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 derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. This paper provides explicit strong solutions and stochastic analogues for distributed-order time-fractional diffusion equations on bounded domains, with Dirichlet boundary conditions.Published by Elsevier Inc.

Retrospective assessment of adverse childhood experiences is widely used in research, although there are concerns about its validity. In particular, recall bias is assumed to produce significant artifacts. Data from a longitudinal cohort (the British National Child Development Study; N=7710) and the retrospective Mainz Adverse Childhood Experiences Study (N=1062, Germany) were compared on 10 adverse childhood experiences and psychological adjustment at age 42 yr. Between the two methods, no significant differences in risk effects were detected. Results held for bivariate analyses on all 10 childhood adversities and a multivariate model; the latter comprises the childhood adversities which show significant long-term sequelae (not always with natural parent, chronically ill parent, financial hardship, and being firstborn) and three covariates. In conclusion, the present data did not show any bias in the retrospective assessment.

a b s t r a c tWe consider the first-exit time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator. An integral form representation and a series representation for the density of the ITS subordinator are obtained. The asymptotic behavior of the q-th order moments of the ITS subordinator are investigated. The limiting form of the ITS density and its k-th order derivatives are derived as the space variable x → 0 + . Finally, the governing pde of the ITS density is also obtained.