Journal of Theoretical Probability, Vol. 17, No. 3, July 2004 (© 2004)
On Symmetric Stable-like Processes: Some Path Properties and Generators T. Uemura Received June 25, 2002; revised October 25, 2003
We derive some path properties of symmetric stable-like processes constructed via Dirichlet form theory and then sufficient conditions in order that the generators of the forms contain a nice functions space, are given.
KEY WORDS: Symmetric stable-like processes; exceptionality of points; transience; recurrence; generators.
1. INTRODUCTION We consider the following quadratic form for a measurable function α(·, ·) defined on Rd × Rd : 2 (u(x) − u(y)) α d 2 dx dy < ∞ , D[E ] = u ∈ L (R ; dx) : |x − y|d+α(x,y) d d R ×R (1) (u(x) − u(y))(v(x) − v(y)) α E (u, v) = dx dy. |x − y|d+α(x,y) Rd ×Rd
In Ref. 9, we considered the special case that α(x, y) = α(x) for all x, y ∈ Rd , that is, the function α(x, y) depends only on the one variable x. Since the measure µ(dx, dy) = |x − y|−d−α(x,y) dx dy is not necessarily a symmetric measure on Rd × Rd , it seems that the form (Eα , D[Eα ]) is a non-symmetric form on L2 (Rd ) at first sight. As a functional of functions 1 School
of Business Administration, University of Hyogo, Nishi, Kobe, Japan, 651-2197. E-mail
[email protected] 541 0894-9840/04/0700-0541/0 © 2004 Springer Science+Business Media, Inc.
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on L2 (Rd ) × L2 (Rd ), the form is indeed symmetric. In particular, the jump (or L´evy) measure corresponding to the form is given by j (dx, dy) =
1 1 dx dy. + |x − y|d+α(x,y) |x − y|d+α(y,x)
This form (1) is a generalization of the Dirichlet forms corresponding to symmetric stable processes, and the associated Markov process (if it exists) is called “symmetric stable-like process” (see Ref. 9). Of course, though we adopt the techniques of the theory of Dirichlet forms, there are many generalizations of symmetric stable processes using other methods (these generalized processes are called stable-like processes); the martingale problems, stochastic differential equations with jumps, pseudo differential operators and so on (see e.g. Ref. 1, 3, 4 and 8). We should note that the processes generated by using these techniques are in general non-symmetric. In this short paper, after we summarize and extend some path properties for the symmetric stable-like processes obtained in Ref. 9, by using an idea developed in Ref. 5, sufficient conditions that the domains of the infinitesimal generators contain C02 (Rd ), the family of twice differentiable functions with compact support, are given. In fact, for u in C02 (Rd ), we can see that the generator is given by the following under some conditon (see Theorem 4.1): Au(x) =
Rd
(u(y) − u(x) − ∇u(x) · (y − x)1|x−y|1 )
×(|x − y|−d−α(x,y) + |x − y|−d−α(y,x) )dy,
x ∈ Rd .
2. CLOSABILITY OF THE SYMMETRIC STABLE-TYPE FORMS In the same way of the argument developed in Ref. 9, we can show the following theorem: Theorem 2.1. If {(x, y) ∈ Rd × Rd : α(x, y) = −∞ or ∞} is a set of zero Lebesgue measure, then the form (Eα , D[Eα ]) is a Dirichlet form on L2 (Rd ; dx) in the wide sense. Then we also want to know when the domain D[Eα ] contains the space of all uniformly Lipschitz continuous functions with compact support. As for this question, we can state the following theorem: lip C0 (Rd ),
On Symmetric Stable-like Processes
543 lip
Theorem 2.2. The domain D[Eα ] contains C0 (Rd ) if and only if the following conditions are satisfied: for any compact set K and relatively compact set G with K ⊂ G, (1) |x − y|2−d−α(x,y) dx dy < ∞, K×K
(2)
(|x − y|−d−α(x,y) + |x − y|−d−α(y,x) )dx dy < ∞.
K×(Rd −G)
The proof of this theorem is similar to that of Theorem 2.1 in Ref. 9 but a bit easier, so we omit it. Remark 2.1. (i)
If the function α(x, y) depends only on the one variable, say x, then the two conditions in the theorem can be reduced to the following three conditions (see Theorem 2.2 in Ref. 9). That is, when there exists a function α(·) on Rd satisfying α(x, y) = α(x) lip for all x, y ∈ Rd , the domain D[Eα ] contains C0 (Rd ) if and only if the following three conditions are satisfied: (1)
0 < α(x) < 2 a.e.,
(2)
1 1 , ∈ L1loc (Rd ), 2 − α(x) α(x)
(3)
there exists a compact set K such that |x|−d−α(x) dx < ∞. Rd −K
(ii)
In the case that there exists a measurable function α(·) ˜ defined ˜ − y|), the on R+ = [0, ∞) with finite values so that α(x, y) = α(|x conditions in the theorem are equivalent to the following condition: ∞ ˜ (1 ∧ u2 )u−1−α(u) du < ∞. (2) 0
In this case, we have a translation invariant Dirichler form. So if there exists a Markov process associated with the form, then it is a symmetric L´evy process. We set that Eα1 (u, v) ≡ Eα (u, v) + (u, v)L2 ,
u, v ∈ D[Eα ].
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Two non-negative functionals f1 , f2 on a function space H is called equivalent (designated f1 ≈ f2 ) if there exists a positive constant A such that Af1 (u) f2 (u) A−1 f1 (u)
for all u ∈ H.
Proposition 2.1 shows that if the function α(x, y) equals to the constant function β(0 < β < 2) near the diagonal {(x, x): x ∈ Rd }, then the form Eα1 is equivalent to the norm, which corresponds to the symmetric β-stable process, added to the L2 -norm. Proposition 2.1. Suppose that either one of the following conditions holds: (i)
there exist ε 1, 0 β < 2 and γ > 0 so that α(x, y) = β α(x, y) γ
(ii)
if |x − y| ε, if |x − y| ε,
there exist 0 < ε < 1, 0 β < 2, γ1 > 0 and γ2 > 0 so that α(x, y) = β α(x, y) γ1 α(x, y) γ2
if |x − y| ε, if ε < |x − y| < 1, otherwise.
lip
Then, for any u ∈ C0 (Rd ), it holds that Eα1 (u, u) ≈ Rd ×Rd
(u(x) − u(y))2 dx dy + ||u||2L2 . |x − y|d+β
Proof. It suffices to show the case (ii) only. So we assume (ii). Take lip u ∈ C0 (Rd ). Then we see that 2 (u(x) − u(y)) + dx dy Eα (u, u) = d+α(x,y) |x − y| |x−y|ε
ε<|x−y|
(u(x) − u(y)2 dx dy |x − y|d+β |x−y|ε
1 1 dx dy u2 (x) + +2 |x − y|d+α(x,y) |x − y|d+α(y,x) ε<|x−y|
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545
(u(x) − u(y))2 dxdy |x − y|d+β |x−y|ε 1 u2 (x)dx dx dy +4 d+γ1 Rd ε|x−y|<1 |x − y| 1 +4 u2 (x)dx dx dy. d+γ2 Rd 1|x−y| |x − y|
So, for some constant C1 > 0, we have Eα1 (u, u) |x−y|ε
(u(x) − u(y))2 dx dy + C1 ||u||2L2 . |x − y|d+β
(3)
Conversely, it holds that Eα (u, u) = Rd ×Rd
(u(x) − u(y))2 dx dy |x − y|d+α(x,y)
|x−y|ε
(u(x) − u(y))2 dx dy. |x − y|d+β
lip
We thus have, for all u ∈ C0 (Rd ), Eα1 (u, u) ≈ |x−y|ε
(u(x) − u(y))2 dx dy + ||u||2L2 . |x − y|d+β lip
On the other hand, a similar calculation shows that, u ∈ C0 (Rd ), Rd ×Rd
(u(x) − u(y))2 dx dy + ||u||2L2 ≈ |x − y|d+β
|x−y|ε
(u(x) − u(y))2 dx dy + ||u||2L2 . |x − y|d+β
Therefore the desired equivalence holds. 3. SOME PATH PROPERTIES OF THE PROCESSES Under the conditions (1) and (2) in Theorem 2.2, denote by Fα the lip closure of C0 (Rd ) with respect to the Eα1 -metric. (Note that we assume a priori that −∞ < α(x, y) < +∞ a.e. x, y ∈ Rd ). Then (Eα , Fα ) is a (symmetric) regular Dirichlet form on L2 (Rd ). So there exists a symmetric Hunt process Mα = (Xt , Px ) associated with (Eα , Fα ) (see Ref. 2). We also call this process symmetric stable-like process with the variable index α.
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In this section, we consider some path properties for the symmetric stable-like process when the function α satisfies the conditions in Remark 2.1((1) –(3) or (2)): (i) α depends only on the one variable (i.e., α(x, y) = α(x)); (ii) α depends only on the distance between x and y (i.e., α(x, y) = α(|x ˜ − y|)). Now we give some results on whether points are exceptional or not under the respective conditions for α, though we have already shown some results about it when α depends only on the one variable and d = 1 in Ref. 9. Theorem 3.1. (i)
Let x0 ∈ Rd .
The first case: α(x, y) = α(x). (1)
Assume that d = 1 (see Theorem 3.1. in Ref. 9). (1.a)
If there exists an open set G with x0 ∈ G so that α(x) 1
(1.b)
a.e. on G,
then {x0 } is exceptional. If there exist an open set G with x0 ∈ G and a constant α0 > 1 so that α(x) α0
a.e. on G,
then {x0 } is not exceptional (that is, the process hits the point x0 ). (2)
Assume that d 2. If there exist an open set G with x0 ∈ G and a constant 0 < α1 < 2 such that α(x) α1
a.e. on G,
then {x0 } is exceptional. In particular, when d 2 and the function α is continuous, then any point in Rd is exceptional provided that 0 < α(x) < 2 for all x. (ii) The second case: α(x, y) = α(|x ˜ − y|). (1)
Assume that d = 1. (1.a)
If there exists a u0 > 0 so that α(x) ˜ 1
a.e. on [0, u0 ),
then {x0 } is exceptional.
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If there exist a u0 > 0 and a constant α˜ 0 > 1 so that
(1.b)
α(u) ˜ α˜ 0
a.e. on [0, u0 ),
then {x0 } is not exceptional (that is, the process hits the point x0 ). (2)
Assume that d 2. Then any point is exceptional.
The second case means that all points are either exceptional or not exceptional according to the behavior of the function α˜ at the origin. Remark 3.1. The proofs (i-2) and (ii-1) of this theorem are quite the same as that in Ref. 9 (there we considered the case d = 1 though). Note that, in using the idea of Ref. 9, we take the part form on Bx0 (1/2) ∩ G (resp. Bx0 ((1/2) ∧ u0 )) when we assume the first case (resp. the second case). So we omit the proof. On the other hand, when we assume that α(x, y) = α(|x ˜ − y|), then the associated process is a symmetric L´evy process. So (ii-2) follows from Theorem 43.9 in Ref. 6. For the assumption (2), we have the characteristic component (or symbol): ˜ (1 − cos(ξ · y))|y|−d−α(|y|) dy. −ψ(ξ ) = Rd
It is known (see e.g. Remark 43.6 in Ref. 6) that a point is not exceptional if and only if
1 dz < ∞ Re q − ψ(z) Rd for some (or equivalently, for all) q > 0. But in general, it is quite difficult to estimate the characteristic component. In the next theorem, we also give a criteria about the recurrence/ transience of the processes. Theorem 3.2. (1)
Assume that α(x, y) = α(x).
Recurrence: Theorem 3.2 in Ref. 9. If d = 1 and the following two conditions are satisfied;
1 1 (i) lim sup + R −α(x) dx < ∞, 2 − α(x) α(x) R→∞ B(R)
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(ii)
lim sup R→∞
(2)
B(2R)
dx|x − y|−1−α(x) < ∞,
dy Rd −B(3R)
then the process (or the form (Eα , Fα )) is recurrent. Transience. If there exists a constant β(0 < β < 2 when d 2, and 0 < β < 1 when d = 1) so that α(x) β a.e., then the process (or the form (Eα , Fα )) is transient.
Proof. (1) is shown in Ref. 9 as Theorem 3.2. So we show (2). We first lip estimate the form as follows under the assumption in (2): for u ∈ C0 (Rd ), (u(x) − u(y))2 (u(x) − u(y))2 α E (u, u) = dx dy dx dy |x − y|d+α(x) |x − y|d+α(x) |x−y|>1
Rd ×Rd
|x−y|>1
(u(x) − u(y))2 dx dy ≡ |x − y|d+β
(u(x) − u(y))2 ν(dx, dy), Rd ×Rd
where the measure ν(dx, dy) on Rd × Rd is: ν(dx, dy) =
1Rd −B(1) (x − y) |x − y|d+β
dx dy.
lip
A form (η, C0 (Rd )) defined by η(u, v) = (u(x) − u(y))(v(x) − v(y))ν(dx, dy),
lip
u ∈ C0 (Rd )
Rd ×Rd
is a closable symmetric form on L2 (Rd ) and the closure of the form lip (η, C0 (Rd )) corresponds to a symmetric L´evy process on Rd having ν(dx) = 1Rd −B(1) (x) |x|dx evy measure. d+β as the L´ When d 3, since the L´evy process having the L´evy measure ν(dx) is genuinely d-dimensional (see Proposition 24.17 and Definition 24.18 in Ref. 6), the process is transient (Theorem 37.8 in Ref. 6). On the other hand, when d = 1 or d = 2, we can show that the L´evy process is transient directly, along the line of the argument of Remark 37.15 in Ref. 6. Therefore the transience of the form (Eα , Fα ) now follows from using the comparison theorem applied to the forms Eα and η. A similar argument of proof of the theorem above gives us the following theorem:
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549
Theorem 3.3. Assume that α(x, y) = α(|x ˜ − y|). (1)
Recurrence. If
R ˜ u1−α(u) du < ∞, lim sup R −2+d R→∞ 0 ∞ ˜ lim sup R d u1−α(u) du < ∞, R→∞
R
then the process is recurrent (These make sense only when d = 1, 2.) ˜ < β˜ < 2 when (2) Transience. If there exist constants K > 0 and β(0 d = 2, and 0 < β˜ < 1 when d = 1) so that α(u) ˜ β˜
a.e. on [K, ∞),
then the process (or the form (Eα , Fα )) is transient. When d 3, the form (or the process) is transient because the process is a genuinely d-dimensional L´evy process. Outline of the proof. From the note after the statement of Theorem 3.2. in Ref. 9, in order to prove the recurrence of the process, it is enough to construct a sequence of functions φn of Fα so that φn ↑ 1 a.e.
and lim sup Eα (φn , φn ) < ∞. n→∞
Let ϕ1 be the function defined in Ref. 9. Then, putting φn (x)=ϕ1 (|x|/n), x∈ Rd , the same argument in Ref. 9 is valid. The proof of (2) is the same as that in Theorem 3.2 (2). Remark 3.2. As is noted in Remark 2.1 (ii), the assumption (2) for the function α gives us a L´evy process. Moreover it is well-known that a L´evy process is transient if and only if for the characteristic component ψ, 1/ψ(ξ ) is locally integrable. Suppose that there exist constants K > 0 and β(0 < β < 2 when d 2, and 0 < β < 1 when d = 1) so that α(x, y) β
if |x − y| K.
Then we can see that the form (Eα , Fα ) is transient in the same way of the proof of Theorem 3.2. On the other hand, suppose further that d = 1 and there exist constants 1 β < 2, K > 0, 0 < α1 < 1 and 0 < ε < K so that α(x, y) = β α(x, y) α1
if |x − y| ε, if |x − y| > K.
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Then, the form (Eα , Fα ) is also transient in this case, though, as in the Proposition 2.1, it holds that (u(x) − u(y))2 lip Eα1 (u, u) ≈ dx dy + ||u||2L2 , u ∈ C0 (R). |x − y|d+β R×R
This shows that, although the form is transient, the form (added the L2 norm) is equivalent to the form (also added the L2 -norm) of symmetric β-stable process which is recurrent. 4. INFINITESIMAL GENERATORS OF THE SYMMETRIC STABLE-LIKE PROCESSES We derive infinitesimal generators of the Dirichlet forms (Eα , Fα ) in this section under the conditions in Theorem 2.2: Theorem 4.1. For the function α(x, y) satisfying the conditions in Theorem 2.2, put φ(x) ≡ (1 ∧ |x − y|) |x − y|−d−α(x,y) + |x − y|−d−α(y,x) dy, x ∈ Rd Rd
and ν(x, y) = |x − y|−d−α(x,y) + |x − y|−d−α(y,x) ,
x, y ∈ Rd .
Suppose the following condition (1) (resp. (2)) is satisfied: (1) φ ∈ L1loc (Rd ), (2) there exists a measurable function α˜ defined on [0, ∞) satisfying (2) so that α(x, y) = α(|x ˜ − y|),
x, y ∈ Rd .
Then the following equation holds: for any u, v ∈ C02 (Rd ), Eα (u, v) = −(u, Av)L2 (Rd ) , where
Av(x) =
(resp.
Av(x) =
Rd
Rd
(v(y) − v(x))ν(x, y)dy,
x ∈ Rd .
u(y) − u(x) − ∇u(x) · (x − y)1B(1) (x − y)
On Symmetric Stable-like Processes
×ν(x, y)dy,
551
x ∈ Rd .)
Moreover, under the condition (2), the operator A maps from the functions in C02 (Rd ) into L2 (Rd ). That is, the domain of the L2 -generator of the form Eα contains C02 (Rd ). On the other hand, when the condition (1) is satisfied, a sufficient condition that the operator A maps C02 into L2 (Rd ) is the following: 2 d φ ∈ Lloc (R ) and ν(x, y)dy ∈ L2 (Rd − B(R)) B(r)
for any R, r > 0 with R − r > 1.
(4)
Proof. Following an idea given in the proof of Theorem 6.4 in Ref. 5, we first estimate the following: for u, v ∈ C02 (Rd ), (u(x) − u(y))(v(x) − v(y))|x − y|−d−α(x,y) dx dy Rd ×Rd
(u(x) − u(y))(v(x) − v(y))|x − y|−d−α(x,y) dx dy
= lim
ε→0 Rd ×Rd \ ε
≡ lim ε , ε→0
where ε = {(x, y) : |x − y| < ε}. We then rewrite ε as follows: ε = − u(y) v(x) − v(y) − ∇y v(y) · (x − y)1B O (1) (x − y) Rd ×Rd \ ε
−
×|x − y|−d−α(x,y) dx dy u(x) v(y) − v(x) − ∇x v(x) · (y − x)1B O (1) (x − y) Rd ×Rd \ ε
−
×|x − y|−d−α(x,y) dx dy u(y)∇y v(y) · (x − y)1B O (1) (x − y)|x − y|−d−α(x,y) dx dy Rd ×Rd \ ε
− Rd ×Rd \ ε
=− Rd ×Rd \ ε
u(x)∇x v(x) · (y − x)1B O (1) (x − y)|x − y|−d−α(x,y) dx dy u(y) v(x) − v(y) − ∇y v(y) · (x − y)1B O (1) (x − y)
× |x − y|−d−α(x,y) + |x − y|−d−α(x,y) dx dy
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−
u(y)∇y v(y) · (x − y)1B O (1) (x − y) Rd ×Rd \ ε
× |x − y|−d−α(x,y) + |x − y|−d−α(y,x) dx dy
1,ε
≡ − 1,ε − 2,ε . = u(y) Rd
|x−y|>1
v(x) − v(y) − ∇y v(y) · (x − y)1B O (1) (x − y)
×ν(x, y)dx + v(x) − v(y) − ∇y v(y) · (x − y)1B O (1) (x − y) 1|x−y|>ε
×ν(x, y)dx dy
≡
Rd
u(y)(I1 (y) + I1,ε (y))dy
and 2,ε =
Rd
u(y)
≡
Rd
u(y)
d ∂v i=1 d i=1
∂yi
(x − y )ν(x, y)dx dy i
(y)
i
1|x−y|ε
∂v i (y)I2,ε (y)dy. ∂yi
We now estimate I1 and I1,ε . I1 is locally integrable, since the following estimates are satisfied: |I1 (y)| = | (v(x) − v(y))ν(x, y)dx| 2 v L∞ ν(x, y) dx. |x−y|>1
|x−y|>1
In either case, the last term is locally integrable. The Taylor expansion for the function v implies that |x − y|2 ν(x, y) dx,
|I1,ε (y)| C(v) ε|x−y|1
where the positive constant C(v) depends only on the second derivative of the function v. This last term is also locally integrable in each case, because |x − y|2 |x − y| if |x − y| < 1. Combining these calculations, the Lebesgue-dominated convergence theorem tells us that the following limit exists:
On Symmetric Stable-like Processes
lim 1,ε =
Rd
ε→0
553
u(y)
(v(x) − v(y) − ∇y v(y) · (x − y)1BO (1) (x − y))
Rd
×ν(x, y) dx dy.
i Next we estimate I2,ε for i = 1, 2, . . . , d. When (1) is satisfied, it is clear that i |I2,ε (y)| |x − y|ν(x, y)dx φ(y) |x−y|1
i (y) is locally integrable by the assumption. Hence we can and, then I2,ε also see that the following limit exists. lim 2,ε = u(y) ∇y v(y) · (x − y) 1BO (1) (x − y)ν(x, y) dx dy. ε→0
Rd
Rd
On the other hand, when α(x, y) satisfies (2), (that is, α(x, y) = α(|x ˜ − y|)), i , (x i − y i )ν(x, y), is an odd function. Here we then the integrand of I2,ε also say that a function f (x) on Rd , is odd if f (x) = −f (−x) for any x . So the integral of the function on the annulus {ε |x − y| 1}, that is , i , disappears. Hence I2,ε 2,ε also disappears. Therefore, in any case, we have for any u, v ∈ C02 (Rd ), (u(x) − u(y))(v(x) − v(y)) dx dy Eα (u, v) = |x − y|d+α(x,y) Rd ×Rd
= − lim ( 1,ε + 2,ε ) ε→0 v(x) − v(y) − ∇y v(y) · (x − y)1BO (1) (x − y) u(y) =− Rd
Rd
×ν(x, y) dx dy d ∂v + u(y) (y) (x i − y i )1BO (1) (x − y)ν(x, y) dx dy i d d ∂y R R i=1
= −(u, Av)L2 (Rd ) . Finally, when the conditions (4) are satisfied (resp. the condition (2) holds), it is easy to check that Au is in L2 (Rd ) provided that u ∈ C02 (Rd ).
Remark 4.1. (i)
While the first condition in (4) seems to be natural, the second one might be a little bit technical. But, it is quite difficult to estimate the integral in general.
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(ii)
In Ref. 7, Schilling has given an example of generators of L´evy processes using the theory of pseudo-differential operators (see Corollary 3.7). In there, some conditions are given similar to (1) and (2) mentioned above. Of course, his example has been treated as pseudo-differential operators which are generated by sub-Markovian semigroups and satisfy the positive maximum principle.
Example 4.1. Assume there exists a measurable function α on Rd satisfying (1) –(3) in section 2 so that α(x, y) = α(x). Under this assumption, a sufficient condition that (1) or (2) in Theorem 4.1 is satisfied is one of the following conditions: (1) (2)
0 < α α(x) < 1 a.e. and 1/(1 − α(x)) ∈ L1loc (Rd ). α(x) ≡ β, 0 < β < 2.
Furthermore (4) holds true when we replace (1) with the following condition: 0 < α α(x) α¯ < 1
a.e.
ACKNOWLEDGMENTS The author thanks Prof. M. Tomisaki for an idea in Ref. 5. The author also thanks Dr. R.L. Schilling and Dr. Y. Isozaki for valuable suggestions and discussions. Part of this work was done while the author was visiting the School of Mathematical Sciences, University of Sussex, Brighton, UK. Financial supports by the Royal Society Japan Ex-Agreement Study Visit to UK 2002–2003 and the Hyogo Prefectural short-term study (abroad) program are gratefully acknowledged.
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6. Sato, K. (1999). L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. 7. Schilling, R. L. (2001). Dirichlet operators and the positive maximum principle. Integr. Eq. Operator Theory 41, 74–92. 8. Tsuchiya, M. (1992). L´evy measure with generalized polar decomposition and the associated SDE with jumps. Stochastics Stochastics Rep. 38, 95–117. 9. Uemura, T. (2002). On some path properties of symmetric stable-like processes for one dimension. Potential Anal. 16, 79–91.
