Feller Processes Generated by Pseudo-Differential Operators: On the ...

Journal of Theoretical Probability, Vol. 11, No. 2, 1998

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths1 Rene L. Schilling2 Received February 14, 1996; revised December 5, 1996 Let { X t } t>0 be a Feller process generated by a pseudo-differential operator whose symbol satisfies sup A . sR » \q(x, ()\ !/(£). The Hausdorff dimension of the set {X,: teE], £0

In fact, (1.5) is a one-to-one correspondence between (sub)-Markovian (C())-convolution semigroups and Levy processes, [cf. Ref. 1, Section 8, 8.29]. Every Levy process can be fully described in terms of its characteristic exponent i/r R" -> C,

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where \[f is a continuous negative definite function (c.n.d.f., for short), i.e., a function having the following Levy-Khinchine representation

with /eR", geR"*" positive hermitian, and the Levy (jump) measure v carried on IR"\{0} s.t. j (Hjc|| 2 /(l + IWI 2 )) v(dx) < oo. C.n.d.f.s. have positive real parts, \l/(—^) = (j/(^), their square roots are subadditive,

and we have

For more properties of c.n.d.f.s. we refer to the monograph (Ref. 1). Note that (1.8) gives a Peetre-type inequality (cf. also Ref. 10),

Let us finally mention that the infinitesimal generator A of {X,}t>0 (or {St}t>0) is given by

So far to the analytic point of view. Taking Levy processes mainly as stochastic objects their sample path behavior, e.g., has been extensively studied. For a survey we refer to the studies by Fristedt(6) and Taylor.(23) For the Hausdorff dimension, in particular, [see Blumenthal and Getoor(2); Pruitt (19) ; Millar (17) ; Hawkes(8); and Hawkes and Pruitt (9) ], and also the detailed bibliography by Taylor.(24) A somewhat more general setup in the framework of comparable processes is considered in Schilling.(20,21) The Hausdorff dimension of a set A c R" is the unique number 1 where the A-dimensional Hausdorff measure A\A)

changes from + oo to a finite value.

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For the study of the Hausdorff dimension of the sample paths of Levy processes various indices were introduced by Blumenthal and Getoor (2) :

(by definition sup 0 = 0). We have always O^yS" 0. As in Section 2 we set

and write {S,},^Q for the convolution semigroup generated by — q t ( D ) Jacob(13) showed that the operators T, themselves can be considered as pseudo-differential operators whose symbols A,(x, £) are given by

where { Y,}t^0 is the Feller process corresponding to {T,}t>(> and e^(x) = e~'xf. (Note that T,e^ is uniquely defined as T, can be extended to the set

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B h (R n ), see, e.g., Courrege.(4) Under the condition that ef be in the domain of the operator —q(x, D), the relation

was shown by Jacob.(13) As a matter of fact, it is enough to assume (2.7)-(2.10.n + 1). This was done in [Ref. 22, Thm. 3.1]. These conditions are almost met in the situation of Theorem 2 where the existence of an extension of —q(x, D) to the generator of a Feller semigroup is shown. Let us observe a simple consequence of (3.3): Lemma 2. Assume that —q(x,D] generates a Feller semigroup Tt = A,(x, D) and that (2.7)-(2.10.n +1) hold. Then

holds for all t > 0, p e [0, 1 ], and x, £, e Rn with an absolute constant C> 0. Proof.

Clearly,

Now choose some ^,eC™(Rn), x\ ^(0)SU and define %k(x)=x1(x/k). Then Jtk(t) = knUkZ) and we find

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By the semigroup and contractivity properties of { T t } t > 0 we get for ueD(-q(x,D})

and since u = x/f^^sD(— q(x, D)} we find with these calculations, the fact that A,(x, £) = e _ ( ( x ) T,e^(x)> and by dominated convergence

Now, for pe [0, 1],

where we used (2.9), (2.10.0), and Remark 2 in the last step.

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One can use Lemma 2 to obtain immediately comparison results for the Feller semigroup. The technique is similar to [Ref. 15, Lemma 5.9] where under heavier assumptions only the case p = 1 was treated. We will see, in particular, that the pseudo-differential operator e-"^x''^ with symbol e-"i1-*-^ is a reasonable approximation of Tt. Proposition 1. Assume that —q(x,D] generates a Fellerian semigroup {T,}1>0 and that (2.7)-(2.10.n+1) hold. Then we have for ueC,°°(R n ), t^G, and xeU"

with the constant C from Lemma 2.

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Jacob and Schilling(15) also used Lemma 2 to get a pointwise comparison of {T,}t>0 and {S,}t>0,

with constants d', d" which, essentially, depend on the perturbation Z|a|s; n + 3 II^JLi, see [Ref. 15, Thm. 4.2]. One can, in fact, prove a better result without resorting to Lemma 2, that is without (3.3). In order to do so, we need an auxiliary result. Lemma 3. For all x>0 and a>0 we have e'x^e(a./e)at (1/(1 + x))a. Proof. The inequality follows directly from the fact that f ( x ) = (1 + x) e~x/a- attains for x > — 1 its maximum at a — 1. D D Theorem 3. Suppose that —q(x,D) generates a Feller semigroup {Tt}t>0 and satisfies (2.10.n + 1). Then we have for all ueCc(Rn), p e (0,"l], and t < 1 (if p = 1, even for all t >0)

with the constant

Proof. Note that Ss: H"2-r(U") -> H"2''\nn) for all r^Q and s>0. A standard perturbation result for operator semigroups, see e.g., [Ref. 5, Lemma 6.2] therefore yields

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Writing

Observe now that \lt__x(x, £)| < 1,

Lemma 3 shows for t < 1 and a = 1 — p e [0, 1)

Note that, by convention, 0° = 1. Moreover, if p = 1, this calculation holds for all t > 0. Using Lemma 1 with m = n + l we get

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From Gradshteyn et al. (7) [formula (8.380.3) and (8.384.1)] we get

and the assertion follows.

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Remark 3. Clearly, Proposition 1 and Theorem 3 hold not only for ue C c M (R n ), but whenever the r.h.s. of the estimates (3.4) or (3.7), respectively, is finite. The above estimates are, therefore, most useful if one chooses the right function u such that ||(1 + a 2 } p u\\ L\ is finite. 4. SOME FOURIER TRANSFORMS We indicated in the previous section that the proper choice of the function u in formula (3.4) and (3.7) can be essential. Here we will compile some preparatory results. The formula of the following Lemma are taken from [Ref. 7, (17.33.16), (17.34.14)]. Lemma 4. Let A > — 1 be a real number. Then the function

has the following Fourier cosine and exponential transform:

Corollary 1. Let A, i// A , be as in Lemma 4, and put

The functions !F*y: Rn -> R, j = 1,..., n,

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have the Fourier transforms

Corollary 2. Let A, i i, gk, and *P\ ;. be as in Lemma 4 and Corollary 1. If a2 is a continuous negative definite function with index B e (0, 2], then we have for any A > ft

with the constant

Here, B ( . , •) denotes the Euler Beta function and eB,A is from (4.8). Proof. Denote by B(x, y) the usual Euler Beta function with the convention that B(x, y) = i whenever x < 0 or y < 0. Note that

cf. [Ref. 7, (3.251.2)]. Since X > B , we may choose B < p < l < 3 , p = (A+ B)/2 say, such that by the definition of the index B there exists a constant cB, A satisfying

Set

Then

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Therefore by Corollary 1

and we are done.

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Remark 4. It is easy to see that ||(1 +a2) ¥"| .}\,,» < co and that therefore any L^-norm in (4.6) is finite. But only in the case p = 1 we can estimate dk uniformly for all k. We will finally combine Corollary 2 and Theorem 3. Lemma 5. Suppose that -~q(x,D], q(x,S,) as in Sections 2 and 3 generates a Feller semigroup {T,}t>0 and satisfies (2.10.n+ 1). Denote by {St}t>0 the convolution semigroup generated by —q^D). Then we have for the functions

and A>B—here B = Bq1 is the index of q 1 —that

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holds for all t > 0 with the constant

Proof.

We have with the notation of Corollary 1

and

Thus, the shift by y does not affect the constant dk(n, A, B) of Corollary 2. Since limk->i *F\ f(x — y) — i y , A , j ( x ) monotonically, we find by monotone convergence that

and

hold for all t> 0 and x, ye Rn. Hence, Theorem 3 and Corollary 2 give

where c is the constant (3.8) with p = 1.

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5. AN UPPER BOUND FOR THE HAUSDORFF DIMENSION In our last section we will apply the estimates of Section 3, in particular (3.7) with p = 1, to derive an upper bound for the Hausdorff dimension of the set

where E is a Borel subset of [0, 1]—or any other bounded interval of [0, oo)—and { Y,} t^n is the Feller process corresponding to the pseudo-differential operator — q(x, D] with symbol — q(x, J) as of Sections 2 and 3. For simplicity's sake we will also assume that { Y,} l>0 has infinite life-time which is ensured by the assumption that

hold (0 is from (2.10.0)), see [Ref. 22, Thm. 1.1]. Throughout this section we denote by {T,} ,>0 and { Y,} t>n the Feller semigroup and Feller process generated by —q(x, D), and by {S,}ts.0 and {Xt}t>0 the convolution semigroup and Levy process induced by —q1(D), see Section 2 for the notation. Our first result is, basically, a restatement of Lemma 5 in terms of stochastic processes rather than semigroups. Proposition 2. Suppose that —q(x,D) generates a Feller semigroup {Tt}t>0 and satisfies (2.10.K+1) and (5.2). Denote by {S,}t>0 the convolution semigroup generated by —q1(D) and by { Yt} t>0 and {X,} t>0 the Levy-type and Levy processes corresponding to {T t } t > 0 and { S t } t > 0 , respectively. Write Be(0, 2] for the (upper) index of the process { X t } t > 0 . Then we have for all j = 1,..., n, X > B, and s, t > 0

with Y, = (Y t ,..., Y"t\

and c, d1 from (4.6) and (4.10). In particular, we have

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Thus, (5.5) follows directly from (5.3). In order to prove (5.3), we use (4.10) where we apply the triangle inequality: for yeRn and 0