We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaus...

In this work, we prove that for any dimension d≥1 and γ∈(0,1) super-Brownian motion corresponding to the log-Laplace equation v(t,x)=(Stf)(x)−∫0t(St−svγ(s,·))(x)ds,(t,x)∈R+×Rd, is absolutely continuous with respect Lebesgue measure at fixed time t>0. {St}t≥0 denotes a transition semigroup of standard Brownian motion. Our proof based on properties solutions equation. We also when initial datum v...

Let (ξ(s))s≥0 be a standard Brownian motion in d ≥ 1 dimensions and let (Ds)s≥0 be a collection of open sets in R. For each s, let Bs be a ball centered at 0 with vol(Bs) = vol(Ds). We show that E[vol(∪s≤t(ξ(s) + Ds))] ≥ E[vol(∪s≤t(ξ(s) + Bs))], for all t. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1 2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that the...

We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. These processes can be represented as W ◦A, where W is a standard Brownian motion, and A is a subordinator independent of W . Under some mild general assumption, we give precise estimates (up to a constant multiple in the logarithmic scale) of the small deviation probabilities. The...

Let (" i) i1 be a sequence of independent Rademacher variables, and let (B t) t0 be a standard Brownian motion. Then the conjecture made in [5] that the best constant in the maximal Khintchine inequality:

Let X be a standard 2-dimensional Brownian motion. There exists a.s. t ∈ (0, 1) such that X([0, t)) ∩ X((t, 1]) = ∅. It follows that X([0, 1]) is not homeomorphic to the Sierpiński carpet a.s.

A statistical application to Gene Set Enrichment Analysis implies calculating the distribution of themaximum of a certain Gaussian process, which is a modification of the standard Brownian bridge. Using the transformation into a boundary crossing problem for the Brownian motion and a piecewise linear boundary, it is proved that the desired distribution can be approximated by an n-dimensional Ga...