On the product formula on non-compact Grassmannians

We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ of two orbital measures on the symmetric space SO_0(p,q)/SO(p)\timesSO(q), $q>p$. We prove sharp conditions on $X$, Y\in\a for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for \SO_0(p,q)/\SO(p)\times\SO(q) will also serve for the spaces SU(p,q)/S(U(p)\timesU(q)) and Sp(p,q)/Sp(p)\timesSp(q), $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta_{e^X}^\natural$

On Wiener-Hopf factors for stable processes

We give a series representation of the logarithm of the bivariate Laplace exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in (0,2]$.Comment: 16 pages. to appear in Annales IHP, 201

Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $\Omega$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones

Convolution of orbital measures on symmetric spaces of type CpC_p and DpD_p

We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star\delta_{e^Y}^\natural$ of two orbital measures on the symmetric spaces ${\bf SO}_0(p,p)/{\bf SO}(p)\times{\bf SO}(p)$, \SU(p,p)/{\bf S}({\bf U}(p)\times{\bf U}(p)) and \Sp(p,p)/{\bf Sp }(p)\times\Sp(p). We prove sharp conditions on $X$, Y\in\a for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.Comment: arXiv admin note: text overlap with arXiv:1212.000

Strong solutions of non-colliding particle systems

We study systems of stochastic differential equations describing positions x_1,x_2,...,x_p of p ordered particles, with inter-particles repulsions of the form H_{ij}(x_i,x_j)/(x_i-x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)\leq ...\leq x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.Comment: 19 page

On exit time of stable processes

We study the exit time $\tau=\tau_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^\alpha$ as the Laplace transform of a positive random variable with explicit density. Consequently, $\tau$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $\tau$ are obtained as an application

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.Comment: 28 pages, editorial change