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Register a new userSharp concentration for the largest and smallest fragment in a $k$-regular self-similar fragmentation
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We study the asymptotics of the $k$-regular self-similar fragmentation process. For $alpha > 0$ and an integer $k geq 2$, this is the Markov process $(I_t)_{t geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^alpha$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{t geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| leq 1$ and $|M_t - h(t)| leq 1$ for all sufficiently large $t$. Furthermore, for each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as $n to infty$.
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We show that in homogeneous fragmentation processes the largest fragment at time $t$ has size $e^{-t Phi(bar{p})}t^{-frac32 (log Phi)(bar{p})+o(1)},$ where $Phi$ is the Levy exponent of the fragmentation process, and $bar{p}$ is the unique solution of the equation $(log Phi)(bar{p})=frac1{1+bar{p}}$. We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.
In the nonparametric Gaussian sequence space model an $ell^2$-confidence ball $C_n$ is constructed that adapts to unknown smoothness and Sobolev-norm of the infinite-dimensional parameter to be estimated. The confidence ball has exact and honest asymptotic coverage over appropriately defined `self-similar parameter spaces. It is shown by information-theoretic methods that this `self-similarity condition is weakest possible.
This paper addresses the question of predicting when a positive self-similar Markov process X attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in Glover et al. (2013) under the assumption that X is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to Baurdoux and van Schaik (2013), where the same question is studied for a Levy process drifting to minus infinity. The connection to Baurdoux and van Schaik (2013) relies on the so-called Lamperti transformation which links the class of positive self-similar Markov processes with that of Levy processes. Our approach will reveal that the results in Glover et al. (2013) for Bessel processes can also be seen as a consequence of self-similarity.
For a positive self-similar Markov process, X, we construct a local time for the random set, $Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L{e}vy process associated to X via Lampertis transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L{e}vy process oscillates.
It is likely that most protostellar systems undergo a brief phase where the protostellar disc is self-gravitating. If these discs are prone to fragmentation, then they are able to rapidly form objects that are initially of several Jupiter masses and larger. The fate of these disc fragments (and the fate of planetary bodies formed afterwards via core accretion) depends sensitively not only on the fragments interaction with the disc, but with its neighbouring fragments. We return to and revise our population synthesis model of self-gravitating disc fragmentation and tidal downsizing. Amongst other improvements, the model now directly incorporates fragment-fragment interactions while the disc is still present. We find that fragment-fragment scattering dominates the orbital evolution, even when we enforce rapid migration and inefficient gap formation. Compared to our previous model, we see a small increase in the number of terrestrial-type objects being formed, although their survival under tidal evolution is at best unclear. We also see evidence for disrupted fragments with evolved grain populations - this is circumstantial evidence for the formation of planetesimal belts, a phenomenon not seen in runs where fragment-fragment interactions are ignored. In spite of intense dynamical evolution, our population is dominated by massive giant planets and brown dwarfs at large semimajor axis, which direct imaging surveys should, but only rarely, detect. Finally, disc fragmentation is shown to be an efficient manufacturer of free floating planetary mass objects, and the typical multiplicity of systems formed via gravitational instability will be low.
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