Solving a non-linear stochastic pseudo-differential equation of ... - Core

Stochastic Processes and their Applications 120 (2010) 2447–2467 www.elsevier.com/locate/spa

Solving a non-linear stochastic pseudo-differential equation of Burgers type Niels Jacob, Alexander Potrykus, Jiang-Lun Wu ∗ Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom Wales Institute of Mathematical and Computational Sciences, United Kingdom Received 2 November 2009; received in revised form 20 August 2010; accepted 20 August 2010 Available online 6 September 2010

Abstract In this paper, we study the initial value problem for a class of non-linear stochastic equations of Burgers type of the following form ∂t u + q(x, D)u + ∂x f (t, x, u) = h 1 (t, x, u) + h 2 (t, x, u)Ft,x for u : (t, x) ∈ (0, ∞) × R → u(t, x) ∈ R, where q(x, D) is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, f, h 1 , h 2 : [0, ∞) × R × R → R are measurable functions, and Ft,x stands for a L´evy space-time white noise. We investigate the stochastic equation on the whole space R in the mild formulation and show the existence of a unique local mild solution to the initial value problem by utilising a fixed point argument. c 2010 Elsevier B.V. All rights reserved. ⃝ MSC: 60H15; 35R60 Keywords: Non-linear stochastic pseudo-differential equations; L´evy space-time white noise; Transition density; Mild equations

1. Introduction Starting with a given complete probability space (Ω , F, P; {Ft }t≥0 ), we are concerned with the inital value problem for the following non-linear stochastic equation in one space dimension ∗ Corresponding author at: Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom. Tel.: +44 1792 602757; fax: +44 1792 295843. E-mail addresses: [email protected] (N. Jacob), [email protected] (A. Potrykus), [email protected] (J.-L. Wu).

c 2010 Elsevier B.V. All rights reserved. 0304-4149/$ - see front matter ⃝ doi:10.1016/j.spa.2010.08.007

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 ∂t u(t, x; ω) + q(x, Dx )u(t, x; ω) + ∂x f (t, x, u(t, x; ω)) = h 1 (t, x, u(t, x, ω)) + h 2 (t, x, u(t, x, ω))Ft,x (ω), (t, x, ω) ∈ (0, ∞) × R × Ω  u(0, x, ω) = u 0 (x, ω), (x, ω) ∈ R × Ω

(1.1)

on the domain [0, ∞)×R, where q(x, Dx ) is a pseudo-differential operator with negative definite symbol of variable order of the form q(x, k) := −|k|α(x) for α : R → (0, 2) being measurable, which generates a stable-like process with transition density {G(s, x; t, y)}0(t−s) θ

(3.5)

and |∂z G(s, z; t, x)| = |∂x G(s, z; t, x)|  ≤ C[1 + (t − s)β ] (t − s)

− θ2

+

C(t − s)λ |x − z|θ . (1 + |x − z|1+θ )2

 t − s +  1 1 1 1 |x − z|2+θ |x−z|>(t−s) θ |x−z|≤(t−s) θ (3.6)

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  1 Now by using (in turn) the translation x − z → x ′ , the facts that |x| ≤ (t − s) θ ∪       1 1 1 1 |x| > (t − s) θ = R and |x| ≤ (t − s) θ ∩ |x| > (t − s) θ = φ, the scaling x = (t −s) θ x ′ , and the inequality (a + b)2 ≤ 2a 2 + 2b2 , we have ∫   β 2 2  + (t − s)(1 + (t − s) ) 1  (t − s)− θ + (t − s)β− θ 1 1 1 2+θ θ θ |x − z| |x−z|>(t−s) |x−z|≤(t−s) R 2 (t − s)λ |x − z|θ + dx (1 + |x − z|1+θ )2  2 ∫ 2 2 (t − s)λ |x|θ  (t − s)− θ + (t − s)β− θ + =  dx 1 (1 + |x|1+θ )2 |x|≤(t−s) θ +

(t − s)(1 + (t − s)β ) (t − s)λ |x|θ + |x|2+θ (1 + |x|1+θ )2 ∫ 3 dx + (t − s)2β− θ 

∫

  1 |x|>(t−s) θ

 3 ≤ 2 (t − s)− θ

2 dx

|x|≤1

∫ dx |x|2θ dx 2λ + 2(t − s) (1 + (t − s) ) + 2(t − s) 4+2θ 1+θ )4 |x|>1 |x| R (1 + |x|   3 3 ≤ C(t − s)− θ 1 + (t − s)2β + (1 + (t − s)β )2 + (t − s)2λ+ 2θ , β 2

− θ3

∫

where in the above derivation we have used the facts that ∫ ∫ 1 dx < ∞, dx < ∞, 4+2θ |x| |x|>1 |x|≤1 and ∫ 1 ∫ ∞ |x|2θ dx 2α ≤2 x dx + 2 x −4−2α dx < ∞. 1+θ )4 R (1 + |x| 0 1 The above calculation indicates that these integrals with respect to x are bounded by constants depending on α L , i.e., the constants are independent of θ . Now in order to prove inequality (2.8), by Minkowski’s inequality (cf. e.g. p. 47 of [30]) and the inequality (3.6) in turn, we have 2 ∫ ∫ t ∫ [∂z G(s, z; t, x)]u(s, z)dzds dx ∫

R

R

0

∫ ∫ t

∫ |u(s, z)|

≤ R

0

R

∫ ∫ t ≤C

|u(s, z)| 0

R

∂z |G(s, z; t, x)| dx 2

∫   R

2

1

2

dzds

 2 2 (t − s)− θ + (t − s)β− θ 1

1

|x−z|≤(t−s) θ

(t − s)(1 + (t − s)β )  (t − s)λ |x − z|θ + + 1 1 |x − z|2+θ (1 + |x − z|1+θ )2 |x−z|>(t−s) θ



1

2

2

2

dx

dzds

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∫ t ∫ ≤C R

0

 3 |u(s, z)|(t − s)− 2θ 1 + (t − s)2β + (1 + (t − s)β )2

3 2λ+ 2θ

+ (t − s)

1

2

2 dzds

]2   1 [∫ t ∫ 3 3 2 |u(s, z)|(t − s)− 2θ dzds ≤ C 1 + T 2β + (1 + T β )2 + T 2λ+ θ 0 t

[∫ ≤C

3 − 2θ

(t − s)

R

]2

∫ |u(s, z)|dzds R

0

which roves inequality (2.8). Inequality (2.9) can be verified by utilizing Fubini’s theorem, Minkowski’s inequality, inequality (2.6), and Young’s inequality (cf. e.g. inequality (4) on p. 99 of [30] with p = r = 2 and q = 1 there). In fact, we have 2 ∫ ∫ t ∫ G(s, z; t, x)u(s, z)dzds dx R 0 R ∫ ∫ ∫  t

R

0

|u(s, z)| (t − s)− θ (1 + (t − s)β )1 1

≤C R

1

|x−z|≤(t−s) θ



 2  1 2 2 (t − s)(1 + (t − s)β )  (t − s)λ  dz dx + 1 + ds 1 |x − z|1+θ 1 + |x − z|1+θ |x−z|>(t−s) θ ∫ ∫  t

R

0

(t − s)− θ (1 + (t − s)β )1 1

≤C

1

|x−z|≤(t−s) θ



  (t − s)(1 + (t − s)β )  (t − s)λ  + + 1 dx 1 |x − z|1+θ 1 + |x − z|1+θ |x−z|>(t−s) θ [∫ ] 1 2 2

|u(s, z)|2 dz

×

ds

R

∫ ≤C

t

β

λ

[1 + (t − s) + (t − s) ]

0

∫ [∫ t

2

|u(s, z)| dz

≤C 0

[∫

2

|u(s, z)| dz

2

]1 2

ds

R

2

]1

2

ds

R

as 0 ≤ t − s ≤ T , where we have used the following calculation in deriving the third inequality: 1 by shifting and scaling x − z = (t − s) θ x ′ we obtain ∫  β 1  + (t − s)(1 + (t − s) ) 1  (t − s)− θ (1 + (t − s)β )1 1 1 1+θ θ θ |x − z| |x−z|≤(t−s) |x−z|>(t−s) R    (t − s)λ + dx ≤ C 1 + (t − s)β + (t − s)λ . 1+θ 1 + |x − z|

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Finally, inequality (2.10) can be derived as follows. By inequality (2.6), the Fubini theorem, the Schwarz inequality, and then again by shifting and scaling, we have  ∫ t ∫    G(s, z; t, x)u(s, z)dzds   0 R  ∫ ∫ t

|u(s, z)| (t − s)− θ (1 + (t − s)β )1 1

≤C 0

R



1

|x−z|≤(t−s) θ

 (t − s)(1 + (t − s)β )  (t − s)λ  dzds + 1 + 1 |x − z|1+θ 1 + |x − z|1+θ |x−z|>(t−s) θ   1 ∫ ∫ ∫ t 2  2 β − θ1 ≤C |u(s, z)| dz  1 (t − s) (1 + (t − s) ) 0

+ (t − s)(1 + (t − s)β )

∫

|u(s, z)|2 dz

1

+ (t − s)λ

∫

|u(s, z)|2 dz

 1 ∫

∫ t ∫

R

2

1  2

|u(s, z)| dz

=C 0

1 − 2θ

(t − s)

(1 + (t − s) )

β

(1 + (t − s) )

∫ |z|>1

t

1

 dz 

 12  dz 



2

  ds 1

∫

2

dz |z|≤1

+ (t − s)

≤C

β

R 1 1+ 2θ

∫

R

1



(|x − z|1+θ )2

dz (1 + |x − z|1+θ )2

2

R

 

1

|x−z|>(t−s) θ

∫

2

R

1



2

|x−z|≤(t−s) θ

R

R

1

1 − 2θ

(t − s)

0

[∫

2

|u(s, z)| dz

dz |z|2+2θ

1 2

λ

+ (t − s)

∫ R

dz (1 + |z|1+θ )2

1  2

ds

]1

2

ds. 

R

4. Proof of Theorem 2.3 Our proof to Theorem 2.3 follows the argumentation in [40], but here we have to treat some extra terms as the estimates of the fundamental solution in Proposition 2.1 are very different from the case when α(x) is a constant which was handled in [40]. To this end, we need some preparation. For any fixed n ∈ N, let the mapping   ∫ 1 πn : L 2 (R) → Bn := u ∈ L 2 (R) : ‖u‖ L 2 :=

be defined by  u, if ‖u‖ L 2 ≤ n nu πn (u) = , if ‖u‖ L 2 > n.  ‖u‖ L 2

R

u 2 (x)dx

2

≤n

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Clearly, for any n ∈ N, we have ‖πn (u)‖ L 2 ≤ n. Moreover, it is clear that the norm ‖πn ‖Lip(L 2 ) :=

sup u 1 ,u 2

∈L 2 ,u

1 ̸=u 2

‖πn (u 2 ) − πn (u 1 )‖ L 2 ≤ 1, ‖u 2 − u 1 ‖ L 2

i.e., πn : L 2 (R) → L 2 (R) is a contraction since ‖πn (u)‖ L 2 = ‖πn (u) − πn (0)‖ L 2 ≤ ‖πn ‖Lip(L 2 ) ‖u − 0‖ L 2 ≤ ‖u‖ L 2 . Notice that if u is a solution to Eq. (2.19), then u is an L 2 (R)-valued, {Ft }-progressive process. Thus, by Theorem 2.1.6 in [15] (page 55), for any n ∈ N,   ∫ 2 2 τn (ω) := inf t ∈ [0, T ] : u (t, x, ω)dx ≥ n , ω ∈ Ω R

defines a stopping time. It is clear that {τn }n∈N is an increasing sequence of stopping times determined by u. Moreover, for any fixed n ∈ N, the stopped process u(t ∧ τn ) satisfies the following equation ∫ u(t, x, ω) = G(0, z; t, x)u 0 (z, ω)dz R ∫ t∫ + [∂z G(s, z; t, x)] f (s, z, (πn u)(s, z, ω))dzds 0 R ∫ t∫ + G(s, z; t, x)h 1 (s, z, (πn u)(s, z, ω))dzds 0 R ∫ t∫ + G(s, z; t, x)h 2 (s, z, (πn u)(s, z, ω))W (ds, dz) R

0

t+ ∫

∫

∫

+ R U

0

G(s, z; t, x)h(s, z, (πn u)(s−, z, ω); y)M(ds, dz, dy, ω). (4.1)

On the other hand, any solution of Eq. (4.1) is a local solution of Eq. (2.19). Therefore, the existence of a unique local solution of Eq. (2.19) is equivalent to the existence of a unique solution of Eq. (4.1). Hence, we will focus our attention to showing the existence of a unique solution of Eq. (4.1). Proof of Theorem 2.3. We will carry out the proof by the following three steps. Step 1. Suppose that u : [0, T ] × R × Ω → R is an L 2 (R)-valued, {Ft }-adapted, c`adl`ag process. For any fixed n ∈ N, set ∫ 4 − (J u)(t, x, ω) := G(s, z; t, x)u 0 (z, ω)dz + (Jk u)(t, x, ω) R

k=1

with (J1 u)(t, x, ω) :=

∫ t∫ 0

R

[∂z G(s, z; t, x)] f (s, z, (πn u)(s, z, ω))dzds

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∫ t∫

(J2 u)(t, x, ω) :=

G(s, z; t, x)h 1 (s, z, (πn u)(s, z, ω))dzds

R

0

∫ t∫

(J3 u)(t, x, ω) :=

G(s, z; t, x)h 2 (s, z, (πn u)(s, z, ω))W (ds, dz, ω)

R

0

and ∫

(J4 u)(t, x, ω) :=

t+ ∫

∫

R U

0

G(s, z; t, x)h(s, z, (πn u)(s, z, ω); y)M(ds, dz, dy, ω).

By (2.8) in Lemma 2.2, the condition (2.20), and the Schwarz inequality, we have 2 ∫ t ∫ ∫ 3 2 − 2θ f (s, z, (πn u)(s, z, ω))dzds [(J1 u)(t, x, ω)] dx ≤ C (t − s) R

R

0 t

[∫ ≤C ∫ ≤C

3

(t − s)− 2θ

0 t

3 − 2θ

(t − s)

∫ ∫

ds

0

R

 ]2 |(πn u)(s, z, ω)|2 dzdz ds

3 − 2θ

(t − s)

0

∫

K (z)dz

× R

≤ Ct

R t

K (z)dz + C

∫

3 1− 2θ

t

∫

2

∫ +C R

3 − 2θ

(t − s)

∫ R

0 3

|(πn u)(s, z, ω)| dzdz 2

K (z)dz

2  ds



2 + Cn

4

ds

3

≤ Ct 2− θ ≤ C T 2− θ < ∞. Notice that here and in what follows the constant C also depends on n (of course C depends on T as well). By inequality (2.9) in Lemma 2.2, we get ∫ ∫  1 2 ∫ t 2 2 2 [(J2 u)(t, x, ω)] dx ≤ C |h 1 (s, z, (πn u)(s, z, ω))| dz ds R

R

0

≤C

∫ [∫  t R

0

∫ ∫ t ≤C R

0

L 1 (z) + C[(πn u)(s, z, ω)]

2

L 1 (z)dz + Cn 2



2

]1 2

dz

ds

2

1 2

ds

≤ Ct 2 ≤ C T 2 < ∞.

On the other hand, by Itˆo’s isometry property for stochastic integrals with respect to (both continuous and c`adl`ag) martingales (cf. e.g. [19]), we have [∫ ] 2 E |(J3 u)(t, x, ·)| dx R [∫ ∫ t ∫  ] 2 2 =E G (s, z; t, x)h 2 (s, z, (πn u)(s, z, ·))dzds dx R

0

R

and [∫ E R

 ∫ ∫ ∫ t |(J4 u)(t, x, ·)| dx = E G 2 (s, z; t, x) 2

]

R

0

R

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∫ ×

   h (s, z, (πn u)(s, z, ·); y)ν(dy) dzds dx . 2

U

Thus, by inequality (2.7) and the condition (2.22), we get [∫ ] ∫ 2 2 |(J3 u)(t, x, ·)| dx + |(J4 u)(t, x, ·)| dx E R R [∫ t ∫ ∫ 1 ≤ CE (t − s)− θ h 22 (s, z, (πn u)(s, z, ·)) 0 R R  ] ∫ 2 h (s, z, (πn u)(s, z, ·); y)ν(dy) dzds + U  ∫ ∫ t ∫ 2 − θ1 E[(πn u)(s, z, ·)] dz ds ≤C (t − s) L 2 (z)dz + C R 0 R ∫  1 1 1 ≤ Ct 1− θ L 2 (z)dz + Cn 2 ≤ Ct 1− θ ≤ C T 1− θ < ∞. R

Therefore, we obtain that ∫    3 1 E |(J u)(t, x, ·)|2 dx ≤ C T 2− θ + T 2 + T 1− θ < ∞ R

for any fixed t ∈ [0, T ]. 2 (R)-valued, {F }-predictable, and square Step 2. Now let ρ > 0 be arbitrary but fixed. For any L t integrable process u : [0, T ] × R × Ω → R (i.e., E R u 2 (t, x, ·)dx < ∞ for all t ∈ [0, T ], with initial condition u(0, x, ω) = u 0 (x, ω)), we define ∫  ∫ T 2 −ρt 2 ‖u‖ρ := e E u (t, x, ·)dx dt. 0

R

Clearly, ‖·‖ρ Let B denote the completion of the collection of all L 2 (R)-valued, {Ft }predictable processes u : [0, T ] × R × Ω → R with the initial condition u(0, x, ω) = u 0 (x, ω), such that ∫  ∫ T ‖u‖2ρ = e−ρt E u 2 (t, x, ·)dx dt < ∞ is a norm.2

0

R

with respect to the norm ‖ · ‖ρ , namely, (B, ‖ · ‖ρ ) is a Banach space. Obviously, we have that our initial data u 0 ∈ B. Now for all u ∈ B, J u is well defined and for any fixed t ∈ [0, T ] ∫    3 1 2 E |(J u)(t, x, ·)| dx ≤ C T 2− θ + T 2 + T 1− θ < ∞. R

Thus, by the formula ∫ ∞ t λ−1 e−st dt = Γ (λ)s −λ 0

2 cf. e.g. page 794 of [16], or one can verify the definition of a norm directly from elementary properties of the Laplace transform, see e.g. page 145 of [14].

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we have ‖J u‖2ρ

∞

∫ ≤C

t

2− θ3

2

+t +t

1− θ1



−ρt

e

[

dt ≤ C ρ

  − 3− θ3

+ρ

−3

+ρ

 ] − 2− θ1

< ∞,

0

i.e., J u ∈ B, which implies that J (B) ⊂ B. Step 3. Now for any u, v ∈ B, by (2.8)–(2.10) in Lemma 2.2 and the Lipschitz condition (2.23), together with Fubini’s theorem, and the Schwarz inequality, we get for any t ∈ [0, T ] ∫  2 E |(J1 u)(t, x, ·) − (J1 v)(t, x, ·)| dx R

t

∫

≤ CE (t − s) ∫ 0 t

≤ CE

∫

3 − 2θ

R 3

(t − s)− 2θ

∫ R

0

| f (s, z, (πn u)(s, z, ·)) − f (s, z, (πn v)(s, z, ·))|dzds

2

(L 3 (z) + C[(πn u)2 (s, z, ·) 2

1 2

+ (πn v) (s, z, ·)]) |(πn u)(s, z, ·) − (πn v)(s, z, ·)|dzds 2

t

∫ ≤C

3 − 2θ

(t − s)

t

∫ dsE 0

0

3 − 2θ

(t − s)

∫ R

(L 3 (z) + C[(πn u)2 (s, z, ·) 2

1 2

+ (πn v) (s, z, ·)]) |(πn u)(s, z, ·) − (πn v)(s, z, ·)|dz 2

∫

t

3 − 2θ

(t − s)

ds

∫

(L 3 (z) + C[(πn u)2 (s, z, ·) ∫ |(πn u)(s, z, ·) − (πn v)(s, z, ·)|2 dzds + (πn v)2 (s, z, ·)])dz  ∫ R ∫ t 3 − 2θ ≤C (t − s) |u(s, z, ·) − v(s, z, ·)|2 dz ds, E ≤ CE

R

0

R

0

where in the derivation of the fourth inequality above, we have used the fact that for θ >     ∫ t 3 3 3 3 3 − 2θ 1− 2θ (t − s) ds = 1 − t ≤ 1− T 1− 2θ < ∞. 2θ 2θ 0 Moreover, ∫  E |(J2 u)(t, x, ·) − (J2 v)(t, x, ·)|2 dx R ∫ ∫ ∫ t

G(s, z; t, x)[L 3 (z) + C((πn u)2 (s, z, ·)

≤E R

0

R

1 2

+ (πn v) (s, z, ·))] |(πn u)(s, z, ·) − (πn v)(s, z, ·)|dzds 2

∫ ∫ t ∫ ≤ CE R 0

R

G(s, z; t, x)[L 3 (z) + C((πn u)2 (s, z, ·)

2 dx

3 2

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2

1 2

+ (πn v) (s, z, ·))] |(πn u)(s, z, ·) − (πn v)(s, z, ·)|dz 2

∫ t ∫ ∫ = CE R

0

R

dsdx

G(s, z; t, x)[L 3 (z) + C((πn u)2 (s, z, ·) 2

1 2

+ (πn v)2 (s, z, ·))] |(πn u)(s, z, ·) − (πn v)(s, z, ·)|dz ] ∫

∫ t ∫ [∫ ≤ CE

G(s, z, ; t, x)dz R

0

R

R

dxds

G(s, z, ; t, x)[L 3 (z) 

+ C((πn u) (s, z, ·) + (πn v) (s, z, ·))]|(πn u)(s, z, ·) − (πn v)(s, z, ·)| dz dxds 2

2

∫ t ∫ ∫ = CE R

0

R

2

G(s, z, ; t, x)dx[L 3 (z) + C((πn u)2 (s, z, ·) 

+(πn v) (s, z, ·))]|(πn u)(s, z, ·) − (πn v)(s, z, ·)| dz ds 2

2

∫ t∫

[L 3 (z) + C((πn u)2 (s, z, ·) 0 R ∫ |(πn u)(s, z, ·) − (πn v)(s, z, ·)|2 dzds + (πn v)2 (s, z, ·))]dz R ∫ t ∫ ≤C E |u(s, z, ·) − v(s, z, ·)|2 dzds ≤ CE

0

R

where we have used Fubini’s theorem and the property that ∫ ∫ G(s, z, ; t, x)dx = 1, 0 ≤ s < t < ∞ G(s, z, ; t, x)dz = R

R

in the above derivation. Furthermore, by Itˆo’s isometry for both stochastic integrals with respect to W (ds, dz) and M(ds, dz, dy, ω), we have ∫ E |(J3 u)(t, x, ·) + (J4 u)(t, x, ·) − (J3 v)(t, x, ·) − (J4 v)(t, x, ·)|2 dx R  ∫ ∫ t ∫ 2 2 ≤ CE G (s, z; t, x)|(πn u)(s, z, ·) − (πn v)(s, z, ·)| dsdz dx R 0 R  ∫ ∫ t ∫ 2 2 ≤ CE G (s, z; t, x)|u(s, z, ·) − v(s, z, ·)| dzds dx R 0 R  ∫ t ∫ ∫ 2 = CE G (s, z; t, x)dx |u(s, z, ·) − v(s, z, ·)|2 dzds 0 R R ∫ t ∫ − θ1 ≤C (t − s) E |u(s, z, ·) − v(s, z, ·)|2 dzds. 0

R

Combining all the above together, we obtain

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N. Jacob et al. / Stochastic Processes and their Applications 120 (2010) 2447–2467

‖J (u − v)‖2ρ =

∫

T

e−ρt E

∫ R

0

∫

T

∫

T

 |(J u)(t, x, ·) − (J v)(t, x, ·)|2 dx dt

∫ t



3

1

3 − 2θ

− θ1

(t − s)− 2α + 1 + (t − s)− α e−ρt 0 0 ∫  ×E |u(s, z, ·) − v(s, z, ·)|2 dz dsdt

≤C

R

[∫

T

e

=C 0

(t − s)



] dt

s

∫ ×E R T

∫



+ 1 + (t − s)  2 |u(s, z, ·) − v(s, z, ·)| dz ds −ρt

[∫

∞



3 − 2θ

− θ1

0

(t − s)



]

+ 1 + (t − s) dt  ×E |u(s, z, ·) − v(s, z, ·)|2 dz ds R ∫ ∞  ∫ ∞ ∫ ∞ 3 −ρt − 2θ −ρt −ρt − θ1 =C e t dt + e tdt + e t dt e

≤C

−ρt

s

∫

0 T

0

0

 |u(s, z, ·) − v(s, z, ·)|2 dz ds e−ρs E 0 R     3 Γ 1 − 2θ Γ 1 − θ1 1  ‖u − v‖2ρ ≤ C + + 1 3 ρ ρ 1− θ ρ 1− 2θ   1 1 1 ‖u − v‖2ρ . ≤C + + 3 1− 2θ 1− θ1 ρ ρ ρ ∫

∫

×

Here the constant C also depends on both u and v. Notice that θ ∈ [α L , αU ] ⊂ 3 2

< θ < 2 and hence 1 − > 0 and 1 −   1 1 1 + + C 0. Thus, let us take ρ large enough so that

which implies that J : B → B is a contraction. Therefore there must be a unique fixed point in B for J and this fixed point is the unique solution for Eq. (4.1). In order to see that this gives us a local solution to Eq. (2.19), let us denote by u n the unique solution of the Eq. (4.1) for each n ∈ N. For this u n , let us set the stopping time   ∫ τn (ω) := inf t ∈ [0, T ] : u 2n (t, x, ω)dx ≥ n 2 , ω ∈ Ω . R

Clearly by the contraction property of J , we have for all j ≥ n and for almost all ω ∈ Ω u j (t, ·, ω) = u n (t, ·, ω), Therefore we define u(t, x, ω) := u n (t, x, ω)

for all (t, ω) ∈ [0, τn ) × Ω .

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N. Jacob et al. / Stochastic Processes and their Applications 120 (2010) 2447–2467

for any (t, x, ω) ∈ [0, τn ) × R × Ω and τ∞ (ω) := sup τn (ω). n∈N

Then {u(t, x, ω) : (t, x, ω) ∈ [0, τ∞ ) × R × Ω } is a local solution of Eq. (2.19). Finally, for the uniqueness of the local solution to Eq. (2.19), suppose that there are two local solutions u and v to Eq. (2.19). Then u and v must satisfy Eq. (4.1) for any fixed n ∈ N. On the other hand, by the uniqueness of the solution to Eq. (4.1), we get u(t, x, ω) = v(t, x, ω),

for all (t, x, ω) ∈ [0, τn ) × R × Ω .

Now let n → ∞, we deduce u(t, x, ω) = v(t, x, ω),

for all (t, x, ω) ∈ [0, τ∞ ) × R × Ω .

Hence we obtain the uniqueness.



Acknowledgements The authors are grateful to Aubrey Truman for many fruitful discussions. The second named author thanks RCUK for the financial support. The third named author wishes to thank Institut Mittag–Leffler of the Royal Swedish Academy of Sciences, for the hospitality and excellent working conditions where part of this work was done. The authors would like to thank the associated editor and two referees for their constructive comments and useful suggestions which led to a big improvement in the presentation of the manuscript. References [1] S. Albeverio, B. R¨udiger, J.-L. Wu, Invariant measures and symmetry property of L´evy type operators, Potential Anal. 13 (2000) 147–168. [2] S.S. Antman, The equations for large vibrations of strings, Amer. Math. Monthly 87 (1980) 359–370. [3] R.F. Bass, Uniqueness in law for pure jump type Mark processes, Probab. Theory Related Fields 79 (1988) 271–287. [4] R.F. Bass, M. Kassmann, H¨older continuity of harmonic functions with respect to operators of variable order, Comm. Partial Differential Equations 30 (7–9) (2005) 1249–1259. [5] R.F. Bass, D.A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. 354 (2002) 2933–2953. [6] P. Biler, T. Funaki, W.A. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998) 9–46. [7] P. Biler, T. Funaki, W.A. Woyczynski, Interacting particle approximation for nonlocal quadratic evolution problems, Probab. Math. Statist. 19 (1999) 321–340. [8] P. Biler, G. Karch, W.A. Woyczynski, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999) 231–252. [9] P. Biler, G. Karch, W.A. Woyczynski, Multifractal and L´evy conservation laws, C. R. Acad. Sci. Paris S´er. I Math. 330 (2000) 343–348. [10] P. Biler, G. Karch, W.A. Woyczynski, Asymptotics for conservation laws involving L´evy diffusion generators, Studia Math. 148 (2001) 171–192. [11] P. Biler, G. Karch, W.A. Woyczynski, Critical nonlinear exponent and self-similar asymptotics for L´evy conservation laws, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 18 (2001) 613–637. [12] P. Biler, W.A. Woyczynski, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math. 59 (1999) 845–869. [13] Z.-Q. Chen, T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl. 108 (2003) 27–62.

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