No abstract uploaded!

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the rst moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, suf cient conditions for the second moment to be bounded or unbounded are proposed.

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ ,$β$>0, and$λ$∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not$β$-porous in scale$λ$^{$n$}for$n$from a set with positive density amongst natural numbers.

We show that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

We show that there exist $(d-1)$ -Ahlfors regular compact sets $E \subset\mathbb{R}^{d}$ , $d\geq2$ such that for any $t< d-1$ , we have $$\sup_{T} \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^{t}}< \infty $$ where the supremum is over all tubes $T$ with width $w(T) >0$ . This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.