The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time. 2 Two equivalent formulationsRecall that the fractional Poisson process (FPP) N β (t) is a renewal process with Mittag-Leffler waiting times (1.1), and the fractal time Poisson process (FTPP) N 1 (E(t)) is Poisson process, with rate λ > 0, time-changed via the inverse stable subordinator (1.7). The proof that the FPP and the FTPP are the same process requires the following simple lemma.Lemma 2.1. Let D(t) be a strictly increasing right-continuous process with left-hand limits, and let E(t) be its right-continuous inverse defined by (1.7). ThenProof. Let t 0 = sup{t > 0 : E(t) < r}. Then there exists a sequence of points t n ↑ t 0 such that E(t n ) < r for all n.Letting n → ∞ shows that D(r−) ≥ t 0 .Since D has left-hand limits, for any r n ↑ r we have D(r n ) → D(r−) as n → ∞. If D(r−) > t 0 , then for some r n < r we have D(r n ) > t 0 . Since E(t) is nondecreasing and continuous, this implies that E(D(r n )) ≥ r, by definition of t 0 . But, E(D(r)) = r for all r > 0 implying that r n ≥ r, which is a contradiction. Thus, (2.1) follows.Theorem 2.2. For any 0 < β < 1, the FTPP N 1 (E(t)) is also a FPP. That is, the waiting times between jumps of the FTPP are IID Mittag-Leffler.Proof. Let W n be an IID sequence with P(W n > t) = e −λt and V n = W 1 + · · · + W n so that the Poisson process N 1 (t) = max{n ≥ 0 : V n ≤ t}. Let (2.2) τ n = sup{t > 0 : N 1 (E(t)) < n} denote the jump times of the FTPP. This definition of the jump times takes into account the fact that E(t) has constant intervals corresponding to the jumps of the process D(t). Using the fact that {N 1 (t) < n} = {V n > t} for the Poisson process, along with (2.2), we haveThen Lemma 2.1 implies that τ n = D(V n −). Define X 1 = τ 1 and X n = τ n − τ n−1 for n ≥ 2, the waiting times between jumps of the FTPP. In order to show that the FTPP is an FPP, it suffices to show that X n are IID Mittag-Leffler, i.e., they are IID with J n . Recall that the Laplace transform of the exponential distribution E(e −sWn ) = λ/(λ + s). Also recall that E(e −sD(t) ) = e −ts β . Since D(t) is a Lévy process, it has no fixed points of discontinuity and hence D(t−), D(t) are identically distributed for all t ≥ 0. (Indeed, D(t) = D(t−) a.s. [1, Lemma 2.3.2]).3
Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time.Comment: Published in at http://dx.doi.org/10.1214/08-AOP426 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.
Abstract. A Brownian time process is a Markov process subordinated to the absolute value of an independent one-dimensional Brownian motion. Its transition densities solve an initial value problem involving the square of the generator of the original Markov process. An apparently unrelated class of processes, emerging as the scaling limits of continuous time random walks, involves subordination to the inverse or hitting time process of a classical stable subordinator. The resulting densities solve fractional Cauchy problems, an extension that involves fractional derivatives in time. In this paper, we will show a close and unexpected connection between these two classes of processes and, consequently, an equivalence between these two families of partial differential equations.
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