On approximation rates for boundary crossing probabilities for the multivariate Brownian motion process

Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, including the Lipschitz condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for "close boundaries" in the univariate case. We also obtained upper bounds for the first boundary crossing time densities.Comment: 15 page

Solution of the Forward Problem in Magnetic-Field Tomography (MFT) Based on Magnetoencephalography (MEG)

This paper presents the methodology and some of the results of accurate solution of the forward problem in magnetic-field tomography based on magnetoencephalography for brain imaging. The solution is based on modeling and computation of magnetic-field distribution in and around the head produced by distributed 2-D cortical and 3-D volume lead current sources. The 3-D finite-element model of the brain incorporates realistic geometry based on accurate magnetic resonance imaging data and inhomogeneous conductivity properties. The model allows arbitrary placement of line, surface, and volume current sources. This gives flexibility in the source current approximation in terms of size, orientation, placement, and spatial distribution

Variations of auroral hydrogen emission near substorm onset

The results of coordinated optical ground-based observations of the auroral substorm on 26 March 2004 in the Kola Peninsula are described. Imaging spectrograph data with high spectral and temporal resolution recorded the Doppler profile of the Hα hydrogen emission; this allows us to estimate the average energy of precipitating protons and the emission intensity of the hydrogen Balmer line. Two different populations of precipitating protons were observed during an auroral substorm. The first of these is associated with a diffuse hydrogen emission that is usually observed in the evening sector of the auroral oval and located equatorward of the discrete electron arcs associated with substorm onset. The average energy of the protons during this precipitation was ~20–35 keV, and the energy flux was ~3x10<sup>–4</sup>Joule/m<sup>2</sup>s. The second proton population was observed 1–2min after the breakup during 4–5min of the expansion phase of substorm into the zone of bright, discrete auroral structures (N-S arcs). The average energy of the protons in this population was ~60 keV, and the energy flux was ~2.2x10<sup>–3</sup>Joule/m<sup>2</sup>s. The observed spatial structure of hydrogen emission is additional evidence of the higher energy of precipitated protons in the second population, relative to the protons in the diffuse aurora. We believe that the most probable mechanism of precipitation of the second population protons was pitch-angle scattering of particles due to non-adiabatic motion in the region of local dipolarization near the equatorial plane.<p><b>Keywords.</b> Auroral ionosphere; Particle precipitation; Storms and substorm

On the infimum attained by a reflected L\'evy process

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of \prob{M(T_u)> u} (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios

Billiards in a general domain with random reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord "picked at random" in ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$.Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains

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

Spherical Model in a Random Field

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations $\sim r^{4-d}$.Comment: 29 page

Scaling of avalanche queues in directed dissipative sandpiles

We simulate queues of activity in a directed sandpile automaton in 1+1 dimensions by adding grains at the top row with driving rate $0 < r \leq 1$. The duration of elementary avalanches is exactly described by the distribution $P_1(t) \sim t^{-3/2}\exp{(-1/L_c)}$, limited either by the system size or by dissipation at defects $L_c= \min (L,\xi)$. Recognizing the probability $P_1$ as a distribution of service time of jobs arriving at a server with frequency $r$, the model represents a new example of the $$ server queue in the queue theory. We study numerically and analytically the tail behavior of the distributions of busy periods and energy dissipated in the queue and the probability of an infinite queue as a function of driving rate.Comment: 11 pages, 9 figures; To appear in Phys. Rev.

Scientific and Professional Degrees in Russia: Developing Traditions into the Future

When studying the history of Russian education, one becomes convinced of the validity of the dialectical principles of spiral development. The period of cyclicality in the history of attestation of scientific personnel is approximately equal to a century. In the first quarter of every century, starting from the time of the tsar Peter the Great, events took place that brought the system to a fundamentally new level. But at the same time, the fundamental foundations laid by the tsar Peter the Great and Lomonosov remain constant. Analyzing the attestation system development spiral, we find ways to solve urgent problems, and the foremost among them is ensuring Russia’s technological sovereignty. A similar problem stood a century ago. It was fully resolved in the USSR, but the foundations for its solution were laid in 1917 in the decree issued by the Russian Provisional Government: “On granting the Petrograd Polytechnic Institute the right to award academic degrees”. In the socioeconomic conditions of the modern Russia, we need new solutions. In this article we offer the