Alfonso Rocha-Arteaga

COVARIANCE-PARAMETER LÉVY PROCESSES IN THE SPACE OF TRACE-CLASS OPERATORS

The paper deals with Levy processes with values in L1(H), the Banach space of trace-class operators in a Hilbert space H. Levy processes with values and parameter in a cone K of L1(H) are introduced and several properties are established. A family of L1(H)-valued Levy processes is obtained via the subordination of K-parameter, L1(H)-valued Levy processes, identifying explicitly their generating triplets.

RANDOM MATRIX MODELS OF STOCHASTIC INTEGRAL TYPE FOR FREE INFINITELY DIVISIBLE DISTRIBUTIONS

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ∫0∞e−1dΨtd, d ≥ 1, where Ψtd is a d × d matrix-valued Lévy process satisfying an Ilog condition.

MatG Random Matrices

A class of infinitely divisible covariance mixtures of Gaussian random matrices is introduced, and a characterization within the class of infinitely divisible left-orthogonally invariant matrix distributions is proved.

Cone-additive processes in duals of nuclear Fréchet spaces

For an additive process ( X t ) t≥0 with values in the dual of a nuclear Fréchet space and for each finite time T > 0, the existence of an equivalent additive process which takes values in a Hilbert space is shown. The additive process takes values in a cone if and only if it has a special Lévy-Khintchine representation and in this case for each T > 0 there exists a pathwise version in some Hilbert space .

On the Process of the Eigenvalues of a Hermitian Lévy process

The dynamics of the eigenvalues (semimartingales) of a Levy process X with values in Hermitian matrices is described in terms of Ito stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of X is also studied. If X has a jump at time t two different situations are considered, depending on the commutativity of X(t) and \(X(t-)\). In the commutative case all the eigenvalues jump at time t only when the jump of X is of full rank. In the noncommutative case, X jumps at time t if and only if all the eigenvalues jump at that time when the jump of X is of rank one.

Stochastic Integral and Covariation Representations for Rectangular Lévy Process Ensembles

A Bercovici-Pata bijection \(\Lambda _{c}\) from the set of symmetric infinitely divisible distributions to the set of ⊞ c -free infinitely divisible distributions, for certain free convolution ⊞ c is introduced in Benaych-Georges (Random matrices, related convolutions. Probab Theory Relat Fields 144:471–515, 2009. Revised version of F. Benaych-Georges: Random matrices, related convolutions. arXiv, 2005). This bijection is explained in terms of complex rectangular matrix ensembles whose singular distributions are ⊞ c -free infinitely divisible. We investigate the rectangular matrix Levy processes with jumps of rank one associated to these rectangular matrix ensembles. First as general result, a sample path representation by covariation processes for rectangular matrix Levy processes of rank one jumps is obtained. Second, rectangular matrix ensembles for ⊞ c -free infinitely divisible distributions are built consisting of matrix stochastic integrals when the corresponding symmetric infinitely divisible distributions under \(\Lambda _{c}\) admit stochastic integral representations. These models are realizations of stochastic integrals of nonrandom functions with respect to rectangular matrix Levy processes. In particular, any ⊞ c -free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type \(\int _{0}^{\infty }e^{-t}\mathrm{d}\Psi (t)\), where \(\left \{\Psi (t): t \geq 0\right \}\) is a rectangular matrix Levy process.