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STUDIA SCI. MATH. HUNGARICA 33, 177–183. Escape rates for Lévy processes* By Davar Khoshnevisan Department of Mathematics The University of Utah Salt Lake City, UT. 84112, U.S.A. davar@math.utah.edu Dedicated to Professor Endre Csáki, on the occasion of his sixtieth birthday Abtract. We prove a space–time estimate for a Lévy process to hit a small set. As an application, we present escape rates for Lévy processes with strictly stable components. §1. Introduction. Let X denote a d–dimensional Lévy process. It is a classical fact that a Borel set A ⊂ Rd is polar for X if and only if A has positive X–capacity; cf. Blumenthal and Getoor BG]. A sharper variant of the aforementioned fact is the consequence of more recent investigations such as those of Benjamini et al. [BPP], Fitzsimmons and Salisbury [FS], Peres [Pe] and Salisbury [Sa]. Roughly speaking, these results provide in a−1variety of different 0 contexts, qualitiative estimates of the type: P (Xt ∈ A, for some t > 0 e (A), where f g implies the existence of some universal C > 1, such that C −1 g ≤ f ≤ Cf pointwise, and e(A) is the X–energy integral associated with A. One of the many uses of such an estimate is that one can often approximate the chance that X ever hits a small set. Wishing to study escape rates, we present a different sort of a qualitative estimate below. Our notation is more or less that of Markov process theory. (1.1) Theorem. Suppose X is a d–dimensional Lévy process. For any b > a > 0 and ε > 0, R 2b−a 0 Rb P (|Xr | ≤ 2ε)dr 1 a P0 (|Xr | ≤ ε)dr 0 ≤ P |Xr | ≤ ε, for some a ≤ t ≤ b ≤ Ra b−a , Rb 2 P0 (|Xr | ≤ ε) P0 (|Xr | ≤ ε)dr 0 0 whenever the integrals exist and are nonzero. The above extends the estimates of Perkins and Taylor [PT], Takeuchi [T1,T2] and Takeuchi and Watanabe [TW], to cite a few examples. To illustrate the use of such a general inequality, let us restrict attention to the class of processes described in Hendricks [H1,H2,H3]. Namely, we consider the case where X is a d–dimensional Lévy process with strictly stable components. In other words, there exists υ, χ ∈ R d+ and α ∈ (0, 2]d , such that for all t > 0 and all ζ ∈ Rd , (1.2) P exp iζ Xt 0 0 d d X X 1/αj = exp −t |ζj χj | −i υj sgn(ζj ) . j=1 * Research partially supported by NSF grant DMS–95–03290 1 j=1 STUDIA SCI. MATH. HUNGARICA 33, 177–183. Throughout, we shall assume that the coordinate processes are not completely asymmetric, i.e,. υ j (1.3) for all j = 1, · · · , d. < tan παj /2 , |χj | > 0, χj Viewed coordinate by coordinate, such processes scale, albeit differently in each. Define, d X 1 β= . α j=1 j (1.4) Our intended application of Theorem (1.1) is the following: (1.5) Theorem. Suppose X is a Lévy process with stable components with parameters given by (1.2)–(1.4). When β < 1, X hits points. When β = 1, singletons are polar, but X is neighborhood recurrent. When β > 1, X is transient. For β ≥ 1, let ϕ : R1+ 7→ R 1+ be an decreasing function and define R ∞ β−1 (t)t−1 dt, if β > 1 1 ϕ J(ϕ) = R . −1 ∞ t| ln ϕ(t)| dt, if β = 1 1 When β ≥ 1, P0 –almost surely, lim inf t→∞ (t→0+ ) ∞, if J(ϕ) < ∞ max1 6 j 6 tϕ(t) j αj d |Xt | = 0, if J(ϕ) = ∞ . When α is a constant vector, the above appears to various degrees of generality in Dvoretsky and Erdős [DE], Spitzer [Sp], Takeuchi [T1,T2] and Takeuchi and Watanabe [TW]. When α is not a constant vector, a different but equivalent formulation can be found in Hendricks [H1] with a longer proof. Our formulation has two distinct advantages over the latter: (1) the large–time results and the small–time results are the same; (2) ours incorporates all the known results as one. Note that the critical case (i.e., β = 1) only applies to two cases: d = 1 and α = 1 (Cauchy process on R 1 ) or d = 2 and α1 = α2 = 2 (planar Brownian motion). Above and throughout, we have used the notation: ln x , loge (x ∨ 1), x > 0. §2. The Proof of Theorem (1.1). Fix 0 < a < b and define T , inf s > 0 : |Xs | ≤ ε . Apply the strong Markov property at time T to see that Z 2b−a Z 2b−a 0 0 P 1 |Xr | ≤ 2ε dr ≥ P 1 |Xr | ≤ 2ε dr T ≤ b · P0 (T ≤ b) a a ≥ inf |x|≤ε ≥ P0 Z P b−a x Z 1 |Xr | ≤ 2ε dr · P0 (T ≤ b) 0 b−a 1 |Xr | ≤ ε dr · P0 (T ≤ b). 0 Divide to obtain the upper bound. The lower bound follows along similar lines. Consider, Z bZ r Z b 2 0 0 P 1 |Xr | ≤ ε dr = 2P 1 |Xr | ≤ ε, |Xs | ≤ ε dsdr a ≤ 2P0 Z a b a Z b Z =2 a (2.1) Z ≤ 2P0 s b Z a r 1 |Xs | ≤ ε 1 |Xr − Xs | ≤ 2ε dsdr a b P0 (|Xs | ≤ ε)P0 (|Xr−s | ≤ 2ε)drds 1 |Xr | ≤ ε dr · a Z b 0 2 P0 (|Xr | ≤ 2ε)dr. STUDIA SCI. MATH. HUNGARICA 33, 177–183. By the Cauchy–Schwartz inequality, P Z 0 b a Z b 0 1 |Xr | ≤ ε dr = P 1 |Xr | ≤ ε dr; T ≤ b s a Z b 2 p 0 ≤ P 1 |Xr | ≤ ε dr · P0 (T ≤ b). a ♦ Use (2.1) and solve to obtain the desired lower bound. (2.2) Remark. An inspection of the proof shows that for any x ∈ R d , P x R 2b−a x P |Xr | ≤ 2ε dr a |Xt | ≤ ε, for some a ≤ t ≤ b ≤ R b−a . P0 |Xr | ≤ ε dr 0 §3. The Proof of Theorem (1.5). Throughout this section, X denotes a Lévy process with strictly stable components given by (1.2)–(1.4). Let us start with a technical lemma. (3.1) Lemma. The random variable |X1 | has a bounded Pa –density, uniformly over all a ∈ Rd . When a = 0, this density g is positive on some neighborhood of 0. Moreover, supx g(x)/g(0) < ∞. Proof. By properties of convolutions, it suffices to show that each component of X has the given properties. The lemma follows from the inversion theorem for Fourier transforms. ♦ For the rest of this section, define, S(x) , max |xj |αj , 1 6j 6d x ∈ Rd . (3.2) Lemma. For all r, a > 0, P0 S(Xr ) ≤ a r−1 a ∧ 1 β . Proof. Since the components X j are independent αj –stable processes, by Lemma (3.1), P0 S(Xr ) 6 ε = d Y P |X1j | 6 ε1/αj r−1/αj j=1 ε β r ∧ 1. ♦ Since αj 6 2 for all j, it is not hard to show that for all x, y ∈ R d , S(x + y) 6 4 S(x) + S(y) . As such, S(x) behaves much like |x|. Going through the proof of Theorem (1.1) and using Lemma (3.2), the following estimate emerges: This proves the lemma. (3.3) Corollary. For all 0 < a < b and all ε > 0 small, P0 S(Xt ) ≤ ε, for some a ≤ t ≤ b h(ε), 3 STUDIA SCI. MATH. HUNGARICA 33, 177–183. where 1, −1 h(ε) = ln(1/ε) , β−1 ε , if β < 1 if β = 1 . if β > 1 We are now ready to prove Theorem (1.5). We shall do so for t → ∞. The case t → 0+ is done similarly. The case β < 1 follows immediately from Corollary (3.3). Let us restrict attention to the case β ≥ 1. We shall assume without loss of generality that ϕ(x) ↓ 0 as x → ∞. (When inf x ϕ(x) > 0, the result is simpler and also follows from the proof given below.) Define tn , 2n , ϕn , ϕ(tn ) and (3.4) En , n inf tn ≤t≤tn+1 o S(Xt ) ≤ 2n ϕn . Note that from Corollary (3.3), P0 (En ) h(ϕn ). (3.5) P From the definition of J(ϕ) given in (1.5), it follows that n P0 (En ) < ∞ if and only if J(ψ) < ∞, when ψ(x) , Aϕ(Bx), for any A, B > 0. Suppose J(ϕ) < ∞. The previous paragraph shows that for any c > 0, X n P0 inf tn ≤t≤tn+1 |Xt | ≤ c2n ϕ(2n−1 ) < ∞. Since c is arbitrary and ϕ is decreasing, by the Borel–Cantelli lemma, lim inf t→∞ |Xt | = ∞, tϕ(t) P0 –a.s. . Now suppose J(ϕ) = ∞. Note that Remark 2.2 holds with |X· | replaced by S(X· ) everywhere. Using the Markov property and Lemma (3.1), for all n large enough, P0 (En ∩ En+k ) ≤ P0 (En ) sup Px (En+k ) P0 (En )P0 (En+k ). x Theorem (1.5) follows from (3.4), (3.5), Kolmogorov’s 0–1 law and the Kochen–Stone lemma ([KS]). ♦ References. [BPP] I. Benjamini, Y. Peres and R. Pemantle (1995). Martin capacity for Markov chains. Ann. Prob., 23, 1332–1346 [BG] R. Blumenthal and R.K. Getoor (1968). 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