Exponential convergence of Markovian semigroups and their spectra on - PowerPoint PPT Presentation

Stochastic Analysis and Applications 2012 Exponential convergence of Markovian semigroups and their spectra on L p -spaces ( joint work with Ichiro Shigekawa) Seiichiro Kusuoka (Kyoto University) 1

0. Introduction ( M, B , m ): a probability space, T t : a Markovian semigroup on L 2 ( m ) i.e. 0 ≤ T t f ≤ 1 for f ∈ L 2 ( m ) and 0 ≤ f ≤ 1. We assume that T t is strong continuous, T t 1 = 1 , T ∗ t is also Markovian and T ∗ t 1 = 1 . Then, { T t } can be extended (or restricted) to the Markovian semigroup on L p ( m ) for p ∈ [1 , ∞ ], and the extension (or the restriction) of { T t } is strong continuous and contractive for p ∈ [1 , ∞ ). 2

∫ M fdm for f ∈ L 1 ( m ). Let ⟨ f ⟩ := We are interested in the index: 1 γ p → q := − lim sup t log ∥ T t − m ∥ p → q t →∞ where m means the linear operator f �→ ⟨ f ⟩ 1 on L p ( m ) and ∥·∥ p → q is the operator norm from L p ( m ) to L q ( m ). In the case that T t is ergodic, γ p → q the exponential rate of the convergence. 3

The index γ p → p is related to the spectra of T t as fol- lows: Rad( T ( p ) − m ) = e − γ p → p t , t ∈ [0 , ∞ ) , t where Rad( A ) is the radius of spectra of A and T ( p ) means the linear operator T t on L p ( m ). t Let A p be the generator of { T ( p ) } . t If { T ( p ) } is an analytic semigroup, then t e tσ ( A p ) \{ 0 } = σ ( T ( p ) − m ) \ { 0 } , t ∈ [0 , ∞ ) , t 1 sup { Re λ ; λ ∈ σ ( A p ) \ { 0 }} = lim t log || T t − m || p → p . t →∞ In this talk, we concern the relation among { γ p → q } . 4

Contents: 1. Properties on γ p → q , 2. Relation between hypercontractivity and γ p → q , 3. Sufficient conditions for L p -spectra to be p -independent, 4. Properties on spectra on L p -spaces of operators symmetric on the L 2 -space, 5. Example that γ p → p depends on p . 5

Define for a linear operator A p on L p ( m ), σ p ( A p ) := { λ ∈ C ; λ − A p is not injective on L p ( m ) } σ c ( A p ) := { λ ∈ C ; λ − A p is injective, but is not onto map, and Ran( λ − A p ) is dense in L p ( m ) } σ r ( A p ) := { λ ∈ C ; λ − A p is injective, but is not onto map, and Ran( λ − A p ) is not dense in L p ( m ) } ρ ( A p ) := { λ ∈ C ; λ − A p is bijective on L p ( m ) } σ p ( A p ), σ c ( A p ), σ r ( A p ) and ρ ( A p ) are disjoint and their union is equal to C . 6

1. Properties on γ p → q Proposition Let p 1 , p 2 , q 1 , q 2 ∈ [1 , ∞ ]. Let r 1 , r 2 ∈ [1 , ∞ ] such that ∃ θ ∈ [0 , 1] satisfying 1 = 1 − θ + θ 1 = 1 − θ + θ and . r 1 p 1 q 1 r 2 p 2 q 2 Then, γ r 1 → r 2 ≥ (1 − θ ) γ p 1 → p 2 + θγ q 1 → q 2 . In particular, s �→ γ 1 /s → 1 /s on [0 , 1] is concave. 7

Theorem The function p �→ γ p → p on [1 , ∞ ] is continuous on (1 , ∞ ). If γ p → p > 0 for some p ∈ [1 , ∞ ], then γ p → p > 0 for all p ∈ (1 , ∞ ). Remark The function γ p → p may not be continuous at p = 1 , ∞ . Indeed, if m has the standard normal distribution and { T t } is the Ornstein-Uhlembeck semigroup, then γ p → p = 1 for p ∈ (1 , ∞ ), γ p → p = 0 for p = 1 , ∞ . 8

Let p ∗ be the conjugate exponent of p , i.e. 1 p + 1 p ∗ = 1. Theorem Assume that { T t } is self-adjoint on L 2 ( m ). Then, γ p → p = γ p ∗ → p ∗ for p ∈ [1 , ∞ ] and p �→ γ p → p is non-decreasing on [1 , 2] and non-increasing on [2 , ∞ ]. In particular, the maximum is attained at p = 2. 9

2. Relation between hypercontractivity and γ p → q If there exist p, q ∈ (1 , ∞ ), K ≥ 0 and C > 0 such that p < q and f ∈ L p ( m ) , || T K f || q ≤ C || f || p , then for any p ′ , q ′ ∈ (1 , ∞ ) such that p ′ < q ′ , there exist K ′ ≥ 0 and C ′ > 0 and f ∈ L p ′ ( m ) . || T K ′ f || q ′ ≤ C ′ || f || p ′ , (If C = 1, we can choose C ′ = 1.) 10

In this talk, we call { T t } hyperbounded, if there exist p, q ∈ (1 , ∞ ), K ≥ 0 and C > 0 such that p < q and f ∈ L p ( m ) . || T K f || q ≤ C || f || p , (1) If (1) holds with C = 1 and some p, q, K , then we call { T t } hypercontractive. 11

Theorem The following conditions are equivalent: 1. { T t } is hyperbounded. 2. γ p → q ≥ 0 for some 1 < p < q < ∞ . 3. γ p → q = γ 2 → 2 for all p, q ∈ (1 , ∞ ). 12

Proposition f ∈ L 2 ( m ) || T K f || r ≤ || f || 2 , for some K > 0 and r > 2. Then, we have || T K f − ⟨ f ⟩|| 2 ≤ ( r − 1) − 1 / 2 || f || 2 , f ∈ L 2 ( m ) , √ √ − t { } || T t f − ⟨ f ⟩|| 2 ≤ r − 1 exp K log r − 1 || f || 2 , f ∈ L 2 ( m ) , t ∈ [0 , ∞ ) . 13

Theorem The following conditions are equivalent: 1. { T t } is hypercontractive. 2. γ p → q > 0 for some 1 < p < q < ∞ . 3. γ p → q = γ 2 → 2 for all p, q ∈ (1 , ∞ ) and γ 2 → 2 > 0. 4. There exist K > 0 and r > 0 such that || T K || 2 → r < ∞ and || T K − m || 2 → 2 < 1 . 14

3. Sufficient conditions for L p -spectra to be p -independent Assume that { T t } is hyperbounded. Let A p be the generator of { T t } on L p ( m ) for p ∈ [1 , ∞ ). Assume that A 2 is a normal operator, i.e. ( A 2 ) ∗ A 2 = A 2 ( A 2 ) ∗ . In this section, we see that the spectra of A p are independent of p . 15

Under the assumption, we can consider the spectral decomposition of − A 2 as follows: ∫ − A 2 = C λdE λ . For a bounded C -valued measurable function φ on C , define an operator φ ( − A 2 ) on L 2 ( m ) by ∫ φ ( − A 2 ) = C φ ( λ ) dE λ . We can regard φ ( − A 2 ) as a linear operator on L p ( m ). 16

Proposition Let h be a C -valued bounded measurable function on C which is analytic on the neighborhood around 0 and define φ ( λ ) := h (1 /λ ). Then, φ ( − A ) is a bounded operator on L p ( m ). Theorem Assume that { T t } is hyperbounded and A 2 is normal. Then, σ ( − A q ) = σ ( − A 2 ) for q ∈ (1 , ∞ ). 17

By a little more calculation, we have the following theorem. Theorem Assume that { T t } is hyperbounded and A 2 is normal. σ p ( − A 2 ) = σ p ( − A p ), σ c ( − A 2 ) = σ c ( − A p ) and Then, σ r ( − A p ) = ∅ for p ∈ (1 , ∞ ). 18

If there exists positive constants K and C such that f ∈ L 1 ( m ) , || T K f || ∞ ≤ C || f || 1 , then { T t } is called ultracontractive. Theorem Assume that { T t } is ultracontractive and that A 2 is a nor- Then, σ ( − A p ) = σ ( − A 2 ) for p ∈ [1 , ∞ ). mal operator. Moreover, σ p ( − A 2 ) = σ p ( − A p ), σ c ( − A 2 ) = σ c ( − A p ) and σ r ( − A p ) = ∅ for p ∈ [1 , ∞ ). 19

4. Properties on spectra on L p -spaces of operators symmetric on the L 2 -space Let A p be a densely defined, closed, and real operator on L p ( m ) for p ∈ [1 , ∞ ). Assume that { A p ; p ∈ [1 , ∞ ) } are consistent, i.e. if p > q , then Dom( A p ) ⊂ Dom( A q ) and A p f = A q f for f ∈ Dom( A p ). A Markovian semigroup { T t } and its generators { A p ; p ∈ [1 , ∞ ) } satisfy the assumption on { A p ; p ∈ [1 , ∞ ) } . Additionally assume that A 2 is self-adjoint on L 2 ( m ), i.e. A 2 = A ∗ 2 . 20

Lemma σ r ( A p ) = ∅ for p ≤ 2. Theorem We have the following. 1. σ p ( A p ) ⊂ σ p ( A q ) for q ≤ p . 2. σ r ( A q ) ⊂ σ r ( A p ) for q ≤ p . 3. σ c ( A p ) ⊂ σ c ( A q ) ∪ σ p ( A q ) for q ≤ p ≤ 2. 4. ρ ( A q ) ⊂ ρ ( A p ) for q ≤ p ≤ 2. σ ( A p ) is decreasing for p ∈ [1 , 2] and increasing for p ∈ [2 , ∞ ). 21

Corollary Let p ∈ [2 , ∞ ). Then the followings hold. 1. σ p ( A p ) ∪ σ r ( A p ) = σ p ( A p ∗ ). 2. σ c ( A p ) = σ c ( A p ∗ ). Corollary σ p ( A p ) ⊂ R for p ∈ [2 , ∞ ). Since A 2 is a self-adjoint operator, by using the gen- eral theory of self-adjoint operators on Hilbert spaces it is obtained that σ ( A 2 ) ⊂ R . However, when p ̸ = 2, it does not always hold. 22

5. Example that γ p → p depends on p Let p ∈ [1 , ∞ ). Define a measure ν on [0 , ∞ ) by ν ( dx ) := e − x dx and a differential operator A p on L p ( ν ) by f ∈ W 2 ,p ( ν ; C ); f ′ (0) = 0 { } Dom( A p ) := , A p := d 2 dx 2 − d dx. Note that A 2 is a self-adjoint operator on L 2 ( ν ). The self-adjointness on L 2 ( ν ) implies that { T t } is analytic semigroup on L p ( m ) for p ∈ (1 , ∞ ). 23

Let p ∈ [1 , 2]. Consider the linear transformation I defined by ( If )( x ) := e − x/ 2 f ( x ) . Then, we have ∫ ∞ ∫ ∞ | If ( x ) | p e ( p 2 − 1 ) x dx = | f ( x ) | p ν ( dx ) , 0 0 and f ′ (0) = 0 if and only if 1 2 ( If )(0) + ( If ) ′ (0) = 0. Hence, I is an isometric transformation ν p := e ( p 2 − 1 ) x dx . from L p ( ν ) to L p (˜ ν p ), where ˜ 24

A p on L p (˜ Define a linear operator ˜ ν p ) by f ∈ W 2 ,p ( ν ; C ); 1 { } 2 f (0) + f ′ (0) = 0 Dom(˜ A p ) := , A p := d 2 dx 2 − 1 ˜ 4 . Then, we have the following commutative diagram. A p L p ( ν ) L p ( ν ) − → I ↓ ↓ I ˜ A p L p (˜ → L p (˜ ν p ) − ν p ) By this diagram we have σ p ( A p ) = σ p (˜ A p ) , σ c ( A p ) = σ c (˜ A p ) , σ r ( A p ) = σ r (˜ A p ) . Hence, to see the spectra of A p , it is sufficient to see the spectra of ˜ A p . 25