Catálogo de publicaciones - libros

The reconstruction of the history of a set of sequences is a central problem in molecular evolutionary biology. Typically this history is summarized in a phylogenetic tree. In current practice the estimation of a phylogenetic tree is a two-step procedure: first a multiple alignment is computed and subsequently a phylogenetic tree is reconstructed, based on the alignment. However, it is well known that the alignment and the tree reconstruction problem are intertwined. Thus, it is of great interest to estimate alignment and tree simultaneously. We present here a stochastic framework for this joint estimation. We discuss a variant of the Thorne-Kishino-Felsenstein model, having equal rates of insertions and of deletions of sequence fragments, for $$\ell $$ ≥ 2 sequences related by a phylogenetic tree. Finally, we review novel approaches to tree reconstruction based on insertion-deletion models.

Branching processes exhibit a particularly rich longtime behaviour when evolving in a random environment. Then the transition from subcriticality to supercriticality proceeds in several steps, and there occurs a second ‘transition’ in the subcritical phase (besides the phase-transition from (sub)criticality to supercriticality). Here we present and discuss limit laws for branching processes in critical and subcritical i.i.d. environment. The results rely on a stimulating interplay between branching process theory and random walk theory. We also consider a spatial version of branching processes in random environment for which we derive extinction and ultimate survival criteria.

Let $$\ell $$ be the projected intersection local time of two independent Brownian paths in $$\mathbb{R}^d $$ for d = 2, 3. We determine the lower tail of the random variable $$\ell $$ ( B (0, 1)), where B (0, 1) is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.

We consider a random dynamical system describing the diffusion of a small-noise Brownian particle in a double-well potential with a periodic perturbation of very large period. According to the physics literature, the system is in stochastic resonance if its random trajectories are tuned in an optimal way to the deterministic periodic forcing. The quality of periodic tuning is measured mostly by the amplitudes of the spectral components of the random trajectories corresponding to the forcing frequency. Reduction of the diffusion dynamics in the small noise limit to a Markov chain jumping between its meta-stable states plays an important role. We study two different measures of tuning quality for stochastic resonance, with special emphasis on their robustness properties when passing to the reduced dynamics of the Markov chains in the small noise limit. The first one is the physicists favourite, spectral power amplification. It is analyzed by means of the spectral properties of the diffusion’s infinitesimal generator in a framework where the system switches every half period between two spatially antisymmetric potential states. Surprisingly, resonance properties of diffusion and Markov chain differ due to the crucial significance of small intra-well fluctuations for spectral concepts. To avoid this defect, we design a second measure of tuning quality which is based on the pure transition mechanism between the meta-stable states. It is investigated by refined large deviation methods in the more general framework of smooth periodically varying potentials, and proves to be robust for the passage to the reduced dynamics.

We study continuity properties of inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. We focus on two cases: (i) the delay time tends to zero and (ii) the intensity of the noise becomes small.

We investigate a system of lcal interacting diffusion processes with attractive interaction. We show how the random walk representation can be used to express the gradient of the semigroup and to estimates for the time-space correlations. In particular we can answer questions dealing with localization, convergence rates to equilibrium and aging properies of the system.

We review recent results on the new concept of worst-case portfolio optimization, i.e. we consider the determination of portfolio processes which yield the highest worst-case expected utility bound if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. They are by construction non-constant ones and thus differ from the usual constant optimal portfolios in the classical examples of the Merton problem. A particular application of such strategies is to model crash possibilities where both the number and the height of the crash is uncertain but bounded. We further solve optimal investment problems in the presence of an additional risk process which is the typical situation of an insurer.

We first explain how to derive the archetypal equation describing the roll motion of a ship in random seaway from first principles. We then present an analytic and numerical case study of two simple nonlinear models of the roll motion using concepts of the theory of random dynamical systems. In contrast to the case of periodic excitation, the incorporation of noise leads to scenarios in which capsizing of the ship (i.e. the disappearance of the random attractor) is not preceded by a series of bifurcations, but happens without announcement “out of the blue sky”.

In the first part of this paper we give an introduction to the contraction method for the analysis of additive recursive sequences of divide and conquer type. Recently some general limit theorems have been obtained by this method based on a general transfer theorem. This allows to conclude from the recursive structure and the asymptotics of first moment(s) the limiting distribution. In the second part we extend the contraction method to max-recursive sequences. We obtain a general existence and uniqueness result for solutions of stochastic equations including maxima and sum terms. We finally derive a general limit theorem for max-recursive sequences of the divide and conquer type.