Meyers inequality and Strong stability for stable-like operators - PowerPoint PPT Presentation

Meyers inequality and Strong stability for stable-like operators Hua Ren Joint work with Prof. Rich Bass September 5, 2013 1 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 1 / 49

Outline Part I: Preliminaries for L´ evy processes with jumps; Part II: Meyers inequality and strong stability results for stable-like operators; I Caccioppoli inequality and Meyers inequality; I Strong stability of semigroups and heat kernels; 2 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 2 / 49

Part I: Preliminaries for L´ evy processes with jumps 3 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 3 / 49

Continuous stochastic processes Continuous stochastic processes have been widely applied in modeling in many areas; for example, the Ornstein-Uhlenbeck process and geometric Brownian motion. 4 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 4 / 49

Brownian motion The most basic continuous stochastic process is the well-known Brownian motion. Given a filtered probability space (Ω , F , {F t } t ≥ 0 , P ), { B t } t ≥ 0 is a standard Brownian motion if: . . . B 0 = 0, P -a.s.; 1 . . . B t has continuous paths; 2 . . . B t − B s has the normal distribution N (0 , t − s ) whenever s < t ; 3 . . . B t − B s is independent of F s whenever s < t ; 4 5 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 5 / 49

Stochastic processes with jumps However, continuous models suffer from some serious defects. For example, stock prices will at times decrease too fast to be followed by a geometric Brownian motion. A model that better fits the data is a geometric Brownian motion with jumps at random times. 6 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 6 / 49

The simplest jump stochastic process is the Poisson process. Given a filtered probability space (Ω , F , {F t } t ≥ 0 , P ), { N t } t ≥ 0 is a Poisson process with parameter λ > 0 if . . . N 0 = 0, P -a.s.; 1 . . . The paths of N t are right continuous with left limits; 2 . . . N t − N s is a Poisson r.v. with parameter λ ( t − s ) whenever s < t ; 3 . . . N t − N s is independent of F s whenever s < t ; 4 Remark: A Poisson process can only have jumps of size 1. 7 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 7 / 49

8 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 8 / 49

L´ evy process Given a probability space (Ω , F , P ), a process X is a L´ evy process if . . . X (0) = 0 P -a.s.; 1 . . d . Stationary increments: X t − X s = X t − s whenever s < t ; 2 . . . Independent increments: X t − X s is independent of σ ( X r : r ≤ s ) 3 whenever s < t ; 9 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 9 / 49

Brownian motions and Poisson processes are both L´ evy process. In this talk, we will study a class of pure jump L´ evy processes— stable and stable-like processes. 10 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 10 / 49

Every L´ evy process has a modification which is right continuous with left limits (c` adl` ag process). Let X t − = lim s ↑ t X s and ∆ X t = X t − X t − . Define � N ( t , A ) = 1 A (∆ X s ) : 0 ≤ s ≤ t the number of the jumps whose size is in the set A ; ν ( A ) = E ( N (1 , A )) : the jump intensity measure of X which is called the L´ evy measure. 11 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 11 / 49

Three ways to study stochastic processes: Infinitesimal generators; Dirichlet forms; Stochastic differential equations (SDEs); 12 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 12 / 49

Infinitesimal generator Standard Brownian motion: L f ( x ) = 1 2 ∆ f ( x ); We can generalize Brownian motion to other continuous diffusions. Two types of operators are common: 13 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 13 / 49

Non-divergence operators: � d � d a ij ( x ) ∂ 2 f b i ( x ) ∂ f L f ( x ) = ( x ) + ( x ); ∂ x i ∂ x j ∂ x i i , j =1 i =1 Divergence operators: � d ∂ i ( a ij ( · ) ∂ f L f ( x ) = ( · ))( x ) . ∂ x j i , j =1 14 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 14 / 49

Poisson process with intensity λ : L f ( x ) = λ [ f ( x + 1) − f ( x )]; Stable process: � � � c L f ( x ) = f ( x + h ) − f ( x ) − 1 {| h | < 1 } ∇ f · h | h | d + α dh , R d \{ 0 } where c is a constant and α ∈ (0 , 2). 15 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 15 / 49

We can also generalize stable processes to stable-like processes. Stable-like processes: � � � L f ( x ) = f ( x + h ) − f ( x ) − 1 {| h | < 1 } ∇ f · h n ( x , dh ) , R d \{ 0 } with some suitable conditions on n . For example: n ( x , dh ) = A ( x , h ) | h | d + α dh . 16 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 16 / 49

Dirichlet forms To make sense of divergence operators when a ij are not differentiable, one looks at the following Dirichlet form: � � d R d a ij ( x ) ∂ f ( x ) ∂ g E ( f , g ) = ( x ) dx . ∂ x i ∂ x j i , j =1 For jump processes, the Dirichlet forms one looks at are of the form � � E ( f , g ) = R d ( f ( y ) − f ( x ))( g ( y ) − g ( x )) J ( dx , dy ) . R d A ( x , y ) For example: J ( dx , dy ) = | x − y | d + α dx dy for α ∈ (0 , 2). 17 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 17 / 49

Stochastic differential equations (SDEs) We can construct other stochastic processes via stochastic differential equations. A one-dimensional SDE driven by Brownian motion is given by dX t = σ ( X t ) dB t . The solution to this equation is a diffusion process. 18 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 18 / 49

We can get jump type SDEs by replacing the Brownian motion by a L´ evy process with jumps. A one-dimensional SDE driven by a stable process of order α ∈ (0 , 2) is given by dX t = σ ( X t − ) dZ t . A c` adl` ag process X is a solution if it satisfies the above equation. 19 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 19 / 49

Part II: Meyers inequality and stability results for stable-like operators 20 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 20 / 49

In this part, we study the stable-like processes associated with the operator � � � A ( x , y ) L f ( x ) = f ( y ) − f ( x ) | x − y | d + α dy , (1) R d where α ∈ (0 , 2), d ≥ 2 and A ( x , y ) satisfies some suitable conditions. The associated Dirichlet form is given by � � A ( x , y ) E ( u , v ) = R d ( u ( y ) − u ( x ))( v ( y ) − v ( x )) | x − y | d + α dy dx , R d α 2 , 2 ( R d ), a certain Sobolev-Besov space. with domain D = W 21 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 21 / 49

Assumptions on A ( x , y ) We suppose: . . . Symmetry: A ( x , y ) = A ( y , x ); 1 . . . Boundedness: there exists a positive number Λ such that 2 Λ − 1 ≤ A ( x , y ) ≤ Λ for all x , y ∈ R d . 22 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 22 / 49

Main Result: Meyers inequality . Theorem . . . Let u be the weak solution to L u = h for h ∈ L 2 . Then there exists p > 2 and a constant c 1 depending on Λ , p , d, and α such that � � 1 2 + � h � L 2 ( R d ) + � u � L 2 ( R d ) � Γ u � L p ( R d ) ≤ c 1 E ( u , u ) , � � � 1 2 . ( u ( y ) − u ( x )) 2 where Γ u ( x ) = dy R d | x − y | d + α . . . . . Remark: If u ∈ D ( L ), then there exists c 2 , such that � � � Γ u � L p ( R d ) ≤ c 2 � h � L 2 ( R d ) + � u � L 2 ( R d ) . 23 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 23 / 49

This is the analogue of the Meyers inequality for divergence form operators. An inequality of Meyers says that if a ij are uniformly elliptic and u is a weak solution to L d u = h for h ∈ L 2 , then not only is ∇ u locally in L 2 but it is locally in L p for some p > 2. Remark: In our jump case, the “gradient” of u is Γ u . 24 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 24 / 49

In the continuous case, to derive the Meyers inequality one uses three main tools: . . . Caccioppoli inequality; 1 . . . Sobolev-Poincar´ e inequality; 2 . . . Reverse H¨ older inequality; 3 25 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 25 / 49