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1 Math. Ann. 309, (1997 Mathematische Annalen c Springer-Verlag 1997 On Feller processes with sample paths in Besov spaces René L. Schilling Mathematisches Institut, Universität Erlangen, Bismarckstraße 1 1/2 D Erlangen, Germany ( schilli@mi.uni-erlangen.de Received: 31 May 1996 / Revised version: 10 December 1996 Abstract. Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real-valued Feller processes in terms of Besov spaces B s pq(r. Our result extends previous papers on the paths of Gaussian, symmetric α-stable [6], [20], and Lévy processes [11]. Mathematics Subect Classification (1991: 60G17, 60J35, 41A15, 35S99, 60J30. 1 Introduction A family of operators {T t } t 0 on the space of continuous functions vanishing at infinity T t : C (R n C (R n satisfying T t T s = T t+s, lim t 0 T t u u C (R n = 0, and 0 T t u 1 whenever 0 u 1 is called Feller semigroup. It is well-known that there is a one-to-one correspondence between Feller semigroups {T t } t 0 and stochastic Feller processes {Y (t} t 0. This relation is given by (1.1 T t u(x =E x u(y(t, u C (R n, t 0, x R n. Here P x (and also E x is the probability measure (resp. expectation for the process under which we have Y (0 = x a.s. Note that (1.1 can be extended to bounded measurable functions u. It is known that one can always choose a càdlàg version of {Y (t} t 0, i.e. a version such that for all ω Ω the sample paths t Y (t,ω are right-continuous and have finite left limits. We will do so without further notice. Financial support by DFG post-doctoral fellowship Schi 419/1 1 is gratefully acknowledged.

2 664 R.L. Schilling The infinitesimal generator (A, D(A of the semigroup {T t } t 0 and/or the process {Y (t} t 0 is given by (1.2 (1.3 Au = T t u u lim, t 0 t D(A = {u C (R n : the limit (1.2 exists in C (R n } Write û(ξ =(2π n/2 e ixξ u(x dx for the Fourier transform. If the test R n functions Cc (R n are contained in D(A, it is due to Ph. Courrège [7] that the generator of a Feller process is necessarily of the form (1.4 p(x, Du = (2π n/2 R n e ixξ p(x,ξû(ξdξ, u C c (R n, i.e. it is a pseudo-differential operator with symbol p : R n R n C. The symbol is locally bounded as a function of (x,ξ and, for fixed x, continuous negative definite as a function of ξ, that is to say that p(x, 0 0 and ξ e tp(x,ξ is for all t > 0 continuous positive definite (in the usual sense or, equivalently, that ξ p(x,ξ admits a Lévy-Khinchine representation. For further results on negative definite functions we refer to the monograph [2]. Recently, several authors constructed Feller processes with prescribed symbol either analytically [14, 15, 19] or as solutions to a martingale problem [18, 12, 13]; for a survey see also [16]. The latter method seems to give the most general class of processes. It is interesting to note that the assumptions of Theorem 1.1 below, in particular (1.6, are always met in these constructions. There are, of course, many other ways how to obtain concrete examples for Feller processes, see [4, 22]. At least one of these, the SDE method i.e. the Feller process is the (strong solution of a (ump-type stochastic differential equation driven by Lévy processes or certain semimartingales is also of interest to us since it is possible to calculate the symbol of the generator from the coefficients and the driving terms of the SDE, cf. e.g. [17, 1]. Let us mention a special class of Feller processes: Lévy processes. They are also known as processes with independent and stationary increments. In fact, Lévy processes are exactly those processes whose generators (1.4 have constant coefficients, i.e. ψ(du(x = (2π n/2 R n e ixξ ψ(ξû(ξ dξ, u C c (R n, with the negative definite characteristic exponent ψ given by E 0 e iξxt = e tψ(ξ, t 0,ξ R n. The properties of Lévy processes are well-studied, especially of their sample paths, see e.g. [9]. The symbol ψ plays a central part. Following [3] we call { ψ(ξ } (1.5 β ψ = inf λ>0 : lim ξ ξ λ =0

3 On Feller processes with sample paths in Besov spaces 665 (upper index of the Lévy process {X (t} t 0. Note that always 0 β ψ 2. We will write β instead of β ψ unless it is too ambiguous. The regularity of the sample paths of Gaussian, α-stable Lévy, and general Lévy processes in terms of Besov spaces was recently investigated in a series of papers [6], [20], [11]. Here we present a theorem for Feller processes that complements these results. Theorem 1.1 Let {Y (t} t 0 be a real-valued Feller process with infinitesimal generator p(x, D as in (1.4. Assume that the symbol p(x,ξ satisfies (1.6 p(x,ξ c(1 + ψ(ξ for all x,ξ R, with a fixed continuous negative definite function ψ : R R. Denote by β ψ the index of the Lévy process {X (t} t 0 generated by ψ(d and by Bpq((0, s 1 the Besov space w.r.t. the parameters 0 < p,0<q,s R.If max{1/p 1, 0} < s < 1 (and, a fortiori, p > 1/2 we have for the restricted sample paths (1.7 t Y (t,ω (0,1 Bpq((0, s 1 a.s. (P x for the following combinations p β ψ, q =, and ps < 1; p >β ψ, p q, and ps < 1; p >β ψ, p<q, and qs < 1. We defer the proof to section 4 below. The condition ps < 1 is optimal in the sense that functions in B s pq(r where ps > 1 are continuous, see [23, 2.3.2(7, 2.7.1(1, 2.7.1(11], whereas the Feller processes under consideration here have typically càdlàg sample paths. Since ξ p(x,ξ is continuous and negative definite, sup x R p(x,ξ c ξ already implies p(x,ξ c(1 + ξ 2 with an absolute constant c = c p > 0, cf. [2, Cor. 7.16]. Clearly, the above theorem holds for any bounded interval (0, T, T > 0. Extending t Y (t,ω onto (, 0] by the constant Y (0,ω, we get Y (t,ω ( ɛ,1 B s pq(( ɛ, 1 for all ɛ>0 and all admissible p, q, s as in Theorem 1.1. Notation We write X for the norm in the space X. C c (R n, C (R n, C b (R n, and B b (R n denote the continuous functions with compact support, vanishing at infinity, bounded, and the bounded Borel measurable functions, respectively; superscripts refer to differentiability properties. û(ξ =(2π n/2 e ixξ u(x dx is

4 666 R.L. Schilling the Fourier transform and E x ( = dp x stands for the mean value; the superscript x indicates that the process started a.s. at x. When dealing with random variables we often suppress ω and write Y (t for Y (t,ω. The bracket x stands for the integer part of x, x + for the positive part max{x, 0}. All other notations are standard or should be clear from the context. 2 Some comparison results for Feller semigroups In this section we will compare the semigroup {T t } t 0 generated by the pseudodifferential operator p(x, D as in (1.4 and satisfying p(x, ξ c(1 + ψ(ξ with the convolution semigroup {S t } t 0 generated by ψ(d Proposition 2.1 Assume that p(x, D of (1.4 extends to a generator of a Feller semigroup {T t } t 0. Furthermore assume that the symbol satisfies p(x,ξ c(1+ ψ(ξ with a fixed continuous negative definite function ψ 0. Let {S t } t 0 denote the convolution semigroup generated by ψ(d. Then (2.1 T t u(x S t u(x ct (1 + ψû L 1 (R n holds uniformly in x for all t 0 and all u C (R n s.t. the right hand side is finite. Proof. It is enough to prove (2.1 for a dense subset, Cc (R n, say. Note that Cc (R n D(ψ(D. Therefore, t T t u u C (R n = T s p(, Du( ds C (R n 0 t p(, Du( C (R n = (2π n/2 t e i ξ p(,ξû(ξdξ C (R n R n ct (1 + ψû L 1 (R n. The estimate for S t is similar and the assertion follows from an application of the triangle inequality. In the sequel we will use (2.1 only for functions of the form (2.2 u λ (x = x λ e λ x, x R,λ>0. Fortunately, the Fourier transform of u λ is explicitly known, cf. [10, ]: 2 ( û λ (ξ = π Γ(λ+ 1(λ2 + ξ 2 λ+1 2 cos (λ + 1 arctan ξ (2.3. λ If β = β ψ is the index of the function ψ from Proposition 2.1, we have from the very definition (1.5 that

5 On Feller processes with sample paths in Besov spaces 667 (2.4 ψ(ξ c λ,β (1 + ξ 2 β 4 + λ 4, λ > β, (strictly speaking, c λ,β = c λ,β,ψ, but the ψ-dependence for small values of ξ is of no interest to us thus the right hand side in (2.1 is finite: (1 + ψû λ L 1 (R n c λ,β const. <, λ > β. This is the key to the following result. Proposition 2.2 In the situation of Proposition 2.1 denote by {Y (t} t 0 the Feller process associated with {T t } t 0 and by {X (t} t 0 the Lévy process associated with {S t } t 0. Then (2.5 E x u λ (Y (t Y (s c λ,β t s for all s, t 0 and λ>β,βbeing the index of {X (t} t 0. Proof. Let t s. By the triangle inequality we get from (2.1 for all y R E y u λ (Y (t s y = T t s u λ ( y(y c(t s (1 + ψû λ L 1 (R n + S t s u λ ( y(y since the Fourier transform of u λ ( y satisfies û λ ( y=e i y û λ (. Now, S t s u λ ( y(y =E y u λ (X(t s y=e 0 u λ (X(t s by the translation invariance of Lévy processes. Let P 0 X (t s denote the distribution of the random variable X (t s and δ 0 the unit mass at 0. By Plancherel s theorem we find Therefore, E 0 u λ (X (t s = = = E 0 u λ (X (t s c (t s This proves u λ (x P 0 X (t s(dx u λ (x(p 0 X(t s(dx δ 0 (dx û λ (ξ(p 0 X(t s(dx δ 0 (ξ dξ = (2π 1/2 û λ (ξ ( e (t sψ(ξ 1 dξ. û λ (ξ ψ(ξ dξ c (t s (1 + ψû λ L 1 (R. (2.6 E y u λ (Y (t s y c λ,β (t s (1 + ψû λ L 1 (R.

6 668 R.L. Schilling Integrating both sides of (2.6 against the distribution P x Y (s(dyofy(s s.t. Y (0 = x a.s., gives (2.5 because of the Markov property of {Y (t} t 0. Remark 2.3 (A The above results also hold in higher dimensions. The proof of Proposition 2.1 carries over literally, whereas for Proposition 2.2 one has to consider the coordinate processes. For technical details we refer to [21, Proposition 5.1]. (B It is interesting to note that the Feller property and Proposition 2.2 give maximal inequalities of the form ( E x sup u λ (Y (t Y (s c λ,β b a, λ > β, a s,t b for b a 1. This can be done (also for n dimensions as in [21]. We do not need this result here. 3 Basics on Besov spaces Throughout this section we will use the notation of Triebel [23, 24]. We begin with the Fourier-analytic definition of the spaces Bpq(R s n. Let {φ } be a smooth dyadic partition of unity in Rn, i.e., φ C (R n and { supp φ0 {ξ R n : ξ 2}; (3.1 supp φ {ξ R n :2 1 ξ 2 +1 }, N; (3.2 sup ξ Rn ; N 0 2 α D α φ (ξ <, α N n 0 ; (3.3 φ (ξ =1, ξ R n. In the following definition φ (D is a pseudo-differential operator like (1.4. Definition 3.1 (A Let 0 < p,0<q,s R, and {φ } be some smooth dyadic partition of unity in R n. Then ( u Bpq(R s n = 2 sq φ (Du L p (R n q 1/q (3.4 (with the usual interpretation if p = and/or q = defines a (quasi-norm on the space (3.5 Bpq(R s n ={u S (R n : u Bpq(R s n < }. The spaces Bpq(R s n are called Besov spaces with parameters p, q, and s.

7 On Feller processes with sample paths in Besov spaces 669 (B Let U R n be a bounded open domain with C -boundary and p, q, sasin (A. Then Bpq(U s is the restriction of Bpq(R s n on U equipped with the (quasi- norm (3.6 u Bpq(U s = inf g Bpq(R s n where the infimum is taken over all g Bpq(R s n such that g U = f in the sense of distributions D (U. Note that the above definition of the norm (3.4 is in fact independent of the particular partition of unity. For this and further properties of the scale Bpq(R s n, e.g. embeddings, interpolation results..., we refer to the monographs [23, 24]. Here, we only need an atomic characterisation of Besov spaces which is due to Frazier and Jawerth, [8]. Again we follow the exposition of Triebel [24, 1.9.2]. Denote by Q k R n, N 0, k Z n an affine copy of the unit cube with side-length 2 and center 2 k.byrq k, r 0 we mean the dilated cube (relative to its center. Definition 3.2 Let 0 < p,s R, and K, L Z such that { ( 1 } (3.7 K ( s +1 + and L max n p + 1 s, 1. A function m : R n C is called s-atom if (3.8 supp m 5Q 0k for some k Z n and (3.9 D α m(x 1 for all α N n 0, α K hold. The function m is called (Q k, s, p-atom, N 0, k Z n, relative to K and L if (3.10 supp m 5Q k (3.11 D α m(x (2 n 1 p + s n α n for all α N n 0, α K (3.12 x β m(x dx = 0 for all β N n 0, β L R n are satisfied. (Condition (3.12 is meaningless if L = 1. For our considerations the following result is essential. Theorem 3.3 ([8, Theorem 7.1], [24, 1.9.2] Let 0 < p,0 < q, s R, and K, L Z be as in Definition 3.2. A tempered distribution u S (R n is contained in B s pq(r n if and only if (3.13 u = k Z n (s k m k (x+ =0 s,k m,k (x (in S (R n

8 670 R.L. Schilling with s-atoms m k relative to Q 0k, (Q k, s, p-atoms m,k, and coefficients s k, s,k C such that ( 1/p ( ( q/p 1/q (3.14 s k p + s,k p <. k Z n k Z n Remark 3.4 Below we will only consider the case where K = 1 and L = 1. A close investigation of the proof of the above theorem, ([8, Theorems 3.1 and 7.1], in particular the key Lemma 3.3, proof of estimate (3.4 reveals that it is indeed enough to assume that m,k is Lipschitz continuous rather than differentiable. Of course, the Lipschitz constant has to satisfy a smallness condition similar to (3.11. This is surely the case if the Lipschitz functions m,k fail to be differentiable at a finite number of points but have otherwise bounded derivatives satisfying ( Proof of the main result The idea to use atomic representations for Besov spaces in order to show results like Theorem 1.1 is due to Ciesielski, Kerkyacharian, and Roynette. In their paper [6] they showed an isomorphism between certain sequence spaces and the spaces B s pq([0, 1]. We will go along similar lines but use the atomic decomposition introduced in Section 3. In order to prove Theorem 1.1 we need some preparations. The Haar functions on [0, 1] are given by H 1 (t =1 [0,1] (t, H,k (t =2 /2 1 [ 2k 2 2k 1 (t 2 /2 1 [ 2k 1 2+1, k (t, 2 +1, 2 +1 for N 0, k =1,2,...,2. By taking primitives we obtain the Schauder functions S 0 (t =1 [0,1] (t, S 1 (t =t1 [0,1] (t, S,k (t = t 0 H,k (rdr, for N 0, k =1,2,...,2. Clearly, H,k L (R = 2 /2 and S,k L (R = /2. For technical reasons it is necessary to transform S 0 and S 1 into Lipschitz functions without changing their values on [0, 1]. Therefore we put S 0 (t =(t+ 11 [ 1,0 (t+s 0 (t+ (2 t1 (1,2] (t and S 1 (t =S 1 (t+(2 t1 (1,2] (t. We need a result on the pointwise approximation of càdlàg functions. The proof of the following Lemma can be found in [6, p. 201]. Lemma 4.1 Let u :[0,1] R beacàdlàg function. On [0, 1] the sequence

9 On Feller processes with sample paths in Besov spaces 671 (4.1 where u n (t =u 0 S 0 (t+u 1 S 1 (t+ n 2 u,k S,k (t =0 u 0 = u(0, u 1 = u(1 u(0, u,k = 2 /2( ( 2k 1 2u 2 +1 u converges a.e. (Lebesgue boundedly to u(t. ( 2k 2 +1 u ( 2k , Extending the above u outside [ 1, 2] by 0 and interpolating linearly on [ 1, 0] [1, 2] we still have u n u a.e. (Lebesgue on R. Due to the compactness of [0, 1], the sequence (4.1 converges also in S (R. We will use (4.1 to obtain an atomic representation (in the sense of Section 3 for càdlàg functions. Lemma 4.2 Let S 0, S 1, S,k be the (modified Schauder functions. For 0 < p and s R the functions m 0 (t = S 0 (t, m 1 (t = S 1 (t on Q = [ 1,2]; [ ] m,k (t = 2 /2 2 ( 1 p s 2k 1 S,k (t on Q k = 2, 2k are a.e. s-atoms, resp., (Q k, s, p-atoms relative to K =1and L = 1 (cf. Definition 3.2 and Remark 3.4. Here a.e. means that (3.11 for α =1, fails to hold at finitely many points in [0, 1]. Proof. On [ 1, 2] the functions m 0 and m 1 clearly fulfill (3.8, (3.9 almost everywhere. Moreover, for t [0, 1] and m,k (t =2 /2 2 (1 p s S,k (t 2 /2 2 (1 p s /2 (2 s 1 p m,k (t 2 /2 2 (1 p s H,k L (R =2 /2 2 (1 p s 2 /2 =(2 s 1 p 1, where m,k is the usual derivative (where it exists, otherwise the sup of the difference quotient. Thus, (3.10 and (3.11 follow. By Remark 3.4 a.e. atoms are enough for our purposes. Using Lemma 4.1 we find the following atomic representation of a càdlàg function u on [0,1]: (4.2 u(t =(u 0 m 0 (t+u 1 m 1 (t + 2 ( 2 /2 2 ( 1 p s u,k m,k (t with u 0, u 1, and u,k as of Lemma 4.1.

10 672 R.L. Schilling Remark 4.3 The choice of K = 1 and L = 1 for the atoms implies that ( s and max { (1/p 1 + s, 1} = 1. This is the same as (4.3 ( 1 p 1 + < s < 1 and, a fortiori, p > 1 2. We are now ready to proceed to the proof of Theorem 1.1. Proof of Theorem 1.1. Put Ω R = {ω Ω : sup t 1 Y (t,ω R}and note that by the càdlàg property of the sample paths lim R P x (Ω R = 1. Recall that u λ (x = x λ e λ x.onω R estimate (2.5 reads Y (t Y (s λ e 2λR dp x u λ (Y (t Y (s dp x c λ,β t s, λ > β, Ω R Ω hence (4.4 Y (t Y (s λ dp x c λ,β,r t s, λ > β. Ω R Since t Y (t,ωiscàdlàg we have for every ω Ω an atomic representation of the form (4.2. For t R we put (4.5 f (t,ω=y 0 (ωm 0 (t+y 1 (ωm 1 (t+ 2 2 /2 2 ( 1 p s y,k (ωm,k (t =0 (convergence in S (R with atoms m 1, m,k as of Lemma 4.2 and coefficients y 0 (ω = Y(0,ω, y 1 (ω = Y(1,ω Y(0,ω, y,k (ω = 2 /2( ( 2k 1 ( 2k ( 2k 2 2Y 2 +1,ω Y 2 +1,ω Y 2 +1,ω. By construction, f (t,ω (0,1 = Y (t,ω (0,1, and the theorem is established if we can show f (,ω Bpq(R s for p, q, s as in the statement of the theorem. We will treat the different cases separately. Case 1: p β ψ, q =, ps < 1. Because of the atomic characterization of Besov spaces (Theorem 3.3 it is enough to show that a.s. (P x which follows from (4.6 ( 2 sup N 0 ( 2 sup N 0 2 (1 ps 2 p /2 y,k (ω p 1 ΩR (ω < 2 (1 ps y,k (ω p 1 ΩR (ω < where y,k (ω =Y( k 2,ω Y ( k 1 2,ω. We will prove (4.6 by the Borel-Cantelli Lemma. For an arbitrarily chosen θ>1we get from the inequality a 1 + a a N θ N θ 1 ( a 1 θ + a 2 θ a N θ that

11 On Feller processes with sample paths in Besov spaces 673 (( 2 E x 1 ΩR 2 (1 ps y,k p θ E x ( 1 ΩR 2 (θ 1 2 θ (1 ps y,k pθ 2 2 θps 2 c pθ,β,r 2 2 where we used (4.4 since pθ >β. Thus, (( 2 E x 1 ΩR 2 (1 ps y,k p θ c pθ,β,r 2 (1 pθs and from the Chebychev-Markov inequality we get for any δ>0 P x({ 2 2 (1 ps y,k p >δ } Ω R δ θ c pθ,β,r 2 (1 pθs. Summing over N 0 we find for pθs < 1 by (the easy direction of the Borel- Cantelli Lemma (e.g. [5] that (4.6 holds a.s. (P x onω R. Since θ>1was arbitrary, (1.7 follows first on Ω R a.s. and, as R via a countable number of values, then on Ω a.s. Case 2: p >β ψ,p q,ps < 1. As in case 1 it is enough to check ( 2 q/p 2 (1 ps 2 p /2 y,k (ω p 1 ΩR (ω < a.s. (P x which is implied by (4.7 ( 2 +1 q/p 2 (1 ps y +1,k (ω p 1 ΩR (ω < a.s. (P x (y,k as above. Since q p 1 we get from Lq/p (P x L 1 (P x that E x ( ( 2 +1 q/p 2 (1 ps y +1,k p 1 ΩR = ( (1 ps E x ( y +1,k p q/p 1 ΩR ( (1 ps c p,β,r 2 1 q/p 2 q p (1 ps c q/p p,β,r <

12 674 R.L. Schilling where we used (4.4 in the penultimate step. This proves (4.7 and we may proceed as in case 1. Case 3: p >β ψ,p<q,qs < 1. Clearly q = is not possible as qs < 1. Thus we may assume that q = p + r for some r > 0. As in case 2 we have to check the a.s. finiteness of (4.7. Observe that for y,k (ω =Y( k 2,ω Y ( k 1 2,ω =0 ( 2 +1 q/p 2 (1 ps y +1,k p 1 ΩR ( (1 ps (2R p r/p( 2 2 (1 ps y +1,k p 1 ΩR =2 r/p (2R r rs 2 (1 ps y +1,k p 1 ΩR =2 r/p (2R r 2 (1 qs y +1,k p 1 ΩR 2 +1 and the assertion follows ust as in case 2. Remark 4.4 (A For α-stable Lévy processes sharper results were obtained in [6]: the paths restricted to [0, 1] are a.s. contained in Bp 1/α for 1 p <α, but are a.s. not in this class if p α. The proof (especially of the second assertion relies on the scaling property of α-stable processes, X (t t 1/α X (1 in law. The restriction p 1 ensures that E( X (1 θ, θ>1, is finite. Note that this assumption is not necessary if one applies our truncation method. For (not necessarily α-stable Lévy processes the result of case 1 above was established in [11]. (B Our theorem only covers the case where p >β ψ. Observing that for p β ψ <λ, L p (P x L λ (P x holds we obtain from (4.4 Y (t Y (s p dp x c p/λ p,β,r t s p/λ, Ω R i.e., one can prove versions of Theorem 1.1 for p β ψ. References 1. Bensoussan, A., J.L. Lions: Contrôle impulsionnel et inéquations quasi variationelles, Dunod, collection méthodes mathématiques de l informatique t. 11, Paris Berg, C., G. Forst: Potential Theory on Locally Compact Abelian Groups, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete,II. Ser. vol. 87, Berlin, Heidelberg, New York: Springer Blumenthal, R.M., R.K. Getoor: Sample Functions of Stochastic Processes with Stationary Independent Increments, J. Math. Mech. 10 (1961,

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