# zbMATH — the first resource for mathematics

Local times and related properties of multidimensional iterated Brownian motion. (English) Zbl 0914.60063
Let $$\{W(t), t\in R\}$$ and $$\{B(t), t\geq 0\}$$ be two independent Brownian motions in $$R$$ with $$W(0)=B(0)=0$$ and let $$Y(t)=W(B(t))$$ $$(t=0)$$ be the iterated Brownian motion. Define $$d$$-dimensional iterated Brownian motion by $$X(t)=(X_1(t),\dots,X_d(t))$$ where $$X_1,\dots,X_d$$ are independent copies of $$Y$$. The author investigates the existence, joint continuity and Hölder conditions in the set variable of the local time $$L=\{L(x,B):x\in R^d$$, $$B\in{\mathcal B}(R_+)\}$$ of $$X(t)$$, where $$({\mathcal B}_+)$$ is the Borel $$\sigma$$-algebra of $$R_+$$. Finally these results are applied to study the irregularities of the sample paths and the uniform Hausdorff dimension of the image and inverse images of $$X(t)$$.

##### MSC:
 60J65 Brownian motion
Full Text: