Strong local nondeterminism for fBm

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Let $B^\alpha$ be an $(N,1)$-fractional Brownian motion with index $\alpha\in(0,1).$ Pitt (Local times for Gaussian vector fields, Indiana Univ. Math. J. 1978)

proved that $B^\alpha$ satisfies the following Strong local nondeterminism: there exists a constant $0<c_1<\infty$ such that, for all integers $n\ge 1$ and all $u, t^1,\ldots,t^n\in R^N$,

$$Var(B^\alpha(u)|B^\alpha(t^1),\ldots,B^\alpha(t^n))\ge c_1 \min_{0\le k\le n}|u-t^k|^{2\alpha},$$

where $t^0=0.$

Wu and Xiao (Dimensional properties of fractional Brownian motion, Act Math Sin, 2007) showed that it is equivalent to the following:

There exista a constant $0<c_2<\infty$ such that, for all integers $n\ge 1$ and all $u, v, t^1,\ldots,t^n\in R^N$,

$$Var(B^\alpha(u)-B^\alpha(v)|B^\alpha(t^1),\ldots,B^\alpha(t^n))\ge c_2\min( \min_{0\le k\le n}|u-t^k|^{2\alpha}+\min_{0\le k\le n}|v-t^k|^{2\alpha}, |u-v|^{2\alpha}).$$

Strong local nondeterminism for fBm

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原文地址：http://www.cnblogs.com/jinjun/p/4979338.html