Let BH={BH(t), t∈ℝ+N} be an (N, d)-fractional brownian sheet with index H=(H1, …, HN)∈(0, 1)N defined by BH(t)=(BH1(t), …, BHd(t)) (t∈ℝ+N), where BH1, …, BHd are independent copies of a real-valued fractional brownian sheet B0H. We prove that if d<∑ℓ=1NHℓ−1, then the local times of BH are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields124 (2002)). We also establish sharp local and global Hölder conditions for the local times of BH. These results...

We briefly review Marcinkiewicz's work, on analysis, on probability, and on the interplay between the two. Our emphasis is on the continuing vitality of Marcinkiewicz's work, as evidenced by its influence on the standard works. What is striking is how many of the themes that Marcinkiewicz studied (alone, or with Zygmund) are very much alive today. What this demonstrates is that Marcinkiewicz and Zygmund, as well as having extraordinary mathematical ability, also had excellent mathematical taste.

In this paper, we consider a random entire function $f(s,\omega )$ defined by a random Dirichlet series ${\sum }_{n=1}^{\infty }{X}_n\left(\omega \right){\mathrm{e}}^{-{\lambda }_{n}s}$ where $X_n$ are independent and complex valued variables, $0\le {\lambda }_n\nearrow +\infty$. We prove that under natural conditions, for some random entire functions of order $\left(R\right)$ zero $f(s,\omega )$ almost surely every horizontal line is a Julia line without an exceptional value. The result improve a theorem of J. R. Yu: Julia lines of random Dirichlet series. Bull. Sci. Math. 128 (2004), 341–353, by relaxing condition on the distribution of $X_n$ for such function...