Higher order PDE's and iterated Processes

We introduce a class of stochastic processes based on symmetric $\alpha$-stable processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of a symmetric $\alpha$-stable process. We call them $\alpha$-time processes. They generalize Brownian time processes studied in \cite{allouba1, allouba2, allouba3}, and they introduce new interesting examples. We establish the connection of $\alpha-$time processes to some higher order PDE's for $\alpha$ rational. We also study the exit problem for $\alpha$-time processes as they exit regular domains and connect them to elliptic PDE's. We also obtain the PDE connection of subordinate killed Brownian motion in bounded domains of regular boundary.Comment: 17 page

Lifetime asymptotics of iterated Brownian motion in R^{n}

Let $\tau_{D}(Z)$ be the first exit time of iterated Brownian motion from a domain D \subset \RR{R}^{n} started at $z\in D$ and let $P_{z}[\tau_{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray} \lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}) P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2}$. Here $\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian ${1/2}\Delta$ in $D$, and $\psi$ is the eigenfunction corresponding to $\lambda_{D}$ . We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM), $Z^{1}_{t}=z+X(|Y(t)|)$, where $X_{t}$ and $Y_{t}$ are independent one-dimensional Brownian motions

Isoperimetric-type inequalities for iterated Brownian motion in R^n

We extend generalized isoperimetric-type inequalities to iterated Brownian motion over several domains in \RR{R}^{n}. These kinds of inequalities imply in particular that for domains of finite volume, the exit distribution and moments of the first exit time for iterated Brownian motion are maximized with the ball $D^{*}$ centered at the origin, which has the same volume as $D$Comment: 10 page