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 of 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 o and McKean D ∗ ← → ← → D s , m , k D s h , m h , 0 s h , m h , 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 = ( l 1 , l 2 ), −∞ ≤ l 1 < l 2 ≤ ∞ 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 = ( l 1 , l 2 ), −∞ ≤ l 1 < l 2 ≤ ∞ m : right continuous increasing fnc. on I k : right continuous nondecreasing fnc. on I ◮ G s , m , k : 1-dim diffusion operator with s , m , and k G s , m , k u = dD s u − udk dm

One dimensional diffusion process ◮ We set s : continuous increasing fnc. on I = ( l 1 , l 2 ), −∞ ≤ l 1 < l 2 ≤ ∞ m : right continuous increasing fnc. on I k : right continuous nondecreasing fnc. on I ◮ G s , m , k : 1-dim diffusion operator with s , m , and k G s , m , k u = dD s u − udk dm ◮ D s , m , k : 1-dim diffusion process with G s , m , k [ l 1 is absorbing if l 1 is regular ]

One dimensional diffusion process ◮ p ( t , x , y ) : transition probability w.r.t. dm for D s , m , k If l 1 is ( s , m , k )-regular, � e − λ t ψ o ( x , λ ) ψ o ( y , λ ) d σ ( λ ) , p ( t , x , y ) = t > 0 , x , y ∈ I , [0 , ∞ ) (1) where d σ ( λ ) is a Borel measure on [0 , ∞ ) satisfying � e − λ t d σ ( λ ) < ∞ , t > 0 , (2) [0 , ∞ ) and ψ o ( x , λ ), x ∈ I , λ ≥ 0, is the solution of the following integral equation ψ o ( x , λ ) = s ( x ) − s ( l 1 ) � + { s ( x ) − s ( y ) } ψ o ( y , λ ) {− λ dm ( y ) + dk ( y ) } ( l 1 , x ]

One dimensional diffusion process Proposition 2.1 Assume that l 1 is ( s , m , k ) -entrance and � { s ( c o ) − s ( x ) } 2 dm ( x ) < ∞ . (3) ( l 1 , c o ] Then p ( t , x , y ) is represented as (1) with d σ ( λ ) satisfying (2) and ψ o ( x , λ ) is the solution of the integral equation � ψ o ( x , λ ) = 1 + { s ( x ) − s ( y ) } ψ o ( y , λ ) {− λ dm ( y ) + dk ( y ) } . ( l 1 , x ]

Harmonic transform ◮ We set H s , m , k ,β = { h > 0; G s , m , k h = β h } , for β ≥ 0 For h ∈ H s , m , k ,β , ds h ( x ) = h ( x ) − 2 ds ( x ) , dm h ( x ) = h ( x ) 2 dm ( x )

Harmonic transform ◮ We set H s , m , k ,β = { h > 0; G s , m , k h = β h } , for β ≥ 0 For h ∈ H s , m , k ,β , ds h ( x ) = h ( x ) − 2 ds ( x ) , dm h ( x ) = h ( x ) 2 dm ( x ) ◮ We obtain � p h ( t , x , y ) = e − β t p ( t , x , y ) � G s h , m h , 0 : h transform of G s , m , k h ( x ) h ( y )

Harmonic transform ◮ We set H s , m , k ,β = { h > 0; G s , m , k h = β h } , for β ≥ 0 For h ∈ H s , m , k ,β , ds h ( x ) = h ( x ) − 2 ds ( x ) , dm h ( x ) = h ( x ) 2 dm ( x ) ◮ We obtain � p h ( t , x , y ) = e − β t p ( t , x , y ) � G s h , m h , 0 : h transform of G s , m , k h ( x ) h ( y ) ◮ D s h , m h , 0 : 1-dim diffusion process with G s h , m h , 0 [ l 1 is absorbing if l 1 is regular ]

Harmonic transform ◮ D ∗ s h , m h , 0 : 1-dim diffusion process with G s h , m h , 0 [ l 1 is regular and reflecting boundary ]

Harmonic transform ◮ D ∗ s h , m h , 0 : 1-dim diffusion process with G s h , m h , 0 [ l 1 is regular and reflecting boundary ] ◮ l ( h ∗ ) ( t , ξ ) : local time for D ∗ s h , m h , 0 , that is, � t � l ( h ∗ ) ( t , ξ ) dm h ( ξ ) , f ( X ( u )) du = t > 0 , 0 I for bounded continuous functions f on I .

Harmonic transform ◮ D ∗ s h , m h , 0 : 1-dim diffusion process with G s h , m h , 0 [ l 1 is regular and reflecting boundary ] ◮ l ( h ∗ ) ( t , ξ ) : local time for D ∗ s h , m h , 0 , that is, � t � l ( h ∗ ) ( t , ξ ) dm h ( ξ ) , f ( X ( u )) du = t > 0 , 0 I for bounded continuous functions f on I . ◮ τ ( h ∗ ) ( t ) : inverse local time l ( h ∗ ) − 1 ( t , l 1 ) at the end point l 1

L´ evy measure density Proposition 2.2 (Itˆ o and McKean) Assume the following conditions. l 1 is ( s , m , 0) -regular and reflecting, s ( l 2 ) = ∞ . Then [ τ ∗ ( t ) , t ≥ 0] is a L´ evy process and there is a L´ evy measure density n ∗ ( ξ ) such that � ∞ � � � e − λτ ∗ ( t ) � E ∗ (1 − e − λξ ) n ∗ ( ξ ) d ξ = exp − t l 1 0 where E ∗ l 1 stands for the expectation with respect to P ∗ l 1 , � n ∗ ( ξ ) = e − λξ d σ ( λ ) , x , y → l 1 D s ( x ) D s ( y ) p ( ξ, x , y ) = lim [0 , ∞ ) where p ( t , x , y ) is the transition probability density for D s , m , 0 , and d σ ( 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 D s , m , k . l 1 is ( s h , m h , 0)-regular if and only if one of the following conditions is satisfied. l 1 is ( s , m , k )-regular and h ( l 1 ) ∈ (0 , ∞ ). (4) l 1 is ( s , m , k )-entrance, h ( l 1 ) = ∞ , and | m h ( l 1 ) | < ∞ . (5) l 1 is ( s , m , k )-natural, h ( l 1 ) = ∞ , and | m h ( l 1 ) | < ∞ . (6)

Main theorem Theorem 2.3 Let h ∈ H s , m , k ,β . Assume one of (4), (5), and (6). Further assume that l 1 is reflecting and s h ( l 2 ) = ∞ . Then there exists L´ evy measure density n ( h ∗ ) ( ξ ) . In particular, if (4) is satisfied, then � e − ξλ d σ ( λ ) n ( h ∗ ) ( ξ ) = h ( l 1 ) 2 e − βξ [0 , ∞ ) = h ( l 1 ) 2 e − βξ lim x , y → l 1 D s ( x ) D s ( y ) p ( ξ, x , y ) . If (5) is satisfied, then � e − ξλ d σ ( λ ) n ( h ∗ ) ( ξ ) = D s h ( l 1 ) 2 e − βξ [0 , ∞ ) = D s h ( l 1 ) 2 e − βξ lim x , y → l 1 p ( ξ, x , y ) .

Examples Example 2.4 (Bessel process) Let us consider the following diffusion operator G ( ν ) on I = (0 , ∞ ) . d 2 G ( ν ) = 1 dx 2 + 2 ν + 1 d dx , 2 2 x where −∞ < ν < ∞ . ds ( ν ) ( x ) = x − 2 ν − 1 dx , dm ( ν ) ( x ) = 2 x 2 ν +1 dx . The killing measure is null. The state of the end point 0 depends on 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 . 0 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 D s ( ν ) ( x ) D s ( ν ) ( y ) p ( ν ) ( ξ, x , y ) � ∞ | ν | e − ξλ σ ( ν ) ( λ ) d λ = 2 −| ν | +1 Γ( | ν | ) ξ − ( | ν | +1) . = 0

Examples (2) − 1 < ν < 1. [ 0 : ( s ( ν ) , m ( ν ) , 0)-regular or -entrance, and (3) is satisfied ] For β > 0, we put � | ν | � β 2 x − ν K | ν | ( � h ( x ) = 2 β x ) 2 Then h ( x ) ∈ H s ( ν ) , m ( ν ) , 0 ,β and � √ 2 β x � � d 2 2 β K ′ � = 1 1 d G ( ν ) ν � dx 2 + 2 x + � √ 2 β x dx , h � 2 K ν 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, ∗ ) with G ( ν ) as the generator and with the end h h point 0 being reflecting. Let n ( ν, ∗ ) be the L´ evy measure density h corresponding to the inverse local time at 0 for D ( ν, ∗ ) . h n ( ν, ∗ ) = 2 −| ν |− 1 Γ( | ν | + 1) ξ − ( | ν | +1) e − βξ . h (3) 0 < ν < 1 We put h (0) ( x ) = { s ( ν ) ( ∞ ) − s ( ν ) ( x ) } / { s ( ν ) ( ∞ ) − s ( ν ) (1) } = x − 2 ν . Denote by G ( ν, 0) the harmonic transform of G ( ν ) based on h h (0) ∈ H s ( ν ) , m ( ν ) , 0 , 0 , that is, d 2 = 1 dx 2 + − 2 ν + 1 d G ( ν, 0) dx . h 2 2 x

Examples ds ( ν, 0) ( x ) = h (0) ( x ) − 2 ds ( ν ) ( x ) = x 2 ν − 1 dx , dm ( ν, 0) ( x ) = h (0) ( x ) 2 dm ( ν ) ( x ) = 2 x − 2 ν +1 dx . The end point 0 is ( s ( ν, 0) , m ( ν, 0) , 0)-regular. We consider the diffusion process D ( ν, 0 , ∗ ) with G ( ν, 0) as the generator and with the h h end point 0 being reflecting. Let n ( ν, 0 , ∗ ) be the L´ evy measure h density corresponding to the inverse local time at 0 for D ( ν, 0 , ∗ ) . h ν n ( ν, 0 , ∗ ) = 2 − ν +1 Γ( ν ) ξ − ν − 1 . h