On second order elliptic equations with a small parameter

The Neumann problem with a small parameter $$(\dfrac{1}{\epsilon}L_0+L_1)u^\epsilon(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^\epsilon}{\partial \gamma^\epsilon}(x)|_{\partial G}=0$$ is considered in this paper. The operators $L_0$ and $L_1$ are self-adjoint second order operators. We assume that $L_0$ has a non-negative characteristic form and $L_1$ is strictly elliptic. The reflection is with respect to inward co-normal unit vector $\gamma^\epsilon(x)$. The behavior of $\lim\limits_{\epsilon\downarrow 0}u^\epsilon(x)$ is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.Comment: 28 pages, 1 figure, revised versio

Multiplicative decompositions and frequency of vanishing of nonnegative submartingales

In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y=N+A, where the measure dA is carried by the set of zeros of Y. In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.Comment: Typos corrected. Close to the published versio

On stochasticity in nearly-elastic systems

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.Comment: 35 pages, 12 figures, already online at Stochastics and Dynamic

Relating pseudospin and spin symmetries through charge conjugation and chiral transformations: the case of the relativistic harmonic oscillator

We solve the generalized relativistic harmonic oscillator in 1+1 dimensions, i.e., including a linear pseudoscalar potential and quadratic scalar and vector potentials which have equal or opposite signs. We consider positive and negative quadratic potentials and discuss in detail their bound-state solutions for fermions and antifermions. The main features of these bound states are the same as the ones of the generalized three-dimensional relativistic harmonic oscillator bound states. The solutions found for zero pseudoscalar potential are related to the spin and pseudospin symmetry of the Dirac equation in 3+1 dimensions. We show how the charge conjugation and $\gamma^5$ chiral transformations relate the several spectra obtained and find that for massless particles the spin and pseudospin symmetry related problems have the same spectrum, but different spinor solutions. Finally, we establish a relation of the solutions found with single-particle states of nuclei described by relativistic mean-field theories with scalar, vector and isoscalar tensor interactions and discuss the conditions in which one may have both nucleon and antinucleon bound states.Comment: 33 pages, 10 figures, uses revtex macro

Multiple G-It\^{o} integral in the G-expectation space

In this paper, motivated by mathematic finance we introduce the multiple G-It\^{o} integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-It\^{o} integrals which is a natural extension of the classical result obtained by It\^{o} in 1951.Comment: 9 page

The Dirac-Dowker Oscillator

The oscillator-like interaction is introduced in the equation for the particle of arbitrary spin, given by Dirac and re-written to a matrix form by Dowker.Comment: LaTeX file, 4pp. Preprint EFUAZ 94-0

Comment on ``the Klein-Gordon Oscillator''

The different ways of description of the $S=0$ particle with oscillator-like interaction are considered. The results are in conformity with the previous paper of S. Bruce and P. Minning.Comment: LaTeX file, 5p

The Local Time Distribution of a Particle Diffusing on a Graph

We study the local time distribution of a Brownian particle diffusing along the links on a graph. In particular, we derive an analytic expression of its Laplace transform in terms of the Green's function on the graph. We show that the asymptotic behavior of this distribution has non-Gaussian tails characterized by a nontrivial large deviation function.Comment: 8 pages, two figures (included

Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $\nu$) of the Fresnel equation is also performed and, in detail, some specific cases, like $\nu=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure

On inversions and Doob hh-transforms of linear diffusions

Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.Comment: 19 page