Martingale transforms and their projection operators on manifolds - PowerPoint PPT Presentation

Martingale transforms and their projection operators on manifolds Fabrice Baudoin Purdue University Probability seminar Based on a joint work with Rodrigo Bañuelos To appear in Potential Analysis

Motivation Let M be a smooth manifold endowed with a smooth measure µ .

Motivation Let M be a smooth manifold endowed with a smooth measure µ . Let X 1 , · · · , X d be locally Lipschitz vector fields defined on M .

Motivation Let M be a smooth manifold endowed with a smooth measure µ . Let X 1 , · · · , X d be locally Lipschitz vector fields defined on M . We consider the Schrödinger operator, d L = − 1 � X ∗ i X i + V , 2 i = 1 where X ∗ i denotes the formal adjoint of X i with respect to µ and where V : M → R is a non-positive smooth function.

Motivation Let M be a smooth manifold endowed with a smooth measure µ . Let X 1 , · · · , X d be locally Lipschitz vector fields defined on M . We consider the Schrödinger operator, d L = − 1 � X ∗ i X i + V , 2 i = 1 where X ∗ i denotes the formal adjoint of X i with respect to µ and where V : M → R is a non-positive smooth function. We study the boundedness in L p , 1 < p < ∞ of the operator � ∞ d � P t X ∗ S A f = i A ij ( t , · ) X j P t fdt , 0 i , j = 1

Motivation Let M be a smooth manifold endowed with a smooth measure µ . Let X 1 , · · · , X d be locally Lipschitz vector fields defined on M . We consider the Schrödinger operator, d L = − 1 � X ∗ i X i + V , 2 i = 1 where X ∗ i denotes the formal adjoint of X i with respect to µ and where V : M → R is a non-positive smooth function. We study the boundedness in L p , 1 < p < ∞ of the operator � ∞ d � P t X ∗ S A f = i A ij ( t , · ) X j P t fdt , 0 i , j = 1 where P t = e tL and A ( t , x ) is a matrix of smooth bounded functions.

Motivation ◮ Such operators naturally appear as projections of martingale transforms.

Motivation ◮ Such operators naturally appear as projections of martingale transforms. ◮ For instance, if A ( t , x ) = a ( t ) Id and V = 0, then � ∞ S A f = − 2 a ( t ) LP 2 t fdt 0

Motivation ◮ Such operators naturally appear as projections of martingale transforms. ◮ For instance, if A ( t , x ) = a ( t ) Id and V = 0, then � ∞ S A f = − 2 a ( t ) LP 2 t fdt = Ψ a ( − L ) f , 0

Motivation ◮ Such operators naturally appear as projections of martingale transforms. ◮ For instance, if A ( t , x ) = a ( t ) Id and V = 0, then � ∞ S A f = − 2 a ( t ) LP 2 t fdt = Ψ a ( − L ) f , 0 � ∞ 0 a ( t ) e − 2 λ t dt . where Ψ a ( λ ) = − 2 λ

Motivation ◮ Such operators naturally appear as projections of martingale transforms. ◮ For instance, if A ( t , x ) = a ( t ) Id and V = 0, then � ∞ S A f = − 2 a ( t ) LP 2 t fdt = Ψ a ( − L ) f , 0 � ∞ 0 a ( t ) e − 2 λ t dt . This is a so-called where Ψ a ( λ ) = − 2 λ multiplier of Laplace transform type.

Motivation ◮ If L is the Laplace-Beltrami operator on a Lie group G of compact type and if A is constant, then � d � − 1 � � X 2 S A f = A ij X i X j f . i i , j i = 1

Motivation ◮ If L is the Laplace-Beltrami operator on a Lie group G of compact type and if A is constant, then � d � − 1 � � X 2 S A f = A ij X i X j f . i i , j i = 1 Defining then the Riesz transforms on G by � − 1 / 2 � d � X 2 R j f = − X j f i i = 1 we see that d � S A f = A ij R i R j f . i , j = 1

Probabilistic representation of S A � d The diffusion ( Y t ) t ≥ 0 with generator − 1 i = 1 X ∗ i X i can be 2 constructed via the Stratonovitch stochastic differential equation d � X i ( Y t ) ◦ dB i dY t = X 0 ( Y t ) dt + t , i = 1

Probabilistic representation of S A � d The diffusion ( Y t ) t ≥ 0 with generator − 1 i = 1 X ∗ i X i can be 2 constructed via the Stratonovitch stochastic differential equation d � X i ( Y t ) ◦ dB i dY t = X 0 ( Y t ) dt + t , i = 1 The celebrated Feynman-Kac formula reads � t � � 0 V ( Y s ) ds f ( Y t ) P t f ( x ) = E x e .

Probabilistic representation of S A Theorem In L p , 1 < p < ∞ , we have lim T →∞ S T A = S A , where � T � � � T � t S T 0 V ( Y s ) ds e − 0 V ( Y s ) ds dM t | Y T = x A f ( x ) = E e . 0 and d � A ij ( T − t , Y t )( X j P T − t f )( Y t ) dB i dM t = t i , j = 1

Probabilistic representation of S A Theorem In L p , 1 < p < ∞ , we have lim T →∞ S T A = S A , where � T � � � T � t S T 0 V ( Y s ) ds e − 0 V ( Y s ) ds dM t | Y T = x A f ( x ) = E e . 0 and d � A ij ( T − t , Y t )( X j P T − t f )( Y t ) dB i dM t = t i , j = 1 Since the conditional expectation is a contraction in L p , we now essentially need to control the L p norm of the stochastic integral � T � T � t 0 V ( Y s ) ds e − 0 V ( Y s ) ds dM t e 0

A variation of the BDG inequality Theorem Let T > 0 and ( M t ) 0 ≤ t ≤ T be a continuous local martingale. Consider the process � t � t � s 0 V s ds e − 0 V u du dM s , Z t = e 0 where ( V t ) 0 ≤ t ≤ T is a non positive adapted and continuous process. For every 0 < p < ∞ , there is a universal constant C p , independent of T, ( M t ) 0 ≤ t ≤ T and ( V t ) 0 ≤ t ≤ T such that � p � �� p � � E sup | Z t | ≤ C p E [ M , M ] 2 . T 0 ≤ t ≤ T

Proof of the variation of the BDG inequality By stopping it is enough to prove the result for bounded M . Let q ≥ 2. We have dZ t = Z t V t dt + dM t

Proof of the variation of the BDG inequality By stopping it is enough to prove the result for bounded M . Let q ≥ 2. We have dZ t = Z t V t dt + dM t and from Itô’s formula we have d | Z t | q = q | Z t | q − 1 sgn ( Z t ) dZ t + 1 2 q ( q − 1 ) | Z t | q − 2 d [ M ] t = q | Z t | q V t dt + q sgn ( Z t ) | Z t | q − 1 dM t + 1 2 q ( q − 1 ) | Z t | q − 2 d [ M ] t .

Proof of the variation of the BDG inequality By stopping it is enough to prove the result for bounded M . Let q ≥ 2. We have dZ t = Z t V t dt + dM t and from Itô’s formula we have d | Z t | q = q | Z t | q − 1 sgn ( Z t ) dZ t + 1 2 q ( q − 1 ) | Z t | q − 2 d [ M ] t = q | Z t | q V t dt + q sgn ( Z t ) | Z t | q − 1 dM t + 1 2 q ( q − 1 ) | Z t | q − 2 d [ M ] t . Since V t ≤ 0, as a consequence of the Doob’s optional sampling theorem, we get that for every bounded stopping time τ , �� τ � E ( | Z τ | q ) ≤ 1 | Z t | q − 2 d [ M ] t 2 q ( q − 1 ) E . 0

Lenglart’s domination inequality Theorem (Lenglart) Let ( N t ) t ≥ 0 be a positive adapted right-continuous process and ( A t ) t ≥ 0 be an increasing process. Assume that for every bounded stopping time τ , E ( N τ ) ≤ E ( A τ ) .

Lenglart’s domination inequality Theorem (Lenglart) Let ( N t ) t ≥ 0 be a positive adapted right-continuous process and ( A t ) t ≥ 0 be an increasing process. Assume that for every bounded stopping time τ , E ( N τ ) ≤ E ( A τ ) . Then, for every k ∈ ( 0 , 1 ) ,  � k  �  ≤ 2 − k � � A k E sup N t 1 − k E .  T 0 ≤ t ≤ T

Proof of the variation of the BDG inequality From the Lenglart’s domination inequality, we deduce then that for every k ∈ ( 0 , 1 ) ,  � k  ��� T � k � � | Z t | q  ≤ C k , q E | Z t | q − 2 d [ M ] t sup . E  0 ≤ t ≤ T 0

Proof of the variation of the BDG inequality From the Lenglart’s domination inequality, we deduce then that for every k ∈ ( 0 , 1 ) ,  � k  ��� T � k � � | Z t | q  ≤ C k , q E | Z t | q − 2 d [ M ] t sup . E  0 ≤ t ≤ T 0 We finally compute ��� T � k � | Z t | q − 2 d [ M ] t E 0  � k  � k ( q − 2 ) �� T � ≤ E | Z t | d [ M ] t sup   0 ≤ t ≤ T 0 1 − 2  � kq  q � 2 � � kq q ≤ E sup | Z t | E [ M ] 2 .   T 0 ≤ t ≤ T

Control of S T A Thanks to the previous result, we are let with the problem of controlling, in L p , the quantity � T d � ( X i P T − t f ) 2 ( Y t ) dt . 0 i = 1

Control of S T A Thanks to the previous result, we are let with the problem of controlling, in L p , the quantity � T d � ( X i P T − t f ) 2 ( Y t ) dt . 0 i = 1 We can use the chain rule to easily check that d d � � 1 i + X 0 + ∂ � � ( X i P T − t f ) 2 ( Y t ) ≤ X 2 ( P T − t f ) 2 ( Y t ) 2 ∂ t i = 1 i = 1

Control of S T A From Itô’s formula, the quantity � d � 1 i + X 0 + ∂ � X 2 ( P T − t f ) 2 ( Y t ) 2 ∂ t i = 1 is the bounded variation part of the sub-martingale ( P T − t f ) 2 ( Y t ) .

Control of S T A From Itô’s formula, the quantity � d � 1 i + X 0 + ∂ � X 2 ( P T − t f ) 2 ( Y t ) 2 ∂ t i = 1 is the bounded variation part of the sub-martingale ( P T − t f ) 2 ( Y t ) . From Lenglart-Lépingle-Pratelli inequality, we have therefore