Some properties of the true self-repelling motion

Transcription

1 Some properties of the true self-repelling motion L. Dumaz Statistical laboratory, University of Cambridge Bath-Paris branching structures meeting, June the 9th, / 20

2 What is a (real-valued) self-repelling process? If (X t ) t 0 is a one-dimensional continuous process, define its occupation time measure: For all t 0, for all borel set A, µ t (A) = t 1 {Xs A}ds 0 2 / 20

3 What is a (real-valued) self-repelling process? If (X t ) t 0 is a one-dimensional continuous process, define its occupation time measure: For all t 0, for all borel set A, µ t (A) = t 1 {Xs A}ds 0 Assumption: This measure has a density i.e. µ t ([x, x + ε]) ε 0 ε L t (x) 2 / 20

4 What is a (real-valued) self-repelling process? If (X t ) t 0 is a one-dimensional continuous process, define its occupation time measure: For all t 0, for all borel set A, µ t (A) = t 1 {Xs A}ds 0 Assumption: This measure has a density i.e. µ t ([x, x + ε]) ε 0 ε L t (x) L t : R R + is called local time. 2 / 20

5 What is a (real-valued) self-repelling process? Heuristical (vague) definition: (X t ) t is called self-repulsive when: (X t, L t ( )) t is a Markov process. (X t ) t prefers to go to the less visited places pushed away from x R where L t (x) is large. 3 / 20

6 Process with its local time Note that we can consider the dimensional process (X t, H t ) := (X t, L t (X t )) 4 / 20

7 Process with its local time Note that we can consider the dimensional process (X t, H t ) is a space filling curve. (X t, H t ) := (X t, L t (X t )) 4 / 20

8 Process with its local time Note that we can consider the dimensional process (X t, H t ) is a space filling curve. (X t, H t ) := (X t, L t (X t )) local time H t := L t (X t ) Time t L t (x) position X t x Figure : Picture at time t: X t and its local time L t ( ) 4 / 20

9 A first example: Brownian polymers Brownian polymers were introduced by Durrett and Rogers in dx t = (b L t (X t + )) 0 dt + db t 5 / 20

10 A first example: Brownian polymers Brownian polymers were introduced by Durrett and Rogers in dx t = (b L t (X t + )) 0 dt + db t Possible case: b = b (ε) is an approximation of the Dirac function at 0. dx t = (b (ε) L t (X t + )) 0 dt + db t 5 / 20

11 A first example: Brownian polymers Brownian polymers were introduced by Durrett and Rogers in dx t = (b L t (X t + )) 0 dt + db t Possible case: b = b (ε) is an approximation of the Dirac function at 0. dx t = (b (ε) L t (X t + )) 0 dt + db t What happens when ε 0? Should converge to the true self-repelling motion (introduced after). 5 / 20

12 A first example: Brownian polymers Time t L t (x) position X t x Figure : Picture at time t: X t and its local time L t ( ) 6 / 20

13 A first example: Brownian polymers Time t L t (x) position X t x Figure : Picture at time t: X t and its local time L t ( ) 7 / 20

14 TSRM definition and first properties Results of the papers of B. Toth - W. Werner 1998 and F. Soucaliuc - B. Toth - W. Werner / 20

15 Short introduction to the true self-repelling motion (TSRM) B. Tóth and W. Werner constructed a one-dimensional continuous self-repelling process (X t, t 0) called the true self-repelling motion (TSRM). 9 / 20

16 Short introduction to the true self-repelling motion (TSRM) B. Tóth and W. Werner constructed a one-dimensional continuous self-repelling process (X t, t 0) called the true self-repelling motion (TSRM). For the construction, they used a family of coalescing reflected Brownian motions in the upper half plane now called Brownian Web. The TSRM is defined as the trace of the contour of the tree of these coalescing Brownian motions. 9 / 20

17 TSRM construction Take the Brownian Web (Λ x,h, (x, h) R R + ). 10 / 20

18 TSRM construction Take the Brownian Web (Λ x,h, (x, h) R R + ). Let us consider the process (X t, H t ) starting at (0, 0) which traces the contour of the tree of these coalescing Brownian motions. space-filling curve. 10 / 20

19 TSRM construction Take the Brownian Web (Λ x,h, (x, h) R R + ). Let us consider the process (X t, H t ) starting at (0, 0) which traces the contour of the tree of these coalescing Brownian motions. space-filling curve. Parametrization: by the area it has swept. For every (x, h) in the upper half plane, the process (X t, H t ) visits the point (x, h) at the random time t = T x,h := R Λ x,h(y)dy TSRM = first coordinate (X t ). 10 / 20

20 Some first properties The TSRM is unusual compared to more classical processes. 11 / 20

21 Some first properties The TSRM is unusual compared to more classical processes. First properties (etablished by Bálint Tóth and Wendelin Werner): Continuity and recurrence. Scaling and local variation: For all a > 0, (X at, t 0) and (a 2/3 X t, t 0) have the same distribution and the TSRM is of finite variation of order 3/2. 11 / 20

22 Some first properties The TSRM is unusual compared to more classical processes. First properties (etablished by Bálint Tóth and Wendelin Werner): Continuity and recurrence. Scaling and local variation: For all a > 0, (X at, t 0) and (a 2/3 X t, t 0) have the same distribution and the TSRM is of finite variation of order 3/2. Local time: The TSRM admits a local time L t ( ) and a.s., for every (x, h) R R +, the Brownian Web curves corresponds to the local time at times T x,h (strong Ray Knight theorem). It implies H t = L t (X t ). 11 / 20

23 Some first properties Markov property: (X t, L t ( )) t is a Markov process. 12 / 20

24 Some first properties Markov property: (X t, L t ( )) t is a Markov process. Localization: Interaction is local: the law of X just after t depends only on L t around the point X t. Moreover, we have a dynamical equation: 1 dx t = lim ε 0 2ε (L t(x t + ε) L t (X t ε)) dt. Limit holds in the probability sense. 12 / 20

25 My contributions 13 / 20

26 Large deviations of the TSRM TSRM is an unusual process and it gives motivation to study some of its finest properties to discover the features it shares/does not share with the other processes. Proposition (L.D.) When x, P(X 1 > x) = exp( κx 3 + O(ln(x))) for some explicit κ (in terms of zeros of Airy function). When h, P(H 1 > h) = exp( 8h 3 /9 + O(ln(h))). 14 / 20

27 Law of the iterated logarithm Pushing forward those results permits to derive a LIL for the TSRM when both t is large and t is small: Proposition (L.D.) a.s., lim sup t 0 t 2/3 (ln(ln(1/t))) 1/3 X t = 1/κ 1/3. 15 / 20

28 Marginal distributions In a joint work with Bálint Tóth, we computed the marginal distributions of this process. Proposition (L.D., B. Tóth) The density of X 1 denoted by ν 1 (x) is equal to: ν 1 (x) = k=1 3 2/3 2 7/3 ( ) Γ(2/3) 2 a k Γ(1/3) 3 f 2/3 (2 1/3 a k x ) where the scaling factors a k are the zeros of the derivative of the Airy function and f 2/3 is the Mittag-Leffler s function. The density of H 1 denoted by ν 2 (h) is equal to: ν 2 (h) = 2 61/3 π Γ(1/3) 2 exp( (8h 3 )/9)U(1/6, 2/3; (8h 3 )/9) where U is the hypergeometric function. 16 / 20

29 Marginal distributions Figure : Density of X 1 (displacement at time 1) 17 / 20

30 A clever (self-repelling) burglar What is the conditional law of the position X 1 knowing L 1 ( )? Figure : On the left (X t, t [0, 1]), and on the right the local time L 1 ( ) 18 / 20

31 A clever (self-repelling) burglar What is the conditional law of the position X 1 knowing L 1 ( )? Figure : On the left (X t, t [0, 1]), and on the right the local time L 1 ( ) 18 / 20

32 A clever (self-repelling) burglar What is the conditional law of the position X 1 knowing L 1 ( )? Fast points Figure : On the left (X t, t [0, 1]), and on the right the local time L 1 ( ) 18 / 20

33 Result Proposition (L.D.) The conditional law of X 1 knowing its local time at time 1, L 1 ( ), is uniform on the interval I defined by: Figure : Definition of I 19 / 20

34 Thank you! 20 / 20