The first-crossing area of a diffusion process with jumps over a constant barrier

For a given barrier $S$ and a one-dimensional jump-diffusion process $X(t),$ starting from $x<S,$ we study the probability distribution of the integral $A_S(x)= \int_0 ^ {\tau_S(x)}X(t) \ dt$ determined by $X(t)$ till its first-crossing time $\tau_S(x)$ over $S.$ In particular, we show that the Laplace transform and the moments of $A_S(x)$ are solutions to certain partial differential-difference equations with outer conditions. The distribution of the minimum of $X(t)$ in $[0, \tau_S(x)]$ is also studied. Thus, we extend the results of a previous paper by the author, concerning the area swept out by $X(t)$ till its first-passage below zero. Some explicit examples are reported, regarding diffusions with and without jumps

Energy Barriers and Activated Dynamics in a Supercooled Lennard-Jones Liquid

We study the relation of the potential energy landscape (PEL) topography to relaxation dynamics of a small model glass former of Lennard-Jones type. The mechanism under investigation is the hopping betweem superstructures of PEL mimima, called metabasins (MB). From the mean durations \tauphi of visits to MBs, we derive effective depths of these objects by the relation \Eapp=\d\ln\tauphi/\d\beta, where \beta=1/\kB T. Since the apparent activation energies \Eapp are of purely dynamical origin, we look for a quantitative relation to PEL structure. A consequence of the rugged nature of MBs is that escapes from MBs are not single hops between PEL minima, but complicated multi-minima sequences. We introduce the concept of return probabilities to the bottom of MBs in order to judge whether the attraction range of a MB was left. We then compute the energy barriers that were surmounted. These turn out to be in good agreement with the effective depths \Eapp, calculated from dynamics. Barriers are identified with the help of a new method, which accurately performs a descent along the ridge between two minima. A comparison to another method is given. We analyze the population of transition regions between minima, called basin borders. No indication for the mechanism of diffusion to change around the mode-coupling transition can be found. We discuss the question whether the one-dimensional reaction paths connecting two minima are relevant for the calculation of reaction rates at the temperatures under study.Comment: 17 pages, 16 figure

Multilevel Monte Carlo methods for applications in finance

Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and suggests directions for future research.Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with arXiv:1106.4730 by other author

Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems

We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.Comment: 37 pages, 11 figure

Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records

We review how the renormalized force correlator Delta(u), the function computed in the functional RG field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a 1-dimensional random-force landscape. The limit of small velocity allows to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Frechet. For each class we obtain analytically the universal function Delta(u), the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w. We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the ABBM model, avalanche distributions and Delta(u) can be computed for any velocity. For 2-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(u), and to extract the anisotropic scaling exponents zeta_x > zeta_y. We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered e.g. in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w.Comment: 41 pages, 35 figure

An explicit Skorokhod embedding for functionals of Markovian excursions

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of Markovian excursions. Our setting allows to solve the Skorokhod embedding problem, in particular, for diffusions and their (signed, scaled) age processes, for Azema's martingale, for spectrally one-sided Levy processes and their reflected versions, for Bessel processes of dimension smaller than 2, and for their age processes, as well as for the age process of excursions of Cox-Ingersoll-Ross processes. This work is a continuation and an important generalization of Obloj and Yor (SPA 110) [35]. Our methodology, following [35], is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Ito measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula in [35] for measures with atoms.Comment: 50 page

Contributions of plasma physics to chaos and nonlinear dynamics

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016