Brownian sheets with values in R d. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 17 / 26

18 Using Theorem 1 Theorem 3 (Khoshnevisan and Shi, 1999) Fix M > 0 and 0 < a l < b l < (l = 1,..., N). Let I = [a 1, b 1] [a N, b N ]. There exists 0 < C < such that for all compact sets A B(0, M), 1 C Cap d 2N(A) P{B(I ) A } C Cap d 2N (A). (2) (Cap denotes Bessel-Riesz capacity) Theorem 4 (Peres, 1999) Property (2) implies that if W 1,..., W k are independent N-parameter Brownian sheets with values in R d and d > 2N, then 1 C Cap k(d 2N) (A) P{W1(I1) W k(i k ) A } C Cap k(d 2N) (A) In particular, the r.h.s. is 0 if A = R d and k(d 2N) = d, i.e. (k 1)d = 2kN. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 18 / 26

19 Creating independence while preserving absolute continuity Theorem 5 Let B be a Brownian sheet and let W 1,..., W k be independent Brownian sheets. Fix M > 0 and k boxes R 1,..., R k, where, for each coordinate axis, the projections of the R i onto this coordinate axis are pairwise disjoint. Then, for all (R 1,..., R k ) R M, the random vectors (B R1,..., B Rk ) and (W 1 R1,..., W k Rk ) (with values in (C(R 1, R d ) C(R k, R d ))) have mutually absolutely continuous probability distributions. Comments. (1) There is no convenient Markov property to use. (2) If the projections of the boxes are not disjoint, then this property would fail. (3) D. et al 2012: used quantitative estimates on the conditional distribution of B Rk given (B R1,..., B Rk 1 ). This was only achieved for certain configurations of boxes, which limited the final result to double points (2 boxes). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 19 / 26

20 First main ingredient in the proof of Theorem 5 Fix M > 0. Define the one-parameter filtration G = (G u, u [0, M]) by { } G u = σ B(t 1,..., t N 1, v) : (t 1,..., t N 1 ) R N 1 +, v [0, u]. Let (Z(s), s R N 1 + [0, M]) be an R d -valued adapted random field: for all s R N 1 + [0, M], Z(s) is G sn -measurable. For u [0, M], define ( L u = exp Z(s) db(s) 1 ) Z(s) 2 ds. R N 1 + [0,u] 2 Theorem 6 (Cameron-Martin-Girsanov) R N 1 + [0,u] If (L u, u [0, M]) is a martingale with respect to G, then the process ( B(t), t R N 1 + [0, M]) defined by B(t 1,..., t N ) = B(t 1,..., t N ) Z(s 1,..., s N ) ds 1 ds N [0,t 1 ] [0,t N ] is an R d -valued Brownian sheet under the probability measure Q, where Q is defined by dq dp = L M. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 20 / 26

21 Using Girsanov to create independence Case N = 1. Let (B t, t R +) be a Brownian motion. Show that Law((B t, t [1, 2]), (B t, t [3, 4]), (B t, t [5, 6])) is mutually absolutely continuous w.r.t. where (W (j) t Law((W (1) t, t [1, 2]), (W (2) t, t [3, 4]), (W (3) t, t [5, 6])) ) are independent Brownian motions. By Girsanov s theorem: (B t, t [0, 6]) Define (h s, s [0, 6]) by h s ( B t = B t t 0 hs ds, t [0, 6] ). B 2 B 4 B 2 B t, t I 1 B t = B t B 2, t I 2 B t B 4, t I 3 Set W (j) t 0 1 I s I I 3 6 = B t, t I i. Then the (W (j) t, t I j ) are independent. Observation. For t I 3, Bt = B t E(B t F(I 1) F(I 2)). For t I 2, Bt = B t E(B t F(I 1)). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 21 / 26

22 Obtain conditional expectations via a Girsanov transformation Case N = 2. Fix k 2 and consider k boxes R 1,..., R k with disjoint projections on each coordinate axis: R j = Ij 1 Ij 2, j = 1,..., k, where, for l = 1, 2, the intervals I1 l,..., Ik l are pairwise disjoint. We assume that I1 2 < I2 2 < < Ik 2. Let S = (I 1 k ) c [0, sup I 2 k 1]. Notice that for j = 1,..., k 1, R j S, and there is some space between S and R k. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 22 / 26

23 Illustration t 2 I 2 k R k u 2 k v 2 k 1 I 2 k 1 R k 1 R 3 S R 2 R 1 Ik 1 0 u 1 t 1 k vk 1 Then for t R k, E(B(t) F(S)) = B(t), where B(t 1, t 2 ) = t1 u 1 k v 1 k u1 k B(vk 1, vk 1) 2 + v k 1 t 1 vk 1 B(uk, 1 vk 1). 2 u1 k Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 23 / 26

24 Creating independent while preserving absolute continuity For s 1 ]0, vk 1 [ and s 2 ]vk 1, 2 uk[, 2 let ( Z(s 1, s 2 ) = 2 s 2 vk 1 2 s 1 s 2 uk 2 v k 1 2 s 1 u 1 k u 1 k ) B(s 1 uk, 1 vk 1) 2, and Z(s 1, s 2 ) = 0 otherwise. Define ˆB(t) = B(t) Z(s 1, s 2 ) ds 1 ds 2. [0,t] Facts. (a) For t R k, ˆB(t) = B(t) E(B(t) F(S)), and the r.h.s. is independent of F(S) (Gaussian process). (b) ˆB = B on R 1 R k 1. (c) Law( ˆB) Law(B). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 24 / 26

25 Concluding the proof We proceed by induction: Law(B R1,..., B Rk ) Law( ˆB R1,..., ˆB Rk 1, ˆB Rk ) Law( ˆB R1,..., ˆB Rk 1, W k Rk ) Law(B R1,..., B Rk 1, W k Rk ) where W k and B are independent. Then we use induction to replace the other terms: Law(W 1 R1,..., W k 1 Rk 1, W k Rk ), where W 1,..., W k are independent Brownian sheets. Use the results of Khoshnevisan & Shi and Peres to conclude. Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 25 / 26

26 References Orey, S. & Pruitt, W.E. Sample functions of the N-parameter Wiener process. Ann. Probab. 1-1 (1973), Rosen, J. Self-intersections of random fields. Ann. Probab (1984), Peres, Y. Probability on Trees: An Introductory Climb. In: Lectures on Probability Theory and Statistics, Saint-Flour (1997), Lect. Notes in Math. 1717, Springer, Berlin (1999), pp Khoshnevisan, D. Multiparameter processes. An introduction to random fields. Springer-Verlag, New York, Khoshnevisan, D. & Shi, Z. Brownian sheet and capacity. Ann. Probab (1999), Dalang, R.C., Khoshnevisan, D., Nualart, E., Wu, D. & Xiao, Y. Critical Brownian sheet does not have double points. Ann. Probab (2012), Dalang, R.C. & Mueller, C. Multiple points of the Brownian sheet in critical dimensions. Ann. Probab. (2014?). Multiple points of the Brownian sheet in critical dimensions Robert C. Dalang 26 / 26