Infinitely divisible processes and their potential theory. II

Port, Sidney C.; Stone, Charles J.

doi:10.5802/aif.398

Cited by 41 publications

(64 citation statements)

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“…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.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Stable processes with drift on the line

Port¹

1989

Trans. Amer. Math. Soc.

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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,

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Section: Statement Of Resultsmentioning

confidence: 99%

“…The explicit form of the kernel a(x) and the fact that p(t,x) ~ j(l + b2)~x enable one to easily apply the general theory in [6] to obtain explicit asymptotic formulas analogous to those in the above theorems. Since this amounts to just plugging in to the formulas in [6] we omit these details.…”

Section: Eb(t A) -Tpb(a) ~ C(b)pb(a) ^-J (¿Y) Tmentioning

confidence: 99%

Stable processes with drift on the line

Port¹

1989

Trans. Amer. Math. Soc.

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“…Namely, let u (0) t (x) := u 0 (x), and then define iteratively 27) for t > 0, x ∈ R, and n ≥ 0, where the stochastic convolution is defined in (2.8). Clearly,…”

Section: Proof Of Proposition 21mentioning

confidence: 99%

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Conus

Khoshnevisan

2010

Probab. Theory Relat. Fields

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We study the stochastic heat equation ∂ t u = Lu + σ(u)Ẇ in (1 + 1) dimensions, whereẆ is space-time white noise, σ : R → R is Lipschitz continuous, and L is the generator of a symmetric Lévy process that has finite exponential moments, and u 0 has exponential decay at ±∞. We prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting. [See, however,[19,20] for the analysis of the location of the peaks in a different model.]Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

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“…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.…”

Section: Applications To Stable Processesmentioning

confidence: 99%

Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

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Abstract. The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

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Infinitely divisible processes and their potential theory. II

Cited by 41 publications

References 1 publication

Stable processes with drift on the line

Stable processes with drift on the line

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Asymptotic estimates of multi-dimensional stable densities and their applications

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