Chung's Functional Law of the Iterated Logarithm for Increments of a Brownian Motion with Respect to ${ {(r,p)}}$-capacities on an Abstract Wiener Space
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We prove that there exists a constant $a(A) \in (0, \infty)$ such that $\lim \inf_{t \rightarrow \infty} (\log \log t/t)\sup_{0 \leq s \leq t}|\int^s_0\langle AW_u, dW_u\rangle | = a(A)$ with probability 1, where $A$ is a skew-symmetric $d \times d$ matrix, $A \neq 0$, and $\{W_t\}_{t\geq 0}$ is a $d$-dimensional Wiener process.
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Suppose $X_t$ is a diffusion, reflecting at 0, with speed measure $m(dx)$. We show, under a mild regularity condition on $m$, that $\lim\sup_{t\rightarrow 0} X_t/h^{-1}(t) = c$, a.s., where $c$ is a nonzero constant and $h(t) = tm\lbrack 0, t\rbrack/\log|\log t|$. The analogue to Chung's law of the iterated logarithm is also obtained.
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An iterated process Z= {Z(t)=BH,K(Y(t)),t0}obtained by taking a bifractional Brownian motion{BH,K(t),t∈R}with Hurst index 0α≤2,0H 1,0K≤1and replacing the time parameter with a strictlyα-stable Levy process{Y(t),t≥0}in R independent of{BH,K(t),t∈R}.This paper study the existence and joint continuity of local time of the iterated process X= {X(t),t∈R+}defined by X(t)=(X1(t),…,Xd(t))where t≥0and X1(t),…,Xd(t)are independent copies of Z.
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Let W(t) be a transient Brownian motion. In this paper, we prove that m(T) = inf{|W(t)|; t ≥ T},(T 0) satisfies the law of the iterated logarithm and also discuss the upper and lower class functions.
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Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{\d s \over W(s)} := \lim_{ε\to0} \int_0^t 1_{(|W(s)|> ε)} {\d s \over W(s)} $ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.
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We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure \mu on T, we show that (M^{p, q}_s (T), \mu) and (W^{p, q}_s (T), \mu) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space \ft{b}^s_{p, \infty} (T). Specifically, we prove that the Brownian motion belongs to \ft{b}^s_{p, \infty} (T) for (s-1) p = -1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces B_{p, q}^s, and indicate the endpoint large deviation estimates.
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