Iterated Brownian motion in an open set

Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If \tau is the first exit time of iterated Brownian motion from the solid, then P(\tau>t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(\tau>t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator {1/8}\Delta^2-\frac{\partial}{\partial t}

Uniqueness for diffusions degenerating at the boundary of a smooth bounded set

For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise uniqueness holds when \gamma and g are Lipschitz and \frac{g(1)}{\gamma^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with \gamma=\sqrt2 and g\equiv c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C^3 boundary and suppose h:\barD\to R Lipschitz on \barD, as well as C^2 on a neighborhood of \partial D with Lipschitz second partials there. Also assume h>0 on D, h=0 on \partial D and |\nabla h|>0 on \partial D. An example of such a function is h(x)=d(x,\partial D). We give conditions which ensure pathwise uniqueness holds for dX_t=h(X_t)^{1/2}\sigma(X_t) dB_t+b(X_t) dt in \barD.Comment: Published at http://dx.doi.org/10.1214/009117904000000810 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Iterated Brownian motion in bounded domains in R^n

Let $\tau_{D}(Z)$ is 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 extension of the result in DeBlassie \cite{deblassie}, for $z\in D$ $$P_{z}[\tau_{D}(Z)>t]\approx t^{1/2} \exp(-{3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}), as t\to\infty .$$ We also study asymptotics of the life time 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.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

Conquest, Consequences, Restoration: The Art of Rebecca Belmore

Rebecca Belmore (Ojibwa/Anishinabe, b. 1960 in Upsala, Ontario), embraces three themes in her oeuvre: conquest, consequences and restoration.Through the mediums of performance art, installation, video and photography, Belmore confronts Indigenous issues regarding land theft, identity, gender, racism, stereotypes,memory, contested histories, and the recovery and reclamation of a decolonized self. All of these themes are sub-categories that fall under the larger theme of the consequences of conquest. The most significant component of Belmores work, however, is restoration, which embraces themes of healing, self-determination and sovereignty. Traditional art-historical methodologies can and have been used to analyze Indigenous art. This thesis proposes that Indigenous art is best examined through Native performance traditions as suggested by Courtney Elkin Mohler\u27s theatre praxis. Mohler argues that the goal of Indigenous performance art can be achieved through (1) exposing popularly accepted racial and ethnic stereotypes as identity constructions; (2) rewriting history in a manner that repositions historically marginalized and objectified cultures as active subjects; (3) utilizing residual creative energies that transcend the normative methods for \u27art making,\u27 thereby exposing an alternative indigenous worldview; and (4) destabilizing historical \u27facts\u27 that constitute an essence of \u27timelessness\u27 and edifice of authority for neocolonial and imperialist practices. These four components are an integral part of Belmore\u27s work. Because Belmore utilizes her own body as the primary medium, she becomes at once the text, the victim, the victor, and catapults the performance into the arena of restoration

The influence of a power law drift on the exit time of Brownian motion from a half-line

AbstractThe addition of a Bessel drift 1x to a Brownian motion affects the lifetime of the process in the interval (0,∞) in a well-understood way. We study the corresponding effect of a power −βxp(β≠0,p>0) of the Bessel drift. The most interesting case occurs when β>0. If p>1 then the effect of the drift is not too great in the sense that the exit time has the same critical value q0 for the existence of qth moments (q>0) as the exit time of Brownian motion. When p<1, the influence is much greater: the exit time has exponential moments