Stable processes with drift on the line

Abstract

Abstract. The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets. Statement of resultsLet Xt be a process with stationary independent increments having log characteristic function -i|0|a(l-zAsgn (0)), where 0 < a < 1 or 1 < a < 2, h = ß tan(na/2), and \ß\ < 1. Such processes are called drift free strictly stable processes. The processes with log characteristic functions -i|0|(l + Agn (0)ln|0|) are called drift free Cauchy processes (or stable processes with index a = 1 ). If Xt -X. -bt we say X( is a stable process with drift -b. In this paper we will always take b > 0.There is an enormous literature on the behavior of drift free stable processes. Surprisingly, little has been written on stable processes with drift. In view of the extensive results known for the potential theory [6] and the path behavior [3,4] for infinitely divisible processes in general, the interest today in special processes such as stable processes with drift lies in the fact that for such processes rather explicit results may be obtained.The incorporation of a drift term changes the behavior of the process as compared to a drift free process. However the drift acts substantially different for processes with index a < I , a = I , and a > 1 . In all cases except for a = 1 and ß = 0 (i.e. the symmetric Cauchy process) the processes with drift are transient and hit points with positive probability. This fact follows from the general theory [3 and 6]. This is in contrast to the drift free processes. For drift free processes if a < 1 the processes are transient but do not hit points,

Occupation time and the Lebesgue measure of the range for a Lévy process

“…['♦"-(tH-'ItJ"^T hen it was shown in [4] that EN(a) ~ A^a", a -♦ co, and, as mentioned above, N(a)/k2aa -> N. Using these facts, (1.1), and Theorem 1, we can conclude the following.…”

“…In all other cases [0, co) is a recurrent set. It was shown in [4] that for ß > -1, EN(a) ~ fc2aQ, a -► co, for a specified numerical constant fci, and that One of the motivations for the present paper was to find the limiting distribution for R(a)/ER(a)…”

“…See e.g. [4]. We will illustrate the use of Theorem 1 by finding the limiting distributions of R(a)/ER(a)…”

Asymptotic estimates of multi-dimensional stable densities and their applications

“…Sato and Watanabe [34] obtained the set T for all transient α-semistable processes with d = 1 and classified all transient α-semistable processes into weakly transient and strongly transient. Port [23], [24] completely studied the asymptotic order of ∆ The final proposition shows that the asymptotics of ∆ Proof. Proof is similar to Lemma 4.5 by using (2.3).…”

“…In the case where X t has a purely singular distribution for every t > 0, we make an additional assumption that P x (T B = T intB ) = 1 for almost every x. Let The following lemma is a result obtained by a series of works [36], [8], [23], [24], [25], [26]. Assertion (i) is given in Theorem 14.2 of [26] and assertion (ii) is in Lemmas 3.1 and 3.3 and Theorem 1 of [25] with (3.19) and (14.13) of [26], respectively.…”