Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics HITTING TIMES FOR TRANSIENT STABLE PROCESSES SIDNEY CHARLES PORT Vol. 21, No. 1 November 1967

2 PACIFIC JOURNAL OF MATHEMATICS Vol. 21, No. 1, 1967 HITTING TIMES FOR TRANSIENT STABLE PROCESSES S. C. PORT In this paper we explicitly find the asymptotic behavior, for large t, of the probability that a transient d-dimensional stable process first (last) hits a bounded Borel set during the time interval (t, oo). Assume that X(t) is a stable process on R d (d-dimensional Euclidean space) having exponent a < d and normalized so that the paths are right continuous with left-hand limits at every point. Assume further that [X(t) - X(0)]t~ lla is distributed like X(l) - X(0), and moreover, that X(l) X(0) has a genuinely ώ-dimensional distribution on R d. [In particular, every symmetric stable process on R d with 0 mean (when it exists) satisfies these conditions.] From these assumptions it follows that X(t) X(0) has a bounded, continuous density, f(t % x), which satisfies the well-known scaling property (1.1) f(t, x) = ί- d/α /(i, t-"*χ). For a Borel (more generally, analytic) set BaR d, V B = mf{t>0:x(t)eb} let denote the first hitting time of B. X(t) ί B As usual we set V B = if Our main purpose in this note is to establish the follow- for all t > 0. ing. R d. THEOREM 1. Let B be a bounded Borel (or analytic) subset of Then under the above assumptions on X(t), (1.2) lim P dla) -Ψ x (t <V B <^) = P X = oo)c(b) Γ - lt/(l, 0), *-~ La J where C(B) is the natural capacity of B. Previously, (by using a different method) Joffe [2] established this result for symmetric processes with (d/2) < a < 1 when B has a nonempty interior, and Spitzer [4] (Lemma, p. 114) established this result for arbitrary compact B in the case of 3-dimensional Brownian motion. In the case of recurrent stable processes the analogue of Theorem 1 can be found in [3]. 161

3 162 S. C. PORT It is interesting to compare Theorem 1 with the following, much easier THEOREM 2. Let T B = inf{ί ^ 0: X(s) $ B, all s > t} be the last hitting time of B. Then under the same conditions as Theorem 1, (1.3) lim t^^ψ x (T B > ί) = C(B) ΓA - lt/(l, 0). c-» L a J 2 Proofs* Proof of. Theorem 1. A first passage decomposition yields (2.1) P.(ί < V B < oo) = ί P X > t, X(t) e dy)p y {V B < oo) = j Λd [/(ί, y - x ) ~ \[\MX, ds, dz)f(t -s,y- z)\p y < oo)dy, where here and in the following, H s (x, ds, dz) = P X e ds, X(s) e dz), and B is the closure of B. But it is a known fact ([1] Prop. 18.4) that there is a measure, e B (dy), with support contained in B (the capacitary measure of B) and finite total mass C(B) (the capacity of B), such that (2.2) P y < oo) = \g(u - y)e B (du), where 9(x) = Γ/(*, *)<** Jo is the potential kernel density for the process X(t). Setting and using the fact that (2.3) j Λd /(t, 2/ - aj)flr(w - 2/)% - R(t, u - x), we obtain from (2.1) that

4 HITTING TIMES FOR TRANSIENT STABLE PROCESSES 163 (2.4) P x (t < V B < oo) \\[*( χ > ds > From the scaling property (1.1) and the fact that /(I, x) is continuous, we see that Mm^ t dla f{t,») = /(l,0), uniformly in x on compacts, and thus (2.5) lim t^-ήit, x) = /(I, 0)Γ^ - lv, uniformly in x on compacts. Set β(t) = ί- (^β)+1 Γ - lt 1. Lα J Then from (2.5), (2.6) lim ί m 'l~ and X) e B (dy) = /(I, O)C(B), lim lim ΓΓί ί H B (x, ds, dz)r(t - s,y - z)e B (dy)\r(t)- 1 (2 7) τ ~* '"*" * *-**)* = lim Γfl-^a;, ds, B)C(B)f(l, 0) = P,( Vi, < oo)c(5)/(l, 0). J From (2.4), we see that in order to complete the proof it suffices to show (2.8) lim lim sup Bit)" 1 \ \_[ H B (x, ds, dz)b(t - s,y - z)e B (dy) = 0. T->oo ί->oo JFJBJB S t rt/2 rt-τ ct as +1 + I. Since T J2* Jί/2 Jt-T SUP /(I, X) = if < oo, it follows from the scaling property that J2(ί, x) S KB{t) for all t > 0. Setting A - #<?( ), we obtain ί/2 ft/2 S ^ A \ P,(F 5 e dβ)b(t - 8) ^ AR(t/2)P x (T < V B and thus Next observe that ί/2 S = 0.

5 164 S. C. PORT ^ A [P x eds)r(t - s) ^ AR(T)P x (t/2 < V B < oo). ί/2 Jί/2 By (2.4) this last term is dominated by A 2 R(T)R(t/2), and thus ί-γ S = 0. ί/2 Finally, from (2.2) we see that But Γ ^ Γ ί # B (α, ds, dz) \_g(y - z)e B (dy) ^ Γ P X e ds). P.(ί - T < V B fί t) = \ P. > t - T, X(t - T) e dy)p y ^ Γ) ^ ί f(t-t,y- J R d x)p y S T)dy ^ K(t - Γ)-"' Since the paths X(t) are bounded a.s. on [0, T], we see that for each T there is a sphere S T^B $ such that P tf (X(ί) S τ ) ^ 1/2 for all ί ^ Γ and y e B. But then ( ώ Γ L H ^, dβ, dy)p y (X(T ~s)e S*) R d JOJB Thus This completes the proof. Proof of Theorem 2. Clearly lim Rit)- 1^ ==0. P X (T B >t) = \ f(t, y - x)p y < JRd Using (2.2) and (2.3) we see that P β (T B >t) = from which the theorem follows. JB \R(t,y-x)e B (dy), REMARK. When d/2 < a < d, it is possible to establish Theorem 1 by a much simpler argument. Set

6 HITTING TIMES FOR TRANSIENT STABLE PROCESSES 165 and H (x, dy) = (V λί P,( V B e dt, x(t) e dy) JO R\x) = \~R(t, χ) e - λt dt. Jo Then from (2.4) we obtain (2.9) Q$(χ) It follows from (2.5) that uniformly in x on compacts, lim R\x)X^«- /(I, O)ΓA - ltv(2 - d/α). λio la J Consequently, from (2.9), we see that d '«= /(I, O)C(B)P X = oo)\± - lvr(2 ~ dja). λio La J An appeal to Karamata's theorem, and the fact that P x (t < V B < o) is monotone in ί, then yields (1.2). The above argument breaks down when a < d/2 since lim R\x) < oo, λio and the more complicated proof given previously is needed. REFERENCES 1. G. A. Hunt, Markoff processes and potentials III, Illinois J. Math. 2 (1958), A. Joffe, Sojourn Time for Stable Processes, Thesis, Cornell U., S. C. Port, Hitting times and potentials for recurrent stable processes, J. D' Analyse Mathematique (to appear) 4. F. Spitzer, Electrostatic capacity, heat flow, and Brownian motion, Z. Warscheinlichk 3, (1964). Received June 30, This research is sponsored by the United States Air Force under Project RAND Contract No. AF 49 (638)-1700 monitored by the Directorate of Development Plans, Deputy Chief of Staff, Research and Development, Hq USAF. THE RAND CORPORATION UNIVERSITY OF CALIFORNIA, LOS ANGELES

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8 PACIFIC JOURNAL OF MATHEMATICS H. SAMELSON Stanford University- Stanford, California J. P. JANS University of Washington Seattle, Washington EDITORS J. DUGUNDJI University of Southern California Los Angeles. California RICHARD ARENS University of California Los Angeles, California E. F. BECKENBACH ASSOCIATE EDITORS B. H. NEUMANN F WOLF K. YOSIDA UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY UNIVERSITY OF SOUTHERN CALIFORNIA SUPPORTING INSTITUTIONS STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY CHEVRON RESEARCH CORPORATION TRW SYSTEMS NAVAL ORDNANCE TEST STATION Printed in Japan by International Academic Printing Co., Ltd., Tokyo Japan

9 Pacific Journal of Mathematics Vol. 21, No. 1 November, 1967 Friedrich-Wilhelm Bauer, Der Hurewicz-Satz D. W. Dubois, A note on David Harrison s theory of preprimes Bert E. Fristedt, Sample function behavior of increasing processes with stationary, independent increments Minoru Hasegawa, On the convergence of resolvents of operators Søren Glud Johansen, The descriptive approach to the derivative of a set function with respect to a σ -lattice John Frank Charles Kingman, Completely random measures Tilla Weinstein, Surfaces harmonically immersed in E Hikosaburo Komatsu, Fractional powers of operators. II. Interpolation spaces Edward Milton Landesman, Hilbert-space methods in elliptic partial differential equations O. Carruth McGehee, Certain isomorphisms between quotients of a group algebra DeWayne Stanley Nymann, Dedekind groups Sidney Charles Port, Hitting times for transient stable processes Ralph Tyrrell Rockafellar, Duality and stability in extremum problems involving convex functions Philip C. Tonne, Power-series and Hausdorff matrices