Iterated Brownian motion: Lifetime asymptotics and isoperimetric -type inequalities

In Chapter 1, iterated Brownian motion started at [special characters omitted] is defined by [special characters omitted]where [special characters omitted] is a two-sided Brownian motion and [special characters omitted] and Yt are three independent one-dimensional Brownian motions, all started at 0. In [special characters omitted] one requires X± to be independent n-dimensional Brownian motions. Let τD( Z) be the first exit time of this processes from a domain D ⊂ [special characters omitted], started at z ∈ D. In Chapter 2, we establish sub-exponential decay of large time asymptotics of P z[τD(Z) \u3e t] for several unbounded domains including parabola-shaped domains of the form Pα = {(x, Y) ∈ [special characters omitted]}, for 0 \u3c α \u3c 1, A \u3e 0 and twisted domains in [special characters omitted]2 as defined in [21]. In Chapter 3, we study the lifetime asymptotics of iterated Brownian motion in bounded domains in [special characters omitted] with regular boundary. We find the exact lifetime asymptotics of iterated Brownian motion in bounded domains which was left open in [20]. Moreover, similar results are shown for Brownian-tune Brownian motion which is defined by [special characters omitted] for z ∈ [special characters omitted], where X and Y are independent one-dimensional Brownian motions started at 0. In Chapter 4, we extend generalized isoperimetric-type inequalities to iterated Brownian motion and Brownian-time Brownian motion in several domains including domains with finite volume and convex domains of finite inradius