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

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Meyers inequality and Strong stability for stable-like operators

Hua Ren

Joint work with Prof. Rich Bass

September 5, 2013

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

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Outline

Part I: Preliminaries for L´ evy processes with jumps; Part II: Meyers inequality and strong stability results for stable-like

• perators;

I Caccioppoli inequality and Meyers inequality; I Strong stability of semigroups and heat kernels; Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 2 / 49

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Part I: Preliminaries for L´ evy processes with jumps

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 3 / 49

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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.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 4 / 49

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Brownian motion

The most basic continuous stochastic process is the well-known Brownian motion. Given a filtered probability space (Ω, F, {Ft}t≥0, P), {Bt}t≥0 is a standard Brownian motion if: .

.

.

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B0 = 0, P-a.s.; .

.

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Bt has continuous paths; .

.

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Bt − Bs has the normal distribution N(0, t − s) whenever s < t; .

.

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Bt − Bs is independent of Fs whenever s < t;

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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.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 6 / 49

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The simplest jump stochastic process is the Poisson process. Given a filtered probability space (Ω, F, {Ft}t≥0, P), {Nt}t≥0 is a Poisson process with parameter λ > 0 if .

.

.

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N0 = 0, P-a.s.; .

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The paths of Nt are right continuous with left limits; .

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Nt − Ns is a Poisson r.v. with parameter λ(t − s) whenever s < t; .

.

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Nt − Ns is independent of Fs whenever s < t; Remark: A Poisson process can only have jumps of size 1.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 7 / 49

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

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L´ evy process

Given a probability space (Ω, F, P), a process X is a L´ evy process if .

.

.

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X(0) = 0 P-a.s.; .

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Stationary increments: Xt − Xs

d

= Xt−s whenever s < t; .

.

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Independent increments: Xt − Xs is independent of σ(Xr : r ≤ s) whenever s < t;

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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.

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Every L´ evy process has a modification which is right continuous with left limits (c` adl` ag process). Let Xt− = lims↑t Xs and ∆Xt = Xt − Xt−. Define N(t, A) =

• 0≤s≤t

1A(∆Xs) : 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.

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Three ways to study stochastic processes: Infinitesimal generators; Dirichlet forms; Stochastic differential equations (SDEs);

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Infinitesimal generator

Standard Brownian motion: Lf (x) = 1

2∆f (x);

We can generalize Brownian motion to other continuous diffusions. Two types of operators are common:

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Non-divergence operators: Lf (x) =

d

• i,j=1

aij(x) ∂2f ∂xi∂xj (x) +

d

• i=1

bi(x) ∂f ∂xi (x); Divergence operators: Lf (x) =

d

• i,j=1

∂i(aij(·) ∂f ∂xj (·))(x).

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Poisson process with intensity λ: Lf (x) = λ[f (x + 1) − f (x)]; Stable process: Lf (x) =

• Rd\{0}
• f (x + h) − f (x) − 1{|h|<1}∇f · h
• c

|h|d+α dh, where c is a constant and α ∈ (0, 2).

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We can also generalize stable processes to stable-like processes. Stable-like processes: Lf (x) =

• Rd\{0}
• f (x + h) − f (x) − 1{|h|<1}∇f · h
• n(x, dh),

with some suitable conditions on n. For example: n(x, dh) = A(x,h)

|h|d+α dh.

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Dirichlet forms

To make sense of divergence operators when aij are not differentiable, one looks at the following Dirichlet form: E(f , g) =

d

• i,j=1
• Rd aij(x) ∂f

∂xi (x) ∂g ∂xj (x) dx. For jump processes, the Dirichlet forms one looks at are of the form E(f , g) =

• Rd
• Rd(f (y) − f (x))(g(y) − g(x)) J(dx, dy).

For example: J(dx, dy) =

A(x,y) |x−y|d+α dx dy for α ∈ (0, 2).

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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 dXt = σ(Xt) dBt. The solution to this equation is a diffusion process.

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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 dXt = σ(Xt−) dZt. A c` adl` ag process X is a solution if it satisfies the above equation.

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Part II: Meyers inequality and stability results for stable-like operators

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In this part, we study the stable-like processes associated with the operator Lf (x) =

• Rd
• f (y) − f (x)
• A(x, y)

|x − y|d+α dy, (1) where α ∈ (0, 2), d ≥ 2 and A(x, y) satisfies some suitable conditions. The associated Dirichlet form is given by E(u, v) =

• Rd
• Rd(u(y) − u(x))(v(y) − v(x))

A(x, y) |x − y|d+α dy dx, with domain D = W

α 2 ,2(Rd), a certain Sobolev-Besov space. Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 21 / 49

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Assumptions on A(x, y)

We suppose: .

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Symmetry: A(x, y) = A(y, x); .

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Boundedness: there exists a positive number Λ such that Λ−1 ≤ A(x, y) ≤ Λ for all x, y ∈ Rd.

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Main Result: Meyers inequality

. Theorem . . . . . . . . Let u be the weak solution to Lu = h for h ∈ L2. Then there exists p > 2 and a constant c1 depending on Λ, p, d, and α such that ΓuLp(Rd) ≤ c1

• E(u, u)

1 2 + hL2(Rd) + uL2(Rd)

• ,

where Γu(x) =

Rd (u(y)−u(x))2 |x−y|d+α

dy 1

2 .

Remark: If u ∈ D(L), then there exists c2, such that ΓuLp(Rd) ≤ c2

• hL2(Rd) + uL2(Rd)
• .

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This is the analogue of the Meyers inequality for divergence form operators. An inequality of Meyers says that if aij are uniformly elliptic and u is a weak solution to Ldu = h for h ∈ L2, then not only is ∇u locally in L2 but it is locally in Lp for some p > 2. Remark: In our jump case, the “gradient” of u is Γu.

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In the continuous case, to derive the Meyers inequality one uses three main tools: .

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Caccioppoli inequality; .

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Sobolev-Poincar´ e inequality; .

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.

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Reverse H¨

• lder inequality;

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Difficulties in the jump case

Our proof of the Meyers inequality also begins by proving a Caccioppoli

• inequality. However, as one might expect, our Caccioppoli inequality is not

a local one, which requires the introduction of some new ideas, such as localization, use of the Hardy-Littlewood maximal function, and use of the Sobolev-Besov embedding theorem.

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Caccioppoli inequality

Let u ∈ W 1,2(Rd) be the weak solution to Ldu(x) = h(x), x ∈ Rd, where Ld is the divergence operator and h ∈ L2(Rd). . Theorem (Caccioppoli inequality for divergence operators) . . . . . . . . For all x0 ∈ Rd, and all r, R with 0 < r < R < ∞, we have

• Br(x0)

|∇u|2 dx ≤ c (R − r)2

BR(x0)

|u − uR|2 dx +

• BR(x0)

h2(x) dx

• ,

where uR =

1 |BR(x0)|

• BR(x0) u(x) dx.

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For our stable-like operators, we prove the Caccioppoli inequality for the weak solution to the equation Lu(x) = h(x), x ∈ Rd h ∈ L2(Rd). (2) A function u ∈ W

α 2 ,2(Rd) is called a weak solution to (2) if

E(u, v) = −(h, v) for all v ∈ W

α 2 ,2(Rd),

(3) where (h, v) =

• Rd h(x)v(x) dx.

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Our first key result shows the Caccioppoli inequality for the weak solution to equation (2): . Theorem . . . . . . . . Let x0 ∈ Rd. Suppose u(x) satisfies (3). There exists a constant c1 depending only on Λ, α, and d such that

• BR/2
• Rd(u(y) − u(x))2

A(x, y) |x − y|d+α dy dx ≤ c1

• Rd(u(y) − uR)2ψ(y) dy +
• BR

|h(y)(u(y) − uR)| dy, where ψ(x) = R−α ∧ Rd |x − x0|d+α .

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Fractional Sobolev-Poincar´ e inequality

. Lemma (Fractional Sobolev-Poincar´ e inequality) . . . . . . . . If u ∈ W

α 2 ,q(BR), 1 < q < d, then u ∈ Lp(BR) for p =

2dq 2d−qα, and there

exists a constant c = c(d, q) such that u − uRLp(BR) ≤ c

BR

• BR

(u(y) − u(x))q |x − y|d+ α

2 q

dy dx 1

q .

Recall that f W

α 2 ,q(Ω) = f Lq(Ω) +

Ω

• Ω

(f (y) − f (x))q |x − y|d+ α

2 q

dy dx 1

q . Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 30 / 49

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Based on the fractional Sobolev-Poincar´ e inequality and H¨

• lder’s

inequality, we obtain the following lemma: . Lemma . . . . . . . . There exists q1 ∈ (1, 2) and a constant c1 depending on d, α, and q1 such that if x0 ∈ Rd and R > 0, then u − uRL2(BR) ≤ c1R(α−α1)/2ΓuLq1(BR), where α1 = (2 − q1)d/q1.

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Hardy-Littlewood maximal function

Given a locally integrable function f (x) on Rd, for each r > 0, we define Mf (x) = sup

r>0

1 |B(x, r)|

• B(x,r)

|f (y)| dy. Remark: M is a bounded operator from Lp(Rd) to Lp(Rd) for 1 < p ≤ ∞, i.e., there exists a finite constant c = c(p, d) > 0 such that Mf Lp(Rd) ≤ cf Lp(Rd) for f ∈ Lp(Rd) and p > 1.

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Reverse H¨

• lder inequality

. Theorem (Reverse H¨

• lder inequality)

. . . . . . . . Let r > q > 1, 0 ≤ g ∈ Lq(Ω), 0 ≤ f ∈ Lr(Ω), and suppose for all BR ⊆ Ω, 1 |Br|

• Br(x0)

gq dx ≤ c 1 |BR|

• BR(x0)

g dx q + 1 |BR|

• BR(x0)

f q dx. Then g(x) ∈ Lq+ε(Ω) for some ε > 0. For example: if g(x) ∈ L2(Ω) and g(x)L2(Br) ≤ cg(x)L1(BR), then g(x) ∈ L2+ε(Ω).

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Put everything together

Based on Caccioppoli inequality, we have Γu2

L2(BR/2) ≤ c

• Rd(u(x) − uR)2ψ(x) dx +
• BR

|h(x)(u(x) − uR)| dx ≤ c

• BR

(u(x) − uR)2 dx + c

• Bc

R

u(x)2ψ(x) dx + c

• Bc

R

u2

Rψ(x) dx +

• BR

|h(x)(u(x) − uR)| dx =: J1 + J2 + J3 + J4.

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For term J1, we apply the previous Lemma: J1 =

• BR

(u(x) − uR)2 dx ≤ c

BR

Γu(x)q1 dx 2

q1 Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 35 / 49

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For term J2, J3 and J4, we take use of Hardy-Littlewood maximal function as the main tool: J2 =

• Bc

R

u(x)2ψ(x) dx ≤ c M(u2)(y); J3 =

• Bc

R

u2

Rψ(x) dx ≤ c M(u2)(y);

J4 =

• BR

|h(x)(u(x) − uR)| dx ≤ c

• BR

|h(x)|Mu(x) dx.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 36 / 49

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Combining all these bounds and integrating over y ∈ BR gives

• BR/2

Γu(x)2 dx ≤ c

BR

Γu(x)q1 dx 2

q1

+ c

• BR

M(u2)(x) dx + c

• BR

|h(x)|Mu(x) dx.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 37 / 49

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Let g(x) = Γu(x)q1 and f (x) =

• M(u2)(x) + |h(x)|Mu(x)

q1

2 ,

then 1 |B(x0, R)|

• B(x0,R/2)

g

2 q1 (x) dx

≤ c

• 1

|B(x0, R)|

• B(x0,R)

g(x) dx 2

q1 + c

1 |B(x0, R)|

• B(x0,R)

f

2 q1 (x) dx.

Here is where Reverse H¨

• lder comes into play.

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Strong stability

One of the applications of Meyers inequality is to obtain the stability of solutions to Ldu = h. Suppose one perturbs the coefficients aij slightly. How does this affect the semigroup and fundamental solution associated with the operator Ld?

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 39 / 49

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In the diffusion case, this question has been answered by Chen, Qian, Hu and Zheng (1998).

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Based on our main result — the Meyers inequality for stable-like operators L, strong stability results can be proved along the lines of proof in the diffusion case.

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Let L and L be the stable-like operators of the form (1). Let p(t, x, y) and p(t, x, y) be the corresponding heat kernels, Pt and Pt be the corresponding semigroups.

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An upper bound

. Theorem . . . . . . . . Suppose d > α. There exist q ≥ 2d/α and a constant c1 depending on Λ, d, α, and q such that if f ∈ L2(Rd), then Ptf − Ptf 2

L2 ≤ c1t−

d 2qα GL2qf 2

L2,

(4) where G(x) = supy∈Rd | A(x, y) − A(x, y)|.

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Sketch of the proof

Let u = Ptf − Ptf . We can write Ptf − Ptf 2

L2 = (Ptf −

Ptf , u) = t d ds (Ps Pt−sf , u) ds. = t (−E

• Pt−sf , Psu) +

E( Pt−sf , Psu)

• ds

≤ c t

• E(

Pt−sf , Pt−sf ) 1

2 Γ(Psu)(x)2pG(x)2q ds

=: c · t I × II ds · G(x)2q.

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For I: We can bound this part by using spectral representation theory; For II: This is where we use our Meyers inequality.

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We make use of the following two results for heat kernels: . Theorem (Chen and Kumagai, 2003) . . . . . . . . There are constants c1, c2 > 0 that depend on d, α such that c1 min {t− d

α ,

t |x − y|d+α } ≤ p(t, x, y) ≤ c2 min {t− d

α ,

t |x − y|d+α }, ∀x, y ∈ Rd. . Theorem (Chen and Kumagai, 2003) . . . . . . . . There exist c3 > 0, and γ > 0 such that |p(t, x1, y1) − p(t, x2, y2)| ≤ c3t− d+γ

α (|x1 − x2| + |y1 − y2|)γ. Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 46 / 49

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Strong stability of p(t, x, y)

. Theorem . . . . . . . . Let t > 0. There exist q > 1 and a constant c1 depending on t, Λ, γ, d, α, and q such that for any x, y ∈ Rd |p(t, x, y) − p(t, x, y)| ≤ c1G

γ 2(d+γ)

2q

.

Hua Ren (Joint work with Prof. Rich Bass) Meyers inequality and Strong stability September 5, 2013 47 / 49

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Strong stability of Pt

. Theorem . . . . . . . . Let t > 0. There exist q > 1 and a constant c2 depending on t, Λ, γ, d, α, and q such that for any p ∈ [1, ∞], we have Ptf − Ptf Lp ≤ c2G

γα 2(d+γ)(d+α)

2q

f Lp.

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Thank you!

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