arXiv subjects
Publications and source records attributed to José Luís da Silva.
We prove that the self-intersection local times for generalized grey Brownian motion $B^{β,α}$ in arbitrary dimension $d$ is a well defined object in a suitable distribution space for $dα<2$.
In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman-Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures
Let $σ$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $dσ(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $Λ\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{σ(Λ)}\,σ$ on $Λ$ and the number of particles, $N$, is random and has Poisson distribution with parameter $σ(Λ)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $Λ$ of the Poisson point process on $\mathbb R^d$ with intensity measure $σ$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $Λ^{(n)}$ ($n\in\mathbb N$) are such that $Λ^{(1)}\subsetΛ^{(2)}\subsetΛ^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty Λ^{(n)}=\mathbb R^d$, then the corresponding particle system in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/σ(Λ^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.
In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H > \frac{1}{2} and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one.
In this paper we establish a substitution formula for stochastic differential equation driven by generalized grey noise. We then apply this formula to investigate the absolute continuity of the solution with respect to the Lebesgue measure and the positivity of the density. Finally, we derive an upper bound and show the smoothness of the density.
We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.