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The Wonderful World of Brownian Functionals

Satya N. Majumdar

Laboratoire de Physique Th´ eorique et Mod` eles Statistiques,CNRS, Universit´ e Paris-Sud, France

July 11, 2011

S.N. Majumdar The Wonderful World of Brownian Functionals

Plan

Plan:

• Brownian motion via Path Integral

S.N. Majumdar The Wonderful World of Brownian Functionals

Plan

Plan:

• Brownian motion via Path Integral
• Brownian functionals → Feynman-Kac formula

Several applications: fluctuating (1 + 1)-dimensional interfaces

S.N. Majumdar The Wonderful World of Brownian Functionals

Plan

Plan:

• Brownian motion via Path Integral
• Brownian functionals → Feynman-Kac formula

Several applications: fluctuating (1 + 1)-dimensional interfaces

• First-passage Brownian functional =

⇒ applications

S.N. Majumdar The Wonderful World of Brownian Functionals

Plan

Plan:

• Brownian motion via Path Integral
• Brownian functionals → Feynman-Kac formula

Several applications: fluctuating (1 + 1)-dimensional interfaces

• First-passage Brownian functional =

⇒ applications

• Summary and Conclusion

S.N. Majumdar The Wonderful World of Brownian Functionals

Random walk

n xn xn xn−1 + ηn

=

• Random walk: xn = xn−1 + ηn starting from x0 = 0

S.N. Majumdar The Wonderful World of Brownian Functionals

Random walk

n xn xn xn−1 + ηn

=

• Random walk: xn = xn−1 + ηn starting from x0 = 0

where ηn → i.i.d Gaussian noise with ηn = 0 and ηnηm = σ2δn,m

S.N. Majumdar The Wonderful World of Brownian Functionals

Random walk

n xn xn xn−1 + ηn

=

• Random walk: xn = xn−1 + ηn starting from x0 = 0

where ηn → i.i.d Gaussian noise with ηn = 0 and ηnηm = σ2δn,m = ⇒ x2

n = σ2 n

S.N. Majumdar The Wonderful World of Brownian Functionals

Probability of a Random Walk Path

n xn xn xn−1 + ηn

=

• ηi’s → i.i.d Gaussian variables

Joint distribution of the noise variables: Prob [{ηi}] ∝ exp

• −
• i

η2

i

2σ2

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Probability of a Random Walk Path

n xn xn xn−1 + ηn

=

• ηi’s → i.i.d Gaussian variables

Joint distribution of the noise variables: Prob [{ηi}] ∝ exp

• −
• i

η2

i

2σ2

• Probability of a path of the random walker:

Prob [{xi}] ∝ exp

• −
• i

(xi − xi−1)2 2σ2

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

= ⇒ x2(t) = 2Dt

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

= ⇒ x2(t) = 2Dt

• xi = xi−1 + ηi with ηiηj = σ2δi,j reduces to

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

= ⇒ x2(t) = 2Dt

• xi = xi−1 + ηi with ηiηj = σ2δi,j reduces to

dx dt = η(t) where η(t) = 0 and

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

= ⇒ x2(t) = 2Dt

• xi = xi−1 + ηi with ηiηj = σ2δi,j reduces to

dx dt = η(t) where η(t) = 0 and η(t)η(t′) = 0 if t = t′ = 2D

∆t if t = t′

S.N. Majumdar The Wonderful World of Brownian Functionals

Continuous-time Brownian Motion

• Recall x2

n = σ2n where n → no. of steps

Continuous-time: t = n∆t = ⇒ x2(t) = σ2

∆t t

• As ∆t → 0, we have to take σ2 → 0 keeping the ratio σ2

∆t = 2D fixed

= ⇒ x2(t) = 2Dt

• xi = xi−1 + ηi with ηiηj = σ2δi,j reduces to

dx dt = η(t) where η(t) = 0 and η(t)η(t′) = 0 if t = t′ = 2D

∆t if t = t′

= ⇒ η(t)η(t′) = 2Dδ(t − t′) → Gaussian white noise

S.N. Majumdar The Wonderful World of Brownian Functionals

Probability of a Brownian path:

x(τ)

τ

i

i

x xi = xi−

1+ηi

continuous−time limit

dτ dx = η(τ)

S.N. Majumdar The Wonderful World of Brownian Functionals

Probability of a Brownian path:

x(τ)

τ

i

i

x xi = xi−

1+ηi

continuous−time limit

dτ dx = η(τ)

• Probability of a path of a random walk:

Prob [{xi}] ∝ exp

• −
• i

(xi − xi−1)2 2σ2

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Probability of a Brownian path:

x(τ)

τ

i

i

x xi = xi−

1+ηi

continuous−time limit

dτ dx = η(τ)

• Probability of a path of a random walk:

Prob [{xi}] ∝ exp

• −
• i

(xi − xi−1)2 2σ2

• reduces in the continuous-time limit: σ2 = 2D∆t to

S.N. Majumdar The Wonderful World of Brownian Functionals

Probability of a Brownian path:

x(τ)

τ

i

i

x xi = xi−

1+ηi

continuous−time limit

dτ dx = η(τ)

• Probability of a path of a random walk:

Prob [{xi}] ∝ exp

• −
• i

(xi − xi−1)2 2σ2

• reduces in the continuous-time limit: σ2 = 2D∆t to

Prob [{x(τ)}] ∝ exp

• − 1

4D t dx dτ 2 dτ

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Free propagator

τ

x(τ)

t

x0 x

Free propagator: G(x, t|x0, 0) → Prob. density of reaching x at time τ = t starting from x0 at time τ = 0

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator

τ

x(τ)

t

x0 x

Free propagator: G(x, t|x0, 0) → Prob. density of reaching x at time τ = t starting from x0 at time τ = 0 satisfies diffusion equation ∂tG = D ∂2

xG with G(x, 0|x0, 0) = δ(x − x0)

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator

τ

x(τ)

t

x0 x

Free propagator: G(x, t|x0, 0) → Prob. density of reaching x at time τ = t starting from x0 at time τ = 0 satisfies diffusion equation ∂tG = D ∂2

xG with G(x, 0|x0, 0) = δ(x − x0)

G(x, t|x0, 0) = 1 √ 4πDt exp

• − 1

4Dt (x − x0)2

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

• In the eigenbasis of ˆ

HψE(x) = EψE(x) (Schr¨

• dinger equation)

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

• In the eigenbasis of ˆ

HψE(x) = EψE(x) (Schr¨

• dinger equation)
• x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

• In the eigenbasis of ˆ

HψE(x) = EψE(x) (Schr¨

• dinger equation)
• x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

• For free particle, (Schr¨
• dinger equation) reads:

−D∂2

xψE(x) = EψE(x) → ψk(x) = 1 √ 2πeikx with E = Dk2

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

• In the eigenbasis of ˆ

HψE(x) = EψE(x) (Schr¨

• dinger equation)
• x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

• For free particle, (Schr¨
• dinger equation) reads:

−D∂2

xψE(x) = EψE(x) → ψk(x) = 1 √ 2πeikx with E = Dk2

• Free propagator: G(x, t|x0, 0) = x|e− ˆ

H t|x0 =

dk

2π eik(x−x0) e−Dk2t

S.N. Majumdar The Wonderful World of Brownian Functionals

Free propagator via path integral

• G(x, t|x0, 0) =

x(t)=x

x(0)=x0

Dx(τ) exp

• − 1

4D t ˙ x2(τ)dτ

• identifying the action of a free quantum particle with mass m =

1 2D

• G(x, t|x0, 0) = x|e− ˆ

H t|x0 where ˆ

H ≡ ˆ

p2 2m ≡ −D∂2 x

• In the eigenbasis of ˆ

HψE(x) = EψE(x) (Schr¨

• dinger equation)
• x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

• For free particle, (Schr¨
• dinger equation) reads:

−D∂2

xψE(x) = EψE(x) → ψk(x) = 1 √ 2πeikx with E = Dk2

• Free propagator: G(x, t|x0, 0) = x|e− ˆ

H t|x0 =

dk

2π eik(x−x0) e−Dk2t

=

1 √ 4πDt exp

• − 1

4Dt (x − x0)2

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional

• Brownian path x(τ) propagating from x0 at τ = 0 to x(t) = x at τ = t

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional

• Brownian path x(τ) propagating from x0 at τ = 0 to x(t) = x at τ = t
• Brownian functional: T =

t U (x(τ)) dτ) where U(x) → arbitrary

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional

• Brownian path x(τ) propagating from x0 at τ = 0 to x(t) = x at τ = t
• Brownian functional: T =

t U (x(τ)) dτ) where U(x) → arbitrary

• Clearly T → random variable–varies from path to path

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional

• Brownian path x(τ) propagating from x0 at τ = 0 to x(t) = x at τ = t
• Brownian functional: T =

t U (x(τ)) dτ) where U(x) → arbitrary

• Clearly T → random variable–varies from path to path

Q: What is its probability distribution P(T|x, t, x0) ?

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional

• Brownian path x(τ) propagating from x0 at τ = 0 to x(t) = x at τ = t
• Brownian functional: T =

t U (x(τ)) dτ) where U(x) → arbitrary

• Clearly T → random variable–varies from path to path

Q: What is its probability distribution P(T|x, t, x0) ?

• Example: U(x) = x → T =

t

0 x(τ)dτ → area under the path

τ

x(τ)

t

x0 x

T=Area under the curve

S.N. Majumdar The Wonderful World of Brownian Functionals

Brownian functional: Feynman-Kac formula

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

where ˆ H ≡ ˆ

p2 2m + sU(x) ≡ −D∂2 x + sU(x)

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

where ˆ H ≡ ˆ

p2 2m + sU(x) ≡ −D∂2 x + sU(x)

Functional sU(x) → quantum potential in Shr¨

• dinger equation

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

where ˆ H ≡ ˆ

p2 2m + sU(x) ≡ −D∂2 x + sU(x)

Functional sU(x) → quantum potential in Shr¨

• dinger equation
• 1. Solve (Schr¨
• dinger equation): [−D∂2

x + sU(x)]ψE(x) = EψE(x)

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

where ˆ H ≡ ˆ

p2 2m + sU(x) ≡ −D∂2 x + sU(x)

Functional sU(x) → quantum potential in Shr¨

• dinger equation
• 1. Solve (Schr¨
• dinger equation): [−D∂2

x + sU(x)]ψE(x) = EψE(x)

2. ˜ P(x, t|x0, s) = x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

S.N. Majumdar The Wonderful World of Brownian Functionals

Feynman-Kac formula: main idea

• Prob. distribution of T =

t

0 U(x(τ))dτ is:

P(T|x, t, x0) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ δ

• T −

t U(x(τ))dτ

• Taking Laplace transform: ˜

P(x, t|x0, s) = ∞ P(T|x, t, x0) e−sT dT

• ˜

P(x, t|x0, s) ∝ x(t)=x

x(0)=x0

Dx(τ)e− 1

4D

t

0 ˙

x2(τ)dτ−s t

0 U(x(τ))dτ= x|e− ˆ

Ht|x0

where ˆ H ≡ ˆ

p2 2m + sU(x) ≡ −D∂2 x + sU(x)

Functional sU(x) → quantum potential in Shr¨

• dinger equation
• 1. Solve (Schr¨
• dinger equation): [−D∂2

x + sU(x)]ψE(x) = EψE(x)

2. ˜ P(x, t|x0, s) = x|e− ˆ

H t|x0 =

• E

ψE(x)ψ∗

E(x0) e−Et

• 3. Finally invert the Laplace transform ˜

P(x, t|x0, s) w.r.t s

S.N. Majumdar The Wonderful World of Brownian Functionals

Examples of Brownian functionals

• Residence (Occupation) time: T =

t θ(x(τ))dτ → U(x) = θ(x) P(T, t|x0 = 0) = 1 π 1

• T(t − T)

(L´

evy, ’39, Lamperti, ’57)

important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ...

(Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....)

S.N. Majumdar The Wonderful World of Brownian Functionals

Examples of Brownian functionals

• Residence (Occupation) time: T =

t θ(x(τ))dτ → U(x) = θ(x) P(T, t|x0 = 0) = 1 π 1

• T(t − T)

(L´

evy, ’39, Lamperti, ’57)

important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ...

(Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....)

• Finance: stock price S(τ) = e−βx(τ) where x(τ) → Brownian motion

(Black-Scholes model, 1973)

S.N. Majumdar The Wonderful World of Brownian Functionals

Examples of Brownian functionals

• Residence (Occupation) time: T =

t θ(x(τ))dτ → U(x) = θ(x) P(T, t|x0 = 0) = 1 π 1

• T(t − T)

(L´

evy, ’39, Lamperti, ’57)

important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ...

(Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....)

• Finance: stock price S(τ) = e−βx(τ) where x(τ) → Brownian motion

(Black-Scholes model, 1973) Integrated stock price: T = t

0 e−βx(τ)dτ → U(x) = e−βx (M. Yor, ...)

S.N. Majumdar The Wonderful World of Brownian Functionals

Examples of Brownian functionals

• Residence (Occupation) time: T =

t θ(x(τ))dτ → U(x) = θ(x) P(T, t|x0 = 0) = 1 π 1

• T(t − T)

(L´

evy, ’39, Lamperti, ’57)

important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ...

(Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....)

• Finance: stock price S(τ) = e−βx(τ) where x(τ) → Brownian motion

(Black-Scholes model, 1973) Integrated stock price: T = t

0 e−βx(τ)dτ → U(x) = e−βx (M. Yor, ...)

T → partition function of the Sinai model in a box [0, t]

S.N. Majumdar The Wonderful World of Brownian Functionals

Examples of Brownian functionals

• Residence (Occupation) time: T =

t θ(x(τ))dτ → U(x) = θ(x) P(T, t|x0 = 0) = 1 π 1

• T(t − T)

(L´

evy, ’39, Lamperti, ’57)

important observable with many recent applications: nonequilibrium dynamics of growing domains, ergodicity properties in anomalous diffusion, transport in disordered systems, blinking quantum dots ...

(Barkai, Bouchaud, Bray, Comtet, Godr` eche, Luck, S.M., Monthus, ....)

• Finance: stock price S(τ) = e−βx(τ) where x(τ) → Brownian motion

(Black-Scholes model, 1973) Integrated stock price: T = t

0 e−βx(τ)dτ → U(x) = e−βx (M. Yor, ...)

T → partition function of the Sinai model in a box [0, t]

• Other examples: “Brownian functionals in physics and computer

science” [review by S.M. in Current Science, (2005), cond-mat/0510064]

S.N. Majumdar The Wonderful World of Brownian Functionals

Fluctuating (1 + 1)-dimensional interfaces

• Fluctuating steps on crystals such as Si, Al(111),..
• diploar chains of non-magnetic particles in a ferrofluid ...

S.N. Majumdar The Wonderful World of Brownian Functionals

Fluctuating (1 + 1)-dimensional interface

L x H(x,t)

Fluctuating Interface height

S.N. Majumdar The Wonderful World of Brownian Functionals

Fluctuating (1 + 1)-dimensional interface

L x H(x,t)

Fluctuating Interface height

• Fluctuating interfaces → very well studied

∂tH = ∂2

xH + λ (∂xH)2 + η(x, t) (Kardar, Parisi, Zhang, 1986)

S.N. Majumdar The Wonderful World of Brownian Functionals

Fluctuating (1 + 1)-dimensional interface

L x H(x,t)

Fluctuating Interface height

• Fluctuating interfaces → very well studied

∂tH = ∂2

xH + λ (∂xH)2 + η(x, t) (Kardar, Parisi, Zhang, 1986)

• λ = 0 → Linear interface (Hammersley, ’66, Edwards-Wilkinson, ’81)

S.N. Majumdar The Wonderful World of Brownian Functionals

• Relevant object that measures fluctuations

h(x, t) = H(x, t) − 1 L L H(x, t)dx → relative height

S.N. Majumdar The Wonderful World of Brownian Functionals

• Relevant object that measures fluctuations

h(x, t) = H(x, t) − 1 L L H(x, t)dx → relative height By definition: L h(x, t)dx = 0

S.N. Majumdar The Wonderful World of Brownian Functionals

• Relevant object that measures fluctuations

h(x, t) = H(x, t) − 1 L L H(x, t)dx → relative height By definition: L h(x, t)dx = 0

• Width of the interface:

w(t, L) =

• h2(x, t) ∼ tβ for ξ(t) ∼ t1/z << L

(growing regime) ∼ L1/2 for ξ(t) ∼ t1/z >> L (steady state)

S.N. Majumdar The Wonderful World of Brownian Functionals

• Relevant object that measures fluctuations

h(x, t) = H(x, t) − 1 L L H(x, t)dx → relative height By definition: L h(x, t)dx = 0

• Width of the interface:

w(t, L) =

• h2(x, t) ∼ tβ for ξ(t) ∼ t1/z << L

(growing regime) ∼ L1/2 for ξ(t) ∼ t1/z >> L (steady state)

• β = 1/3 and z = 3/2 (for KPZ, λ > 0)
• β = 1/4 and z = 2

(for EW, λ = 0)

S.N. Majumdar The Wonderful World of Brownian Functionals

• Relevant object that measures fluctuations

h(x, t) = H(x, t) − 1 L L H(x, t)dx → relative height By definition: L h(x, t)dx = 0

• Width of the interface:

w(t, L) =

• h2(x, t) ∼ tβ for ξ(t) ∼ t1/z << L

(growing regime) ∼ L1/2 for ξ(t) ∼ t1/z >> L (steady state)

• β = 1/3 and z = 3/2 (for KPZ, λ > 0)
• β = 1/4 and z = 2

(for EW, λ = 0)

• Full distribution of w 2(t, L) = h2(x, t) → in the steady state

→ Exact solution (G. Foltin, K. Oerding, Z. Racz, R.L. Workman, R.K.P. Zia, 1994)

S.N. Majumdar The Wonderful World of Brownian Functionals

Extreme Relative Height

Recent interest in Extreme height fluctuations (Rare Events) hm = max0≤x≤L{h(x, t)} → maximal relative height (MRH)

S.N. Majumdar The Wonderful World of Brownian Functionals

Extreme Relative Height

Recent interest in Extreme height fluctuations (Rare Events) hm = max0≤x≤L{h(x, t)} → maximal relative height (MRH) In particular, in the steady state regime t >> Lz where the heights at different points are strongly correlated (ξ(t) >> L) Q: Prob. distr. of hm: P(hm, L) =?

S.N. Majumdar The Wonderful World of Brownian Functionals

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact solution via Path Integral method

x L

hm

M

h(x)

• hm = max0≤x≤L{h(x, t)} → maximal relative height
• Cumulative distribution of hm in the steady state (t1/z >> L)

F(M, L) = Prob [hm ≤ M, L] = ⇒ Prob. that the path stays below the level M over x ∈ [0, L]

S.N. Majumdar The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• S.N. Majumdar

The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• Cumulative distribution F(M, L) ≡ Prob. that the path stays below M

S.N. Majumdar The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• Cumulative distribution F(M, L) ≡ Prob. that the path stays below M

F(M, L) ∝ M

−∞

dh0 h(L)=h0

h(0)=h0

Dh(x) e− 1

2

L

0 (∂xh)2dx × δ

L h(x)dx

• ד
• 0≤x≤L

θ (M − h(x)) ”

S.N. Majumdar The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• Cumulative distribution F(M, L) ≡ Prob. that the path stays below M

F(M, L) ∝ M

−∞

dh0 h(L)=h0

h(0)=h0

Dh(x) e− 1

2

L

0 (∂xh)2dx × δ

L h(x)dx

• ד
• 0≤x≤L

θ (M − h(x)) ”

• Identify x ≡ τ and change variable M − h(x) ≡ x(τ):

F(M, L) ∝ ∞ dx0 x(L)=x0

x(0)=x0

Dx(τ) e− 1

2

L

0 (∂τ x)2dτ × δ

L x(τ)dτ − ML

• ד
• 0≤τ≤L

θ (x(τ)) ”

S.N. Majumdar The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• Cumulative distribution F(M, L) ≡ Prob. that the path stays below M

F(M, L) ∝ M

−∞

dh0 h(L)=h0

h(0)=h0

Dh(x) e− 1

2

L

0 (∂xh)2dx × δ

L h(x)dx

• ד
• 0≤x≤L

θ (M − h(x)) ”

• Identify x ≡ τ and change variable M − h(x) ≡ x(τ):

F(M, L) ∝ ∞ dx0 x(L)=x0

x(0)=x0

Dx(τ) e− 1

2

L

0 (∂τ x)2dτ × δ

L x(τ)dτ − ML

• ד
• 0≤τ≤L

θ (x(τ)) ”

• Now apply Feynman-Kac technique

S.N. Majumdar The Wonderful World of Brownian Functionals

Path Integral with Constraint

• Joint distribution of relative heights (steady state) for (1 + 1)-dim.

KPZ interfaces:

• Prob. [{h(x}] ∝ exp
• −1

2 L ∂h ∂x 2 dx

• × δ

L h(x)dx

• Cumulative distribution F(M, L) ≡ Prob. that the path stays below M

F(M, L) ∝ M

−∞

dh0 h(L)=h0

h(0)=h0

Dh(x) e− 1

2

L

0 (∂xh)2dx × δ

L h(x)dx

• ד
• 0≤x≤L

θ (M − h(x)) ”

• Identify x ≡ τ and change variable M − h(x) ≡ x(τ):

F(M, L) ∝ ∞ dx0 x(L)=x0

x(0)=x0

Dx(τ) e− 1

2

L

0 (∂τ x)2dτ × δ

L x(τ)dτ − ML

• ד
• 0≤τ≤L

θ (x(τ)) ”

• Now apply Feynman-Kac technique → take Laplace transform

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0 Schr¨

• dinger eq. in a traingular potential → exactly solvable

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0 Schr¨

• dinger eq. in a traingular potential → exactly solvable
• Prob. density of hm: P(hm, L) = ∂MF(M, L)|M=hm

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0 Schr¨

• dinger eq. in a traingular potential → exactly solvable
• Prob. density of hm: P(hm, L) = ∂MF(M, L)|M=hm

= ⇒ P(hm, L) =

1 √ L f

• hm

√ L

• where

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0 Schr¨

• dinger eq. in a traingular potential → exactly solvable
• Prob. density of hm: P(hm, L) = ∂MF(M, L)|M=hm

= ⇒ P(hm, L) =

1 √ L f

• hm

√ L

• where

∞ f (x) e−s x dx = s √ 2π

∞

• k=1

e−αk 2−1/3 s2/3 where αk’s → zeroes of Airy function Ai(z)

[S.M. and A. Comtet, PRL, 92, 225501 (2004); J. Stat. Phys. 119, 777 (2005)]

S.N. Majumdar The Wonderful World of Brownian Functionals

Exact Results

• ∞

F(M, L) e−sM L dM ∝ ∞ dx0 x0|e− ˆ

H L|x0 = Tr

• e− ˆ

H L

with ˆ H ≡ − 1

2∂2 x + V (x)

where the quantum potential V (x) = s x for x > 0 = ∞ for x < 0 Schr¨

• dinger eq. in a traingular potential → exactly solvable
• Prob. density of hm: P(hm, L) = ∂MF(M, L)|M=hm

= ⇒ P(hm, L) =

1 √ L f

• hm

√ L

• where

∞ f (x) e−s x dx = s √ 2π

∞

• k=1

e−αk 2−1/3 s2/3 where αk’s → zeroes of Airy function Ai(z)

[S.M. and A. Comtet, PRL, 92, 225501 (2004); J. Stat. Phys. 119, 777 (2005)]

• The Laplace transform can be inverted → exact asymptotics

S.N. Majumdar The Wonderful World of Brownian Functionals

The scaling function f (x)

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

S.N. Majumdar The Wonderful World of Brownian Functionals

The scaling function f (x)

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

U(a, b, z) → Kummer’s function and αk → zeroes of Airy function Ai(z)

S.N. Majumdar The Wonderful World of Brownian Functionals

The scaling function f (x)

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

U(a, b, z) → Kummer’s function and αk → zeroes of Airy function Ai(z) Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

The scaling function f (x)

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

U(a, b, z) → Kummer’s function and αk → zeroes of Airy function Ai(z) Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

Airy distribution function

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

S.N. Majumdar The Wonderful World of Brownian Functionals

Airy distribution function

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

Airy distribution function

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

• A number of discrete SOS models → same f (x)

(G. Schehr & S.M, 2006)

S.N. Majumdar The Wonderful World of Brownian Functionals

Airy distribution function

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

• A number of discrete SOS models → same f (x)

(G. Schehr & S.M, 2006)

• Amazingly, the same f (x) appears in a number of problems in graph

theory and computer science:

• total path length in rooted planar trees (Takacs, ’94)
• connected components in a random graph (Flajolet, Knuth, Pittel, ’89)

. . . f (x) → Airy distribution function (Janson & Louchard, 2007)

S.N. Majumdar The Wonderful World of Brownian Functionals

Airy distribution function

f (x) = 2 √ 6 x10/3

∞

• k=1

b2/3

k

e−bk/x2 U(−5/6, 4/3, bk/x2) , bk = 2α3

k/27

Asymptotics: f (x) ≈

8 81α9/2 1

x−5 exp

• − 2α3

1

27x2

• as x → 0 ( α1 = 2.3381..)

≈ 72

√ 6 √π x2 e−6x2

as x → ∞

• A number of discrete SOS models → same f (x)

(G. Schehr & S.M, 2006)

• Amazingly, the same f (x) appears in a number of problems in graph

theory and computer science:

• total path length in rooted planar trees (Takacs, ’94)
• connected components in a random graph (Flajolet, Knuth, Pittel, ’89)

. . . f (x) → Airy distribution function (Janson & Louchard, 2007)

• Other measures of extreme fluctuations → studied recently

(Burkhardt, Gy¨

• rgyi, Moloney, Ra´

cz, 2007, Rambeau & Schehr, 2009)

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian Functional

τ x(τ)

x0

tf tf

Brownian motion till its first−passage through two different realizations

• Brownian motion:

dx dτ = η(τ) where η(τ) → Gaussian white noise

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian Functional

τ x(τ)

x0

tf tf

Brownian motion till its first−passage through two different realizations

• Brownian motion:

dx dτ = η(τ) where η(τ) → Gaussian white noise

• The process, starting at fixed x0, stops when the particle hits x = 0

tf → first-passage time → random variable

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian Functional

τ x(τ)

x0

tf tf

Brownian motion till its first−passage through two different realizations

• Brownian motion:

dx dτ = η(τ) where η(τ) → Gaussian white noise

• The process, starting at fixed x0, stops when the particle hits x = 0

tf → first-passage time → random variable

• first-passage functional: T =

tf U (x(τ)) dτ; Question : P(T|x0) ?

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian Functional

τ x(τ)

x0

tf tf

Brownian motion till its first−passage through two different realizations

• Brownian motion:

dx dτ = η(τ) where η(τ) → Gaussian white noise

• The process, starting at fixed x0, stops when the particle hits x = 0

tf → first-passage time → random variable

• first-passage functional: T =

tf U (x(τ)) dτ; Question : P(T|x0) ?

• Ex: (i) If U(x) = 1, T = tf

(ii) If U(x) = x, T = tf

0 x(τ)dτ → area till first-passage time

S.N. Majumdar The Wonderful World of Brownian Functionals

Queueing theory: busy period

• 2

2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 1

ln = ln−1 + ηn ln

time

n

busy period length queue

• Queue length: ln = ln−1 + ηn

ηn = 1 if a new customer arrives = −1 when a customer gets served

S.N. Majumdar The Wonderful World of Brownian Functionals

Queueing theory: busy period

• 2

2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 1

ln = ln−1 + ηn ln

time

n

busy period length queue

• Queue length: ln = ln−1 + ηn

ηn = 1 if a new customer arrives = −1 when a customer gets served

• T ≡ total waiting time of all customers during the busy period

≡ area under the curve

S.N. Majumdar The Wonderful World of Brownian Functionals

Queueing theory: busy period

• 2

2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 1

ln = ln−1 + ηn ln

time

n

busy period length queue

• Queue length: ln = ln−1 + ηn

ηn = 1 if a new customer arrives = −1 when a customer gets served

• T ≡ total waiting time of all customers during the busy period

≡ area under the curve

• Continuum limit: ln → x(τ) → dx

dτ = η(τ) → Brownian motion

S.N. Majumdar The Wonderful World of Brownian Functionals

Queueing theory: busy period

• 2

2 2 2 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 1

ln = ln−1 + ηn ln

time

n

busy period length queue

• Queue length: ln = ln−1 + ηn

ηn = 1 if a new customer arrives = −1 when a customer gets served

• T ≡ total waiting time of all customers during the busy period

≡ area under the curve

• Continuum limit: ln → x(τ) → dx

dτ = η(τ) → Brownian motion

T ≡ A = tf

0 x(τ)dτ → area till the first-passage time; P(A|x0) = ?

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

• backward approach: [0, tf ] ≡ [0, ∆τ] + [∆τ, tf ]

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

• backward approach: [0, tf ] ≡ [0, ∆τ] + [∆τ, tf ]

˜ Ps(x0) = e−s

tf U(x(τ))dτ

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

• backward approach: [0, tf ] ≡ [0, ∆τ] + [∆τ, tf ]

˜ Ps(x0) = e−s

tf U(x(τ))dτ = e−sU(x0)∆τ ˜

Ps(x0 + ∆x)∆x

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

• backward approach: [0, tf ] ≡ [0, ∆τ] + [∆τ, tf ]

˜ Ps(x0) = e−s

tf U(x(τ))dτ = e−sU(x0)∆τ ˜

Ps(x0 + ∆x)∆x = e−sU(x0)∆τ ˜ Ps(x0 + η(0)∆τ)η0

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

τ x(τ)

x0

tf

Brownian motion till its first−passage through 0

∆τ

+ ∆x x0

• dx

dτ = η(τ);

x(0) = x0

• T =

tf

0 U (x(τ)) dτ

• P(T|x0) =?
• Laplace transform: ˜

Ps(x0) = ∞ P(T|x0) e−s T dT≡ e−s

tf U(x(τ))dτ

• backward approach: [0, tf ] ≡ [0, ∆τ] + [∆τ, tf ]

˜ Ps(x0) = e−s

tf U(x(τ))dτ = e−sU(x0)∆τ ˜

Ps(x0 + ∆x)∆x = e−sU(x0)∆τ ˜ Ps(x0 + η(0)∆τ)η0

• Expand in Taylor series and use η(0) = 0 and η2(0) = 2D

∆τ

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

• Thus ˆ

H Ps(x0) = 0 where quantum Hamiltonian: ˆ H ≡ −D∂2

x0 + s U(x0) → simpler than Feynman-Kac

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

• Thus ˆ

H Ps(x0) = 0 where quantum Hamiltonian: ˆ H ≡ −D∂2

x0 + s U(x0) → simpler than Feynman-Kac

• Finally invert the Laplace transform ˜

Ps(x0) to obtain P(T|x0)

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

• Thus ˆ

H Ps(x0) = 0 where quantum Hamiltonian: ˆ H ≡ −D∂2

x0 + s U(x0) → simpler than Feynman-Kac

• Finally invert the Laplace transform ˜

Ps(x0) to obtain P(T|x0)

• Generlization to the process:

dx dτ = F(x) + η(τ)

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

• Thus ˆ

H Ps(x0) = 0 where quantum Hamiltonian: ˆ H ≡ −D∂2

x0 + s U(x0) → simpler than Feynman-Kac

• Finally invert the Laplace transform ˜

Ps(x0) to obtain P(T|x0)

• Generlization to the process:

dx dτ = F(x) + η(τ)

−D d2Ps dx2 − F(x0)dPs dx0 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

First-passage Brownian functional

• =

⇒ −D d2Ps dx2 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ boundary conditions: (i) Ps(x0 → 0) = 1 (ii) Ps(x0 → ∞) = 0

• Thus ˆ

H Ps(x0) = 0 where quantum Hamiltonian: ˆ H ≡ −D∂2

x0 + s U(x0) → simpler than Feynman-Kac

• Finally invert the Laplace transform ˜

Ps(x0) to obtain P(T|x0)

• Generlization to the process:

dx dτ = F(x) + η(τ)

−D d2Ps dx2 − F(x0)dPs dx0 + s U(x0) Ps(x0) = 0 in 0 ≤ x0 ≤ ∞ [M.J. Kearney and S.M., 2005]

S.N. Majumdar The Wonderful World of Brownian Functionals

Results for the area distribution: D = 1/2

• T = A =

tf

0 x(τ)dτ ≡ area till first-passage time → U(x) = x

S.N. Majumdar The Wonderful World of Brownian Functionals

Results for the area distribution: D = 1/2

• T = A =

tf

0 x(τ)dτ ≡ area till first-passage time → U(x) = x

P(A|x0) = 21/3 32/3 Γ[1/3] x0 A4/3 exp

• −2x3

9A

• (Kearney & S.M, 2005)

S.N. Majumdar The Wonderful World of Brownian Functionals

Results for the area distribution: D = 1/2

• T = A =

tf

0 x(τ)dτ ≡ area till first-passage time → U(x) = x

P(A|x0) = 21/3 32/3 Γ[1/3] x0 A4/3 exp

• −2x3

9A

• (Kearney & S.M, 2005)

result relevant to:

• Waiting-time distribution during busy-period in queueing theory
• Avalanche size distribution in directed Abelian sandpile model (Dhar &

Ramaswamy, ’89)

• Area distribution under staircase polygons → Combinatorics (Prellberg,

Kearney, Richards...)

• Inelastic collapse of a ball bouncing on a noisy vibrating platform (S.M.

and Kearney, 2007)

• DNA breathing dynamics (Bandyopadhyay, Gupta & Segal, 2011)

S.N. Majumdar The Wonderful World of Brownian Functionals

Results for the area distribution: D = 1/2

• T = A =

tf

0 x(τ)dτ ≡ area till first-passage time → U(x) = x

P(A|x0) = 21/3 32/3 Γ[1/3] x0 A4/3 exp

• −2x3

9A

• (Kearney & S.M, 2005)

result relevant to:

• Waiting-time distribution during busy-period in queueing theory
• Avalanche size distribution in directed Abelian sandpile model (Dhar &

Ramaswamy, ’89)

• Area distribution under staircase polygons → Combinatorics (Prellberg,

Kearney, Richards...)

• Inelastic collapse of a ball bouncing on a noisy vibrating platform (S.M.

and Kearney, 2007)

• DNA breathing dynamics (Bandyopadhyay, Gupta & Segal, 2011)
• For drifted Brownian motion:

dx dτ = −µ + η(τ)

(µ > 0) P(A|x0) ∼ x0µ7/4 A3/4 exp

• −
• 8

3 µ3/2√ A

• as A → ∞

(M.J. Kearney, S.M. & R. Martin, 2007)

S.N. Majumdar The Wonderful World of Brownian Functionals

Lifetime of Comets in solar system

S.N. Majumdar The Wonderful World of Brownian Functionals

Lifetime of Comets in solar system

• A comet enters the solar system with a negative energy (per unit mass)

E0 < 0

S.N. Majumdar The Wonderful World of Brownian Functionals

Lifetime of Comets in solar system

• A comet enters the solar system with a negative energy (per unit mass)

E0 < 0

• Had there been no planets, E0 = − GMs

2a

→ elliptical orbit

S.N. Majumdar The Wonderful World of Brownian Functionals

Lifetime of Comets in solar system

• A comet enters the solar system with a negative energy (per unit mass)

E0 < 0

• Had there been no planets, E0 = − GMs

2a

→ elliptical orbit

• energy of the comet, after successive orbitals, gets perturbed by big

planets Jupiter/Saturn and successive perturbations

S.N. Majumdar The Wonderful World of Brownian Functionals

Lifetime of Comets in solar system

• A comet enters the solar system with a negative energy (per unit mass)

E0 < 0

• Had there been no planets, E0 = − GMs

2a

→ elliptical orbit

• energy of the comet, after successive orbitals, gets perturbed by big

planets Jupiter/Saturn and successive perturbations = ⇒ positive energy of the comet which then leaves the solar system

(Lyttleton, ’53)

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

• Time to complete an orbit with E < 0 is

c (−E)3/2 → Kepler’s 3rd law

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

• Time to complete an orbit with E < 0 is

c (−E)3/2 → Kepler’s 3rd law

• Total lifetime of the comet: T =

n∗

• n=1

c (−En)3/2

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

• Time to complete an orbit with E < 0 is

c (−E)3/2 → Kepler’s 3rd law

• Total lifetime of the comet: T =

n∗

• n=1

c (−En)3/2 Continuum limit: → T = tf

c (x(τ))3/2 dτ

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

• Time to complete an orbit with E < 0 is

c (−E)3/2 → Kepler’s 3rd law

• Total lifetime of the comet: T =

n∗

• n=1

c (−En)3/2 Continuum limit: → T = tf

c (x(τ))3/2 dτ =

⇒ U(x) = c x−3/2

S.N. Majumdar The Wonderful World of Brownian Functionals

Random Walk in energy space

• energy
• rbit number

E0

random walk in energy space

• En = En−1 + ηn starting from E0 < 0
• The comet exits at n = n∗ when

En > 0

• Continuum limit: x(τ) ≡ −En

with dx

dτ = η(τ);

x(0) = x0 = −E0

• Time to complete an orbit with E < 0 is

c (−E)3/2 → Kepler’s 3rd law

• Total lifetime of the comet: T =

n∗

• n=1

c (−En)3/2 Continuum limit: → T = tf

c (x(τ))3/2 dτ =

⇒ U(x) = c x−3/2

• Following the general procedure, one gets

P(T|x0) = 64x0 T 3 exp

• −8√x0

T

• (Hammersley, ’61)

S.N. Majumdar The Wonderful World of Brownian Functionals

Maximum till the first-passage time

tf x0

time space M L

• M → maximum till first-passage time tf ;

Question: P(M|x0) = ?

S.N. Majumdar The Wonderful World of Brownian Functionals

Maximum till the first-passage time

tf x0

time space M L

• M → maximum till first-passage time tf ;

Question: P(M|x0) = ?

• Cumulative Prob. Q(x0, L) = Prob.[M ≤ L|x0]

≡ Prob. that the particle exits the box [0, L] through 0 = ⇒ classic Gambler’s Ruin problem

S.N. Majumdar The Wonderful World of Brownian Functionals

Maximum till the first-passage time

tf x0

time space M L

• M → maximum till first-passage time tf ;

Question: P(M|x0) = ?

• Cumulative Prob. Q(x0, L) = Prob.[M ≤ L|x0]

≡ Prob. that the particle exits the box [0, L] through 0 = ⇒ classic Gambler’s Ruin problem

• satisfies d2Q

dx2

0 = 0 with Q(x0 = 0) = 1 and Q(x0 = L) = 0

Solution: Q(x0, L) = 1 − x0

L → P(M|x0) = x0 M2 with M ≥ x0

S.N. Majumdar The Wonderful World of Brownian Functionals

Maximum till the first-passage time

tf x0

time space M L

• M → maximum till first-passage time tf ;

Question: P(M|x0) = ?

• Cumulative Prob. Q(x0, L) = Prob.[M ≤ L|x0]

≡ Prob. that the particle exits the box [0, L] through 0 = ⇒ classic Gambler’s Ruin problem

• satisfies d2Q

dx2

0 = 0 with Q(x0 = 0) = 1 and Q(x0 = L) = 0

Solution: Q(x0, L) = 1 − x0

L → P(M|x0) = x0 M2 with M ≥ x0

• power-law distribution for M with all moments Mn = ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

Multiple Non-interacting Walkers

tf x

time space

1

M

x2

multiple non−interacting walkers L

• maximum M till one of the walkers hits 0

S.N. Majumdar The Wonderful World of Brownian Functionals

Multiple Non-interacting Walkers

tf x

time space

1

M

x2

multiple non−interacting walkers L

• maximum M till one of the walkers hits 0
• P (M|x1, x2, . . . xN) = ?

S.N. Majumdar The Wonderful World of Brownian Functionals

Multiple Non-interacting Walkers

tf x

time space

1

M

x2

multiple non−interacting walkers L

• maximum M till one of the walkers hits 0
• P (M|x1, x2, . . . xN) = ?
• Exact result via path integral method

P (M|x1, x2, . . . xN) ≈ AN x1 x2 . . . , xN MN+1 as M → ∞

S.N. Majumdar The Wonderful World of Brownian Functionals

Multiple Non-interacting Walkers

tf x

time space

1

M

x2

multiple non−interacting walkers L

• maximum M till one of the walkers hits 0
• P (M|x1, x2, . . . xN) = ?
• Exact result via path integral method

P (M|x1, x2, . . . xN) ≈ AN x1 x2 . . . , xN MN+1 as M → ∞ = ⇒ moments up to order (N − 1) finite

S.N. Majumdar The Wonderful World of Brownian Functionals

Multiple Non-interacting Walkers

tf x

time space

1

M

x2

multiple non−interacting walkers L

• maximum M till one of the walkers hits 0
• P (M|x1, x2, . . . xN) = ?
• Exact result via path integral method

P (M|x1, x2, . . . xN) ≈ AN x1 x2 . . . , xN MN+1 as M → ∞ = ⇒ moments up to order (N − 1) finite A1 = 1, A2 =

1 4π2 Γ4[1/4], ...

AN ∼ N 4

π ln(N)

N/2

(Krapivsky, S.M. & Rosso, 2010, S.M. & Bray, 2010)

S.N. Majumdar The Wonderful World of Brownian Functionals

Self-affine Processes: Non-Brownian

tf x0

time space M L

• x(τ) → self-affine x(τ) ∼ τ H

where H → Hurst exponent

• M → maximum till first-passage time
• P(M|x0) = ?

S.N. Majumdar The Wonderful World of Brownian Functionals

Self-affine Processes: Non-Brownian

tf x0

time space M L

• x(τ) → self-affine x(τ) ∼ τ H

where H → Hurst exponent

• M → maximum till first-passage time
• P(M|x0) = ?
• P(M|x0) ∼

xφ Mφ+1 for large M

S.N. Majumdar The Wonderful World of Brownian Functionals

Self-affine Processes: Non-Brownian

tf x0

time space M L

• x(τ) → self-affine x(τ) ∼ τ H

where H → Hurst exponent

• M → maximum till first-passage time
• P(M|x0) = ?
• P(M|x0) ∼

xφ Mφ+1 for large M

φ = θ H where θ → persistence exponent of the process

• Various applications:
• fractional Brownian motion: Using θ = 1 − H (Krug. et. al., 1996),
• ne gets: φ = 1−H

H

• particle in a Sinai potential

.....

[S.M., Rosso & Zoia, PRL, 104, 020602 (2010)]

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

• For Brownian motions over fixed time interval [0, t]

= ⇒ Feynman-Kac formula

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

• For Brownian motions over fixed time interval [0, t]

= ⇒ Feynman-Kac formula Several applications: maximal relative height distribution for fluctuating (1 + 1)-dimensional interfaces

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

• For Brownian motions over fixed time interval [0, t]

= ⇒ Feynman-Kac formula Several applications: maximal relative height distribution for fluctuating (1 + 1)-dimensional interfaces

• First-passage Brownian functional =

⇒ applications

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

• For Brownian motions over fixed time interval [0, t]

= ⇒ Feynman-Kac formula Several applications: maximal relative height distribution for fluctuating (1 + 1)-dimensional interfaces

• First-passage Brownian functional =

⇒ applications Review: “Brownian functionals in Physics and Computer Science”, S.M., Current Science, 89, 2076 (2005), arXiv/cond-mat/0510064

S.N. Majumdar The Wonderful World of Brownian Functionals

Summary and Conclusion

• Path Integral =

⇒ powerful tool to study statistical properties of “constrained” Brownian motions

• For Brownian motions over fixed time interval [0, t]

= ⇒ Feynman-Kac formula Several applications: maximal relative height distribution for fluctuating (1 + 1)-dimensional interfaces

• First-passage Brownian functional =

⇒ applications Review: “Brownian functionals in Physics and Computer Science”, S.M., Current Science, 89, 2076 (2005), arXiv/cond-mat/0510064

• Generalization of Path Integral techniques

= ⇒ Interacting many-particle problem = ⇒ Vicious Walkers

Schehr, S.M., Comtet, Randon-Furling, PRL, 101, 150601 (2008) Nadal & S.M., PRE, 79, 061117 (2009) Forrester, S.M. & Schehr, Nucl. Phys. B 844, 500 (2011)

S.N. Majumdar The Wonderful World of Brownian Functionals

Collaborators:

• C. Nadal (LPTMS, Orsay, France)
• J. Randon-Furling (Univ. Paris-1, France)
• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

S.N. Majumdar The Wonderful World of Brownian Functionals

Collaborators:

• C. Nadal (LPTMS, Orsay, France)
• J. Randon-Furling (Univ. Paris-1, France)
• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
• A. J. Bray (Manchester Univ., UK)
• A. Comtet (LPTMS, Orsay, France)
• D. S. Dean (LPT, Toulouse, France)
• M. J. Kearney (Univ. of Surrey, UK)
• P. L. Krapivsky (Boston Univ., USA)
• R. Martin (Credit Suisse, London, UK)
• A. Rosso (LPTMS, France)
• S. Sabhapandit (RRI, Bangalore, India)
• G. Schehr (LPT, Orsay, France)
• K. J. Wiese (ENS, Paris, France)
• M. Yor (LPMA, Jussieu, France)
• A. Zoia (Saclay, France)

S.N. Majumdar The Wonderful World of Brownian Functionals