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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Brownian motion with variable drift can have isolated zeros

Julia Ruscher

Technical University of Berlin

University of Warwick January 2011

Joint work with Antunovi´ c, Burdzy and Peres.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Outline

1 Introduction 2 Isolated zeros 3 Hausdorff dimension

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Theorem Standard one-dimensional Brownian motion has no isolated zeros almost surely. Question: Can Brownian motion with drift have isolated zeros?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Theorem Standard one-dimensional Brownian motion has no isolated zeros almost surely. Question: Can Brownian motion with drift have isolated zeros?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Let B(t) be a one-dimensional Brownian motion and f : [0, 1] → R be a continuous function. If f (t) = t

0 g(s)ds, g ∈ L2[0, 1] laws of B(t) − f (t) and B(t)

are mutually absolutely continuous. (Cameron-Martin theorem) ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L2).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Let B(t) be a one-dimensional Brownian motion and f : [0, 1] → R be a continuous function. If f (t) = t

0 g(s)ds, g ∈ L2[0, 1] laws of B(t) − f (t) and B(t)

are mutually absolutely continuous. (Cameron-Martin theorem) ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L2).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Let B(t) be a one-dimensional Brownian motion and f : [0, 1] → R be a continuous function. If f (t) = t

0 g(s)ds, g ∈ L2[0, 1] laws of B(t) − f (t) and B(t)

are mutually absolutely continuous. (Cameron-Martin theorem) ⇒ B − f has no isolated zeros almost surely if f is in Dirichlet space (integrals of functions in L2).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Outline

1 Introduction 2 Isolated zeros 3 Hausdorff dimension

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Classical fact: One-dimensional Brownian motion has no isolated zeros almost surely. Proof. For zeros of the form τq = min{t ≥ q : B(t) = 0}, for some q ∈ Q use strong Markov property. For a zero z not of the form τq there is a sequence (qn) ⊂ Q so that τqn ↑ z. Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Classical fact: One-dimensional Brownian motion has no isolated zeros almost surely. Proof. For zeros of the form τq = min{t ≥ q : B(t) = 0}, for some q ∈ Q use strong Markov property. For a zero z not of the form τq there is a sequence (qn) ⊂ Q so that τqn ↑ z. Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Classical fact: One-dimensional Brownian motion has no isolated zeros almost surely. Proof. For zeros of the form τq = min{t ≥ q : B(t) = 0}, for some q ∈ Q use strong Markov property. For a zero z not of the form τq there is a sequence (qn) ⊂ Q so that τqn ↑ z. Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Classical fact: One-dimensional Brownian motion has no isolated zeros almost surely. Proof. For zeros of the form τq = min{t ≥ q : B(t) = 0}, for some q ∈ Q use strong Markov property. For a zero z not of the form τq there is a sequence (qn) ⊂ Q so that τqn ↑ z. Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely. The above result is sharp! Theorem (Antunovi´ c, Burdzy, Peres, R.) For every γ < 1/2 there is a γ-H¨

• lder continuous function f

such that, with positive probability B(t) − f (t) has isolated zeros.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Isolated zeros

Proposition (Antunovi´ c, Burdzy, Peres, R.) If f is 1/2-H¨

• lder continuous, the process B(t) − f (t) has no

isolated zeros almost surely. The above result is sharp! Theorem (Antunovi´ c, Burdzy, Peres, R.) For every γ < 1/2 there is a γ-H¨

• lder continuous function f

such that, with positive probability B(t) − f (t) has isolated zeros.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

How to construct such functions?

t0 f(t) − f(t0) >> |t − t0|1/2 All isolated zeros of B(t) − f (t) are contained in the set

• t : lim

h↓0

f (t + h) − f (t) h1/2 = ±∞

• 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).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

How to construct such functions?

t0 f(t) − f(t0) >> |t − t0|1/2 All isolated zeros of B(t) − f (t) are contained in the set

• t : lim

h↓0

f (t + h) − f (t) h1/2 = ±∞

• 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).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

How to construct such functions?

t0 f(t) − f(t0) >> |t − t0|1/2 All isolated zeros of B(t) − f (t) are contained in the set

• t : lim

h↓0

f (t + h) − f (t) h1/2 = ±∞

• 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).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

How to construct such functions?

t0 f(t) − f(t0) >> |t − t0|1/2 All isolated zeros of B(t) − f (t) are contained in the set

• t : lim

h↓0

f (t + h) − f (t) h1/2 = ±∞

• 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).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

β β 1 2 1 1/2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

β2 β2 β2 β2 1 2 1 1/2 3/4 1/4

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

1 2 1

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

1 2 1

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

1 2 1

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Cantor function fβ

1 2 1 Cantor function fβ is

log 2 log(1/β)-H¨

• lder continuous.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

Theorem (Antunovi´ c, Burdzy, Peres, R.) For β < 1/4 the process B(t) − fβ(t) has isolated zeros with positive probability.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

Theorem (Antunovi´ c, Burdzy, Peres, R.) For β < 1/4 the process B(t) − fβ(t) has isolated zeros with positive probability. Proposition (Antunovi´ c, Burdzy, Peres, R.) The probability that B(t) − fβ(t) = 0 for some t ∈ Cantor(β) is positive if and only if β = 1/4.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

How to use the proposition to prove the theorem? Watch out for the red parts of the graph!

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

How to use the proposition to prove the theorem? Watch out for the red parts of the graph!

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

How to use the proposition to prove the theorem? Watch out for the red parts of the graph!

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

How to use the proposition to prove the theorem? Watch out for the red parts of the graph!

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

How to use the proposition to prove the theorem? Watch out for the red parts of the graph!

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

Proposition (Antunovi´ c, Burdzy, Peres, R.) The probability that B(t) − fβ(t) = 0 for some t ∈ Cantor(β) is positive if and only if β = 1/4. Let’s prove this in the case β = 1/4.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set

Proposition (Antunovi´ c, Burdzy, Peres, R.) The probability that B(t) − fβ(t) = 0 for some t ∈ Cantor(β) is positive if and only if β = 1/4. Let’s prove this in the case β = 1/4.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument. Count how many times the graph of Brownian motion hits blue vertical intervals.

Hits: 3

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument. Count how many times the graph of Brownian motion hits blue vertical intervals.

Hits: 2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument. Count how many times the graph of Brownian motion hits blue vertical intervals.

Hits: 5

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument. Count how many times the graph of Brownian motion hits blue vertical intervals.

Hits: 4

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Proof of the proposition for β = 1/4: Use a second moment argument. Count how many times the graph of Brownian motion hits blue vertical intervals.

Hits: 5

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Let Xn be the number of hits at level n. P(∃ zero of B(t)−f (t) in Cantor(β)) ≥ P(Xn > 0 infinitely often) Get a lower bound on P(Xn > 0) (independent of n) using the second moment method P(Xn > 0) ≥ E(Xn)2 E(X 2

n ) .

Key estimates: E(Xn) > c > 0 and E(X 2

n ) < C < ∞.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Let Xn be the number of hits at level n. P(∃ zero of B(t)−f (t) in Cantor(β)) ≥ P(Xn > 0 infinitely often) Get a lower bound on P(Xn > 0) (independent of n) using the second moment method P(Xn > 0) ≥ E(Xn)2 E(X 2

n ) .

Key estimates: E(Xn) > c > 0 and E(X 2

n ) < C < ∞.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Let Xn be the number of hits at level n. P(∃ zero of B(t)−f (t) in Cantor(β)) ≥ P(Xn > 0 infinitely often) Get a lower bound on P(Xn > 0) (independent of n) using the second moment method P(Xn > 0) ≥ E(Xn)2 E(X 2

n ) .

Key estimates: E(Xn) > c > 0 and E(X 2

n ) < C < ∞.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Let Xn be the number of hits at level n. P(∃ zero of B(t)−f (t) in Cantor(β)) ≥ P(Xn > 0 infinitely often) Get a lower bound on P(Xn > 0) (independent of n) using the second moment method P(Xn > 0) ≥ E(Xn)2 E(X 2

n ) .

Key estimates: E(Xn) > c > 0 and E(X 2

n ) < C < ∞.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Conditioned on hitting the red interval, the expected number of blue

• nes hit in each rectangle is bounded from below (Brownian scaling).

Use the first moment estimates.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Case β = 1/4 already appeared in Taylor & Watson (1985).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Case β = 1/4 already appeared in Taylor & Watson (1985).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Zeros in the Cantor set, β = 1/4

Case β = 1/4 already appeared in Taylor & Watson (1985).

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has isolated zeros almost surely.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has isolated zeros almost surely. 1 4−1 1 2−1

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has isolated zeros almost surely. 1 4−1 4−2 1 2−1 2−2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has isolated zeros almost surely. 1 4−1 4−2 1 2−1 2−2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has isolated zeros almost surely. 1 4−1 4−2 1 2−1 2−2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has only one zero with positive probability.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has only one zero with positive probability. q1 q2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has only one zero with positive probability. q1 q2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Some modifications

Continuous function f for which B(t) − f (t) has only one zero with positive probability. δ q1 q2

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Outline

1 Introduction 2 Isolated zeros 3 Hausdorff dimension

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension

For a set A ⊂ R and 0 < α < ∞: Hα(A) := lim

δ↓0 inf

• k

rα

k :

• k

B(xk, rk) ⊃ A, rk < δ 2

• where B(xk, rk) is an open ball with center xk and radius rk.

Then, the Hausdorff dimension of a set A is defined as dim(A) := inf {α > 0 : Hα(A) = 0} The zero set of one-dimensional Brownian motion has Hausdorff dimension equal to 1/2 almost surely. (Taylor, 55) How to estimate the Hausdorff dimension of the zero set of B(t) − f (t)?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension

For a set A ⊂ R and 0 < α < ∞: Hα(A) := lim

δ↓0 inf

• k

rα

k :

• k

B(xk, rk) ⊃ A, rk < δ 2

• where B(xk, rk) is an open ball with center xk and radius rk.

Then, the Hausdorff dimension of a set A is defined as dim(A) := inf {α > 0 : Hα(A) = 0} The zero set of one-dimensional Brownian motion has Hausdorff dimension equal to 1/2 almost surely. (Taylor, 55) How to estimate the Hausdorff dimension of the zero set of B(t) − f (t)?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension

For a set A ⊂ R and 0 < α < ∞: Hα(A) := lim

δ↓0 inf

• k

rα

k :

• k

B(xk, rk) ⊃ A, rk < δ 2

• where B(xk, rk) is an open ball with center xk and radius rk.

Then, the Hausdorff dimension of a set A is defined as dim(A) := inf {α > 0 : Hα(A) = 0} The zero set of one-dimensional Brownian motion has Hausdorff dimension equal to 1/2 almost surely. (Taylor, 55) How to estimate the Hausdorff dimension of the zero set of B(t) − f (t)?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

General strategy: Find a good (random) cover for the zero set of B(t) − f (t). Subdivide the interval [1, 2] into subintervals of equal length. Construct the cover U using the intervals that contain zeros of B(t) − f (t). Try to get an upper bound on E(

U∈U diam(U)α)

independent of the length of subintervals.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

General strategy: Find a good (random) cover for the zero set of B(t) − f (t). Subdivide the interval [1, 2] into subintervals of equal length. Construct the cover U using the intervals that contain zeros of B(t) − f (t). Try to get an upper bound on E(

U∈U diam(U)α)

independent of the length of subintervals.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

General strategy: Find a good (random) cover for the zero set of B(t) − f (t). Subdivide the interval [1, 2] into subintervals of equal length. Construct the cover U using the intervals that contain zeros of B(t) − f (t). Try to get an upper bound on E(

U∈U diam(U)α)

independent of the length of subintervals.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

General strategy: Find a good (random) cover for the zero set of B(t) − f (t). Subdivide the interval [1, 2] into subintervals of equal length. Construct the cover U using the intervals that contain zeros of B(t) − f (t). Try to get an upper bound on E(

U∈U diam(U)α)

independent of the length of subintervals.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

General strategy: Find a good (random) cover for the zero set of B(t) − f (t). Subdivide the interval [1, 2] into subintervals of equal length. Construct the cover U using the intervals that contain zeros of B(t) − f (t). Try to get an upper bound on E(

U∈U diam(U)α)

independent of the length of subintervals.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

Theorem (Antunovi´ c, Burdzy, Peres, R.) Almost surely the Hausdorff dimension of the zero set of B(t) − f (t) is less or equal than 1/2 if either of the following holds f is 1/2-H¨

• lder continuous,

f is of bounded variation. How important are these conditions?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

Theorem (Antunovi´ c, Burdzy, Peres, R.) Almost surely the Hausdorff dimension of the zero set of B(t) − f (t) is less or equal than 1/2 if either of the following holds f is 1/2-H¨

• lder continuous,

f is of bounded variation. How important are these conditions?

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

B(H)(t) fractional Brownian motion with Hurst index 0 < H < 1 (continuous Gaussian process with B(H)(0) = 0 and E(|Bt − Bs|2) = |t − s|2H) Classical result: dim(zero(B(H))) = 1 − H. If H < 1/2 and B and B(H) are independent then the Hausdorff dimension of the zero set of B(t) − B(H)(t) is 1 − H a.s.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

B(H)(t) fractional Brownian motion with Hurst index 0 < H < 1 (continuous Gaussian process with B(H)(0) = 0 and E(|Bt − Bs|2) = |t − s|2H) Classical result: dim(zero(B(H))) = 1 − H. If H < 1/2 and B and B(H) are independent then the Hausdorff dimension of the zero set of B(t) − B(H)(t) is 1 − H a.s.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - upper bounds

B(H)(t) fractional Brownian motion with Hurst index 0 < H < 1 (continuous Gaussian process with B(H)(0) = 0 and E(|Bt − Bs|2) = |t − s|2H) Classical result: dim(zero(B(H))) = 1 − H. If H < 1/2 and B and B(H) are independent then the Hausdorff dimension of the zero set of B(t) − B(H)(t) is 1 − H a.s.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Hausdorff dimension - lower bounds

Proposition (A general lower bound – Antunovi´ c, Burdzy, Peres, R.) For any continuous function f , with positive probability the zero set of the process B(t) − f (t) has Hausdorff dimension at least 1/2.

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Brownian motion with variable drift can have isolated zeros Julia Ruscher Introduction Isolated zeros Hausdorff dimension

Thank you!

Thank you!

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