The solution of the perturbed Tanaka-equation is pathwise unique

The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.Comment: Published in at http://dx.doi.org/10.1214/11-AOP716 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limits of Zeros of Orthogonal Polynomials on the Circle

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a result of Alfaro and Vigil (which answered a question of Tur\'an): namely, for $n<N$, one can freely prescribe the $n$-th polynomial and $N-n$ zeros of the $N$-th one. We shall also describe all possible limit sets of zeros within the unit disk

Moving and oblique observations of beams and plates

We study the observability of the one-dimensional Schr{\"o}dinger equation and of the beam and plate equations by moving or oblique observations. Applying different versions and adaptations of Ingham's theorem on nonharmonic Fourier series, we obtain various observability and non-observability theorems. Several open problems are also formulated at the end of the paper