Probability tilting of compensated fragmentations

Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a L\'evy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.Comment: 41 pages, 1 figure. This revised version improves the conditions in Theorem 6.

Probabilistic aspects of critical growth-fragmentation equations

The self-similar growth-fragmentation equation describes the evolution of a medium in which particles grow and divide as time proceeds, with the growth and splitting of each particle depending only upon its size. The critical case of the equation, in which the growth and division rates balance one another, was considered by Doumic and Escobedo in the homogeneous case where the rates do not depend on the particle size. Here, we study the general self-similar case, using a probabilistic approach based on L\'evy processes and positive self-similar Markov processes which also permits us to analyse quite general splitting rates. Whereas existence and uniqueness of the solution are rather easy to establish in the homogeneous case, the equation in the non-homogeneous case has some surprising features. In particular, using the fact that certain self-similar Markov processes can enter $(0,\infty)$ continuously from either $0$ or $\infty$, we exhibit unexpected spontaneous generation of mass in the solutions.Comment: 28 pages. v2 adds an expository section 6 and fixes some error

A growth-fragmentation connected to the ricocheted stable process

Growth-fragmentation processes describe the evolution of systems in which cells grow slowly and fragment suddenly. Despite originating as a way to describe biological phenomena, they have recently been found to describe the lengths of certain curves in statistical physics models. In this note, we describe a new growth-fragmentation connected to random planar maps with faces of large degree, having as a key ingredient the ricocheted stable process recently discovered by Budd. The process has applications to the excursions of planar Brownian motion and Liouville quantum gravity.Comment: 12 pages. v3 makes minor descriptive changes and adds Corollary

Linear Perturbations in Brane Gas Cosmology

We consider the effect of string inhomogeneities on the time dependent background of Brane Gas Cosmology. We derive the equations governing the linear perturbations of the dilaton-gravity background in the presence of string matter sources. We focus on long wavelength fluctuations and find that there are no instabilities. Thus, the predictions of Brane Gas Cosmology are robust against the introduction of linear perturbations. In particular, we find that the stabilization of the extra dimensions (moduli) remains valid in the presence of dilaton and string perturbations.Comment: 17 pages, 1 figur

The hitting time of zero for a stable process

For any two-sided jumping $\alpha$-stable process, where $1 < \alpha < 2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pant\'i-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications

A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual

Dynamical decompactification from brane gases in eleven-dimensional supergravity

Brane gas cosmology provides a dynamical decompactification mechanism that could account for the number of spacetime dimensions we observe today. In this work we discuss this scenario taking into account the full bosonic sector of eleven-dimensional supergravity. We find new cosmological solutions that can dynamically explain the existence of three large spatial dimensions characterised by an universal asymptotic scaling behaviour and a large number of initially unwrapped dimensions. This type of solutions enlarge the possible initial conditions of the Universe in the Hagedorn phase and consequently can potentially increase the probability of dynamical decompactification from anisotropically wrapped backgrounds.Comment: 8 figures, JHEP3 styl

Effective Field Theory Approach to String Gas Cosmology

We derive the 4D low energy effective field theory for a closed string gas on a time dependent FRW background. We examine the solutions and find that although the Brandenberger-Vafa mechanism at late times no longer leads to radion stabilization, the radion rolls slowly enough that the scenario is still of interest. In particular, we find a simple example of the string inspired dark matter recently proposed by Gubser and Peebles.Comment: 19 pages, 2 figures, comments adde

The halo mass function through the cosmic ages

In this paper we investigate how the halo mass function evolves with redshift, based on a suite of very large (with N_p = 3072^3 - 6000^3 particles) cosmological N-body simulations. Our halo catalogue data spans a redshift range of z = 0-30, allowing us to probe the mass function from the dark ages to the present. We utilise both the Friends-of-Friends (FOF) and Spherical Overdensity (SO) halofinding methods to directly compare the mass function derived using these commonly used halo definitions. The mass function from SO haloes exhibits a clear evolution with redshift, especially during the recent era of dark energy dominance (z < 1). We provide a redshift-parameterised fit for the SO mass function valid for the entire redshift range to within ~20% as well as a scheme to calculate the mass function for haloes with arbitrary overdensities. The FOF mass function displays a weaker evolution with redshift. We provide a `universal' fit for the FOF mass function, fitted to data across the entire redshift range simultaneously, and observe redshift evolution in our data versus this fit. The relative evolution of the mass functions derived via the two methods is compared and we find that the mass functions most closely match at z=0. The disparity at z=0 between the FOF and SO mass functions resides in their high mass tails where the collapsed fraction of mass in SO haloes is ~80% of that in FOF haloes. This difference grows with redshift so that, by z>20, the SO algorithm finds a ~50-80% lower collapsed fraction in high mass haloes than does the FOF algorithm, due in part to the significant over-linking effects known to affect the FOF method.Comment: v4, 16 pages, 16 colour figures. Changed to match MNRAS print version. NOTE: v1 of this paper has a typo in the fitting function. Please ensure you use the latest versio