L evy measure density corresponding to inverse local time Tomoko - - PowerPoint PPT Presentation

L´ evy measure density corresponding to inverse local time

Tomoko Takemura and Matsuyo Tomisaki

• 2012. 9.27

motivation

We are concerned with L´ evy measure density corresponding to the inverse local time at the regular end point for harmonic transform

• f a one dimensional diffusion process. We show that the L´

evy measure density is represented as a Laplace transform of the spectral measure corresponding to an original diffusion process, where the absorbing boundary condition is posed at the end point if it is regular. h transform Itˆ

• and McKean

Ds,m,k ← → Dsh,mh,0 ← → D∗

sh,mh,0

absorbing absorbing reflecting n∗(ξ)

Tabel contents

• 1. One dimensional diffusion process
• 2. Harmonic transform
• 3. L´

evy measure density

• 4. Main theorem
• 5. Examples

One dimensional diffusion process

◮ We set

s : continuous increasing fnc. on I = (l1, l2), −∞ ≤ l1 < l2 ≤ ∞ m : right continuous increasing fnc. on I k : right continuous nondecreasing fnc. on I

One dimensional diffusion process

◮ We set

s : continuous increasing fnc. on I = (l1, l2), −∞ ≤ l1 < l2 ≤ ∞ m : right continuous increasing fnc. on I k : right continuous nondecreasing fnc. on I

◮ Gs,m,k : 1-dim diffusion operator with s, m, and k

Gs,m,ku = dDsu − udk dm

One dimensional diffusion process

◮ We set

s : continuous increasing fnc. on I = (l1, l2), −∞ ≤ l1 < l2 ≤ ∞ m : right continuous increasing fnc. on I k : right continuous nondecreasing fnc. on I

◮ Gs,m,k : 1-dim diffusion operator with s, m, and k

Gs,m,ku = dDsu − udk dm

◮ Ds,m,k : 1-dim diffusion process with Gs,m,k

[l1 is absorbing if l1 is regular ]

One dimensional diffusion process

◮ p(t, x, y) : transition probability w.r.t. dm for Ds,m,k

If l1 is (s, m, k)-regular, p(t, x, y) =

• [0,∞)

e−λtψo(x, λ)ψo(y, λ) dσ(λ), t > 0, x, y ∈ I, (1) where dσ(λ) is a Borel measure on [0, ∞) satisfying

• [0,∞)

e−λt dσ(λ) < ∞, t > 0, (2) and ψo(x, λ), x ∈ I, λ ≥ 0, is the solution of the following integral equation ψo(x, λ) =s(x) − s(l1) +

• (l1,x]

{s(x) − s(y)}ψo(y, λ){−λ dm(y) + dk(y)}

One dimensional diffusion process

Proposition 2.1

Assume that l1 is (s, m, k)-entrance and

• (l1,co]

{s(co) − s(x)}2 dm(x) < ∞. (3) Then p(t, x, y) is represented as (1) with dσ(λ) satisfying (2) and ψo(x, λ) is the solution of the integral equation ψo(x, λ) = 1 +

• (l1,x]

{s(x) − s(y)}ψo(y, λ){−λ dm(y) + dk(y)}.

Harmonic transform

◮ We set

Hs,m,k,β = {h > 0; Gs,m,kh = βh}, for β ≥ 0 For h ∈ Hs,m,k,β, dsh(x) = h(x)−2ds(x), dmh(x) = h(x)2dm(x)

Harmonic transform

◮ We set

Hs,m,k,β = {h > 0; Gs,m,kh = βh}, for β ≥ 0 For h ∈ Hs,m,k,β, dsh(x) = h(x)−2ds(x), dmh(x) = h(x)2dm(x)

◮ We obtain

Gsh,mh,0 : h transform of Gs,m,k

• ph(t, x, y) = e−βt p(t, x, y)

h(x)h(y)

Harmonic transform

◮ We set

Hs,m,k,β = {h > 0; Gs,m,kh = βh}, for β ≥ 0 For h ∈ Hs,m,k,β, dsh(x) = h(x)−2ds(x), dmh(x) = h(x)2dm(x)

◮ We obtain

Gsh,mh,0 : h transform of Gs,m,k

• ph(t, x, y) = e−βt p(t, x, y)

h(x)h(y)

• ◮ Dsh,mh,0 : 1-dim diffusion process with Gsh,mh,0

[l1 is absorbing if l1 is regular ]

Harmonic transform

◮ D∗ sh,mh,0 : 1-dim diffusion process with Gsh,mh,0

[l1 is regular and reflecting boundary ]

Harmonic transform

◮ D∗ sh,mh,0 : 1-dim diffusion process with Gsh,mh,0

[l1 is regular and reflecting boundary ]

◮ l(h∗)(t, ξ) : local time for D∗ sh,mh,0, that is,

t f (X(u)) du =

• I

l(h∗)(t, ξ) dmh(ξ), t > 0, for bounded continuous functions f on I.

Harmonic transform

◮ D∗ sh,mh,0 : 1-dim diffusion process with Gsh,mh,0

[l1 is regular and reflecting boundary ]

◮ l(h∗)(t, ξ) : local time for D∗ sh,mh,0, that is,

t f (X(u)) du =

• I

l(h∗)(t, ξ) dmh(ξ), t > 0, for bounded continuous functions f on I.

◮ τ (h∗)(t) : inverse local time l(h∗)−1(t, l1) at the end point l1

L´ evy measure density

Proposition 2.2 (Itˆ

• and McKean)

Assume the following conditions. l1 is (s, m, 0)-regular and reflecting, s(l2) = ∞. Then [τ ∗(t), t ≥ 0] is a L´ evy process and there is a L´ evy measure density n∗(ξ) such that E ∗

l1

• e−λτ ∗(t)

= exp

• −t

∞ (1 − e−λξ)n∗(ξ) d ξ

• where E ∗

l1 stands for the expectation with respect to P∗ l1,

n∗(ξ) = lim

x,y→l1 Ds(x)Ds(y)p(ξ, x, y) =

• [0,∞)

e−λξdσ(λ), where p(t, x, y) is the transition probability density for Ds,m,0, and dσ(λ) is the Borel measure appeared in (1) satisfying (2).

Main theorem

Now we give a representation of n(h∗)(ξ) by means of items corresponding to the diffusion process Ds,m,k. l1 is (sh, mh, 0)-regular if and only if one of the following conditions is satisfied. l1 is (s, m, k)-regular and h(l1) ∈ (0, ∞). (4) l1 is (s, m, k)-entrance, h(l1) = ∞, and |mh(l1)| < ∞. (5) l1 is (s, m, k)-natural, h(l1) = ∞, and |mh(l1)| < ∞. (6)

Main theorem

Theorem 2.3

Let h ∈ Hs,m,k,β. Assume one of (4), (5), and (6). Further assume that l1 is reflecting and sh(l2) = ∞. Then there exists L´ evy measure density n(h∗)(ξ). In particular, if (4) is satisfied, then n(h∗)(ξ) =h(l1)2e−βξ

• [0,∞)

e−ξλ dσ(λ) =h(l1)2e−βξ lim

x,y→l1 Ds(x)Ds(y)p(ξ, x, y).

If (5) is satisfied, then n(h∗)(ξ) =Dsh(l1)2e−βξ

• [0,∞)

e−ξλ dσ(λ) =Dsh(l1)2e−βξ lim

x,y→l1 p(ξ, x, y).

Examples

Example 2.4 (Bessel process)

Let us consider the following diffusion operator G(ν) on I = (0, ∞). G(ν) = 1 2 d2 dx2 + 2ν + 1 2x d dx , where −∞ < ν < ∞. ds(ν)(x) = x−2ν−1 dx, dm(ν)(x) = 2x2ν+1 dx. The killing measure is null. The state of the end point 0 depends

• n ν, that is,

it is (s(ν), m(ν), 0)-entrance if ν ≥ 0, it is (s(ν), m(ν), 0)-regular if −1 < ν < 0, it is (s(ν), m(ν), 0)-exit if ν ≤ −1.

Examples

Further 1 {s(ν)(1) − s(ν)(x)}2 dm(ν)(x) < ∞ ⇐ ⇒ |ν| < 1. The end point ∞ is (s(ν), m(ν), 0)-natural for all ν, and in particular, s(ν)(∞) = ∞ ⇐ ⇒ ν ≤ 0. Let D(ν) : the diffusion process on I with G(ν) ( 0 being absorbing if −1 < ν < 0) p(ν)(t, x, y) :the transition probability density w.r.t. dm(ν).

Examples

(1) −1 < ν < 0 [ 0 : (s(ν), m(ν), 0)-regular ] D(ν,∗) : the diffusion process on I with G(ν) ( 0 being reflecting) n(ν,∗) :the L´ evy measure density corresponding to the inverse local time at 0 for D(ν,∗) Since s(ν)(∞) = ∞, n(ν,∗)(ξ) = lim

x,y→0 Ds(ν)(x)Ds(ν)(y)p(ν)(ξ, x, y)

= ∞ e−ξλσ(ν)(λ) dλ = 2−|ν|+1 |ν| Γ(|ν|)ξ−(|ν|+1).

Examples

(2) −1 < ν < 1. [ 0 : (s(ν), m(ν), 0)-regular or -entrance, and (3) is satisfied ] For β > 0, we put h(x) = β 2 |ν|

2

x−νK|ν|(

• 2βx)

Then h(x) ∈ Hs(ν),m(ν),0,β and G(ν)

h

= 1 2 d2 dx2 +

• 1

2x +

• 2β K ′

ν

√2β x

• Kν

√2β x

• d

dx , ds(ν,β)(x) = h(x)−2 ds(ν)(x), dm(ν,β)(x) = h(x)2 dm(ν)(x).

Examples

The end point 0 is (s(ν,β), m(ν,β), 0)-regular. We consider the diffusion process D(ν,∗)

h

with G(ν)

h

as the generator and with the end point 0 being reflecting. Let n(ν,∗)

h

be the L´ evy measure density corresponding to the inverse local time at 0 for D(ν,∗)

h

. n(ν,∗)

h

= 2−|ν|−1Γ(|ν| + 1)ξ−(|ν|+1)e−βξ. (3) 0 < ν < 1 We put h(0)(x) = {s(ν)(∞) − s(ν)(x)}/{s(ν)(∞) − s(ν)(1)} = x−2ν. Denote by G(ν,0)

h

the harmonic transform of G(ν) based on h(0) ∈ Hs(ν),m(ν),0,0, that is, G(ν,0)

h

= 1 2 d2 dx2 + −2ν + 1 2x d dx .

Examples

ds(ν,0)(x) =h(0)(x)−2 ds(ν)(x) = x2ν−1 dx, dm(ν,0)(x) =h(0)(x)2 dm(ν)(x) = 2x−2ν+1 dx. The end point 0 is (s(ν,0), m(ν,0), 0)-regular. We consider the diffusion process D(ν,0,∗)

h

with G(ν,0)

h

as the generator and with the end point 0 being reflecting. Let n(ν,0,∗)

h

be the L´ evy measure density corresponding to the inverse local time at 0 for D(ν,0,∗)

h

. n(ν,0,∗)

h

= 2−ν+1 ν Γ(ν)ξ−ν−1.

Examples

Example 2.5 (Radial Ornstein-Uhlenbeck process)

Let us consider the following diffusion operator G(ν,κ) on I = (0, ∞). G(ν,κ) = 1 2 d2 dx2 + 2ν + 1 2x − κx d dx , where −∞ < ν < ∞ and κ > 0. ds(ν,κ)(x) = x−2ν−1eκx2 dx, dm(ν,κ)(x) = 2x2ν+1e−κx2 dx. The killing measure is null. The state of the end point 0 depends

• n ν, that is,

it is (s(ν,κ), m(ν,κ), 0)-entrance if ν ≥ 0, it is (s(ν,κ), m(ν,κ), 0)-regular if −1 < ν < 0, it is (s(ν,κ), m(ν,κ), 0)-exit if ν ≤ −1.

Examples

Further 1 {s(ν,κ)(1) − s(ν,κ)(x)}2 dm(ν,κ)(x) < ∞ ⇐ ⇒ |ν| < 1. The end point ∞ is always (s(ν,κ), m(ν,κ), 0)-natural for all ν, and s(ν,κ)(∞) = ∞. Let D(ν,κ) : the diffusion process on I with G(ν,κ) ( 0 being absorbing if −1 < ν < 0) p(ν,κ)(t, x, y) :the transition probability density w.r.t. dm(ν,κ).

Examples

(1) −1 < ν < 0 [ 0 : (s(ν,κ), m(ν,κ), 0)-regular ] D(ν,κ,∗) : the diffusion process on I with G(ν,κ) ( 0 being reflecting) n(ν,κ,∗) :the L´ evy measure density corresponding to the inverse local time at 0 for D(ν,κ,∗) Since s(ν,κ)(∞) = ∞, n(ν,κ,∗)(ξ) = lim

x,y→0 Ds(ν,κ)(x)Ds(ν,κ)(y)p(ν,κ)(ξ, x, y)

= 2−|ν|+1 |ν| Γ(|ν|)

• κ

sinh(κξ) |ν|+1 eκ(ν+1)ξ.

Examples

(2) −1 < ν < 1 [ 0 : (s(ν,κ), m(ν,κ), 0)-regular or -entrance, and (3) is satisfied ] For β > 0, we put h(x) = κ

|ν| 2 − 1 2

2 Γ |ν| 2 − ν 2 + β 2κ

• x−ν−1e

κx2 2 W− β 2κ + ν+1 2 , |ν| 2 (κx2).

Then h(x) ∈ Hs(ν,κ),m(ν,κ),0,β and G(ν,κ)

h

= 1 2 d2 dx2 +   − 1 2x + 2κx W ′

− β

2κ + ν+1 2 , |ν| 2

(κx2) W− β

2κ + ν+1 2 , |ν| 2 (κx2)

   d dx , ds(ν,κ,)

h

(x) = h(x)−2 ds(ν,κ)(x), dm(ν,κ,)

h

(x) = h(x)2 dm(ν,κ)(x).

Examples

The end point 0 is (s(ν,κ)

h

, m(ν,κ)

h

, 0)-regular and s(ν,κ)

h

(∞) = ∞. We consider the diffusion process D(ν,κ,∗)

h

with G(ν,κ)

h

as the generator and with the end point 0 being reflecting. Let n(ν,κ,∗)

h

be the L´ evy measure density corresponding to the inverse local time at 0 for D(ν,κ,∗)

h

. n(ν,κ,∗)

h

(ξ) =2−|ν|−1Γ(|ν| + 1)

• κ

sinh(κξ) |ν|+1 e{κ(ν+1)−β}ξ.

Examples

We finally consider the special case β = κ(ν + 1) > 0. Then G(ν,κ)

h

is reduced to G(ν,κ)

h

=1 2 d2 dx2 +   − 1 2x + 2κx W ′

0, |ν|

2

(κx2) W0, |ν|

2 (κx2)

   d dx =1 2 d2 dx2 +

• 1

2x + κx K ′

|ν|/2(κx2/2)

K|ν|/2(κx2/2)

• d

dx , L´ evy measure density corresponding to the inverse local time at 0 for D(ν,κ,∗)

h

is given by n(ν,κ,∗)

h

(ξ) = 2−|ν|−1Γ(|ν| + 1)

• κ

sinh(κξ) |ν|+1 .