Brownian motion with variable Brownian motion with variable drift can have drift can have isolated zeros isolated zeros Julia Ruscher Introduction Julia Ruscher Isolated zeros Technical University of Berlin Hausdorff dimension University of Warwick January 2011 Joint work with Antunovi´ c, Burdzy and Peres. 1 / 25
Outline Brownian motion with variable drift can have isolated zeros 1 Introduction Julia Ruscher Introduction Isolated 2 Isolated zeros zeros Hausdorff dimension 3 Hausdorff dimension 2 / 25
Brownian motion with variable drift can have isolated zeros Theorem Julia Ruscher Standard one-dimensional Brownian motion has no isolated Introduction zeros almost surely. Isolated zeros Hausdorff dimension Question: Can Brownian motion with drift have isolated zeros? 3 / 25
Brownian motion with variable drift can have isolated zeros Theorem Julia Ruscher Standard one-dimensional Brownian motion has no isolated Introduction zeros almost surely. Isolated zeros Hausdorff dimension Question: Can Brownian motion with drift have isolated zeros? 3 / 25
Brownian motion with variable drift can have isolated zeros Let B ( t ) be a one-dimensional Brownian motion and Julia f : [0 , 1] → R be a continuous function. Ruscher � t Introduction 0 g ( s ) ds , g ∈ L 2 [0 , 1] laws of B ( t ) − f ( t ) and B ( t ) If f ( t ) = Isolated are mutually absolutely continuous. (Cameron-Martin theorem) zeros Hausdorff dimension ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L 2 ). 4 / 25
Brownian motion with variable drift can have isolated zeros Let B ( t ) be a one-dimensional Brownian motion and Julia f : [0 , 1] → R be a continuous function. Ruscher � t Introduction 0 g ( s ) ds , g ∈ L 2 [0 , 1] laws of B ( t ) − f ( t ) and B ( t ) If f ( t ) = Isolated are mutually absolutely continuous. (Cameron-Martin theorem) zeros Hausdorff dimension ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L 2 ). 4 / 25
Brownian motion with variable drift can have isolated zeros Let B ( t ) be a one-dimensional Brownian motion and Julia f : [0 , 1] → R be a continuous function. Ruscher � t Introduction 0 g ( s ) ds , g ∈ L 2 [0 , 1] laws of B ( t ) − f ( t ) and B ( t ) If f ( t ) = Isolated are mutually absolutely continuous. (Cameron-Martin theorem) zeros Hausdorff dimension ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L 2 ). 4 / 25
Outline Brownian motion with variable drift can have isolated zeros 1 Introduction Julia Ruscher Introduction Isolated 2 Isolated zeros zeros Hausdorff dimension 3 Hausdorff dimension 5 / 25
Isolated zeros Brownian motion with variable Classical fact: One-dimensional Brownian motion has no drift can have isolated isolated zeros almost surely. zeros Julia Proof. Ruscher For zeros of the form τ q = min { t ≥ q : B ( t ) = 0 } , for some Introduction q ∈ Q use strong Markov property. For a zero z not of the Isolated zeros form τ q there is a sequence ( q n ) ⊂ Q so that τ q n ↑ z . Hausdorff dimension Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no isolated zeros almost surely. 6 / 25
Isolated zeros Brownian motion with variable Classical fact: One-dimensional Brownian motion has no drift can have isolated isolated zeros almost surely. zeros Julia Proof. Ruscher For zeros of the form τ q = min { t ≥ q : B ( t ) = 0 } , for some Introduction q ∈ Q use strong Markov property. For a zero z not of the Isolated zeros form τ q there is a sequence ( q n ) ⊂ Q so that τ q n ↑ z . Hausdorff dimension Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no isolated zeros almost surely. 6 / 25
Isolated zeros Brownian motion with variable Classical fact: One-dimensional Brownian motion has no drift can have isolated isolated zeros almost surely. zeros Julia Proof. Ruscher For zeros of the form τ q = min { t ≥ q : B ( t ) = 0 } , for some Introduction q ∈ Q use strong Markov property. For a zero z not of the Isolated zeros form τ q there is a sequence ( q n ) ⊂ Q so that τ q n ↑ z . Hausdorff dimension Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no isolated zeros almost surely. 6 / 25
Isolated zeros Brownian motion with variable Classical fact: One-dimensional Brownian motion has no drift can have isolated isolated zeros almost surely. zeros Julia Proof. Ruscher For zeros of the form τ q = min { t ≥ q : B ( t ) = 0 } , for some Introduction q ∈ Q use strong Markov property. For a zero z not of the Isolated zeros form τ q there is a sequence ( q n ) ⊂ Q so that τ q n ↑ z . Hausdorff dimension Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no isolated zeros almost surely. 6 / 25
Isolated zeros Brownian motion with variable drift can Proposition (Antunovi´ c, Burdzy, Peres, R.) have isolated zeros If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no Julia Ruscher isolated zeros almost surely. Introduction The above result is sharp! Isolated zeros Hausdorff dimension Theorem (Antunovi´ c, Burdzy, Peres, R.) For every γ < 1 / 2 there is a γ -H¨ older continuous function f such that, with positive probability B ( t ) − f ( t ) has isolated zeros. 7 / 25
Isolated zeros Brownian motion with variable drift can Proposition (Antunovi´ c, Burdzy, Peres, R.) have isolated zeros If f is 1 / 2 -H¨ older continuous, the process B ( t ) − f ( t ) has no Julia Ruscher isolated zeros almost surely. Introduction The above result is sharp! Isolated zeros Hausdorff dimension Theorem (Antunovi´ c, Burdzy, Peres, R.) For every γ < 1 / 2 there is a γ -H¨ older continuous function f such that, with positive probability B ( t ) − f ( t ) has isolated zeros. 7 / 25
How to construct such functions? Brownian motion with variable f ( t ) − f ( t 0 ) >> | t − t 0 | 1 / 2 drift can have isolated zeros Julia Ruscher t 0 Introduction Isolated All isolated zeros of B ( t ) − f ( t ) are contained in the set zeros Hausdorff � f ( t + h ) − f ( t ) � dimension t : lim = ±∞ h 1 / 2 h ↓ 0 Proposition (Antunovi´ c, Burdzy, Peres, R.) For a given f there is a (deterministic) set of Hausdorff dimension at most 1 / 2 that contain all isolated zeros of B ( t ) − f ( t ) . 8 / 25
How to construct such functions? Brownian motion with variable f ( t ) − f ( t 0 ) >> | t − t 0 | 1 / 2 drift can have isolated zeros Julia Ruscher t 0 Introduction Isolated All isolated zeros of B ( t ) − f ( t ) are contained in the set zeros Hausdorff � f ( t + h ) − f ( t ) � dimension t : lim = ±∞ h 1 / 2 h ↓ 0 Proposition (Antunovi´ c, Burdzy, Peres, R.) For a given f there is a (deterministic) set of Hausdorff dimension at most 1 / 2 that contain all isolated zeros of B ( t ) − f ( t ) . 8 / 25
How to construct such functions? Brownian motion with variable f ( t ) − f ( t 0 ) >> | t − t 0 | 1 / 2 drift can have isolated zeros Julia Ruscher t 0 Introduction Isolated All isolated zeros of B ( t ) − f ( t ) are contained in the set zeros Hausdorff � f ( t + h ) − f ( t ) � dimension t : lim = ±∞ h 1 / 2 h ↓ 0 Proposition (Antunovi´ c, Burdzy, Peres, R.) For a given f there is a (deterministic) set of Hausdorff dimension at most 1 / 2 that contain all isolated zeros of B ( t ) − f ( t ) . 8 / 25
How to construct such functions? Brownian motion with variable f ( t ) − f ( t 0 ) >> | t − t 0 | 1 / 2 drift can have isolated zeros Julia Ruscher t 0 Introduction Isolated All isolated zeros of B ( t ) − f ( t ) are contained in the set zeros Hausdorff � f ( t + h ) − f ( t ) � dimension t : lim = ±∞ h 1 / 2 h ↓ 0 Proposition (Antunovi´ c, Burdzy, Peres, R.) For a given f there is a (deterministic) set of Hausdorff dimension at most 1 / 2 that contain all isolated zeros of B ( t ) − f ( t ) . 8 / 25
Cantor function f β Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension 9 / 25
Cantor function f β Brownian 1 motion with variable drift can have isolated zeros Julia Ruscher 1 / 2 Introduction Isolated zeros Hausdorff dimension 1 2 β β 9 / 25
Cantor function f β Brownian 1 motion with variable drift can have isolated 3 / 4 zeros Julia Ruscher 1 / 2 Introduction Isolated 1 / 4 zeros Hausdorff dimension 1 2 β 2 β 2 β 2 β 2 9 / 25
Cantor function f β Brownian 1 motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension 1 2 9 / 25
Cantor function f β Brownian 1 motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension 1 2 9 / 25
Cantor function f β Brownian 1 motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension 1 2 9 / 25
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