MO Yong-xiang, LIU Yong-hong, ZHOU Xia

(School of Mathematics and Computing Science, Guilin University of Electronic Technology;Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation,Guilin 541004, China)

Abstract: In this paper,we obtain the quasi sure local Chung’s functional law of the iterated logarithm for increments of a Brownian motion.As an application, a quasi sure Chung’s type functional modulus of continuity for a Brownian motion is also derived.

Keywords: Brownian motion;increment;local Chung’s law of the iterated logarithm;(r,p)-capacity

1 Introduction and Main Result

Let(B,H,μ)be an abstract Wiener space.The capacity is a set function on B with the property that it sometimes takes positive values even for μ-null sets, while a set of capacity zero has always μ-measure zero.As we know, capacity is much finer than probability.An important difference between the capacity and probability is that the second Borel-Cantelli’s lemma does not hold with respect to capacity Cr,pwhile it holds with respect to probability.Therefore, an interesting problem is to find out what property holds not only almost sure but also quasi sure.In this paper, we discuss this topic.

Many basic properties of Wiener processes were studied by authors (see [1–6]), such as the functional law of iterated logarithm, the functional modulus of continuity and large increments hold not only for μ-a.s.but also for the sense of Cr,p-a.s.

In recent paper[2],Gao and Liu established local functional Chung’s law for increments of Brownian motion.In the present paper, we discuss similar results, but the probability is replaced by (r,p)-capacity.The exact approximation rate for the modulus of continuity of Brownian motion can be viewed as a special case of our results.

We use standard notation and concepts on the abstract Wiener space (B,H,μ), including the H-derivative D, its adjoint D?and the Ornstein-Uhlenbeck operator L = ?D?D.Let Dr,p,r >0,1 p<∞ be Sobolev space of Wiener functionals, i.e.,

where Lpdenotes Lp-space of real-valued functions on (B,μ).For r > 0, p > 1, (r,p)-capacity is defined by Cr,p(O) =μ-a.s.on O}, for open set O ? W, and for any set A ? W, Cr,p(A)=inf{Cr,p(O);A ? O ? W, O is open}.

Let us consider classical Wiener space (W,H,μ) as follows

Let Cddenote the space of continuous functions from[0,1]to Rdendowed with usual supnormK :={f ∈Hd;2I(f) 1}, where

Throughout this paper, let au,bube two non-decreasing continuous functions from (0,1) to(0,e?1) satisfying

(i) aubufor any u ∈ (0,1) and

Let w ∈ W, for u ∈ (0,1), 0 t 1, ?(t,u) denotes the following path

The following theorems are results of this paper.

Theorem 1.1If conditions (i) and (ii) hold, then for any f ∈K with 2I(f) < 1, we have

Theorem 1.2If conditions (i), (ii) and (iii) hold, then for any f ∈K with 2I(f)<1,we have

Corollary 1.1Let Mt,h(x)=For any f ∈ K with 2I(f)<1, we have

2 Proof of Theorem 1.1

Proof of Theorem 1.1 is completed by the following lemmas.

Lemma 2.1(see [3], Lemma 2.2) Let 1k ∈ Z, q1,q2,∈ (1,∞) be given so thatFor any f ∈K, put

Lemma 2.2(see [3], Lemma 2.4) There exists a positive number cdsuch that for any h>0, τ >0, f ∈ K, we have

Lemma 2.3For any f ∈K with 2I(f)<1, we have

Proof Case Ithen bu→ 0 as u → 0 and there exists a 0

For any 0 < ε < 1, choose δ > 0 such that η = ?2δ+2I(f)+By Lemma 2.2,for n large enough, we have

by Borel-Cantelli’s lemma,

On the other hand, for any δ0>0,

by Borel-Cantelli’s lemma

By (2.1)–(2.3), we get

Remark that un→ 0, so for any small enough u, there is a unique n such that u ∈(un+1,un]. Let φt,u(s) = βu(w(t + aus) ? w(t)),s ∈ [0,1],t ∈ [0,bu? au].We define X(u) =By the definition of infimum,for any ε>0, there exists Tn∈ (un+1,un]such that XnX(Tn)? ε.

Noting that

by (2.4),(2.5), (2.7)–(2.9), we get

Since

which ends the proof.

Case IIthen we can choose a nonincreasing sequence{un;n ≥1}withthenand h(n)→ ∞ as n → ∞.Let l(u),knand ti,i=1,2,...,knbe defined by Case I.Then for some constant C >0, if d is chosen in a suitable way, then

which implies that

Similarly to the proof of Case I, the proof of Lemma 2.3 is completed.

Lemma 2.4For any f ∈K with 2I(f)<1, we have

ProofSetIf ρ <1 and bu→ b0 as u → 0, thenIn this case, see Lemma 3.2.Therefore, we only consider the following two cases

(I) ρ <1 and bu→ 0 as u → 0,

(II) ρ=1.

Case Iρ <1 and bu→ 0 as u → 0 If ρ <1 and bu→ 0 as u → 0, then we can choose{uk,k ≥ 1} such that buk+1= buk? auk, k ≥ 1. For any ε > 0, choose δ > 0 such thatSet k =[r]+1, by Lemma 2.1, we have

moreover, by small deviation,

Noting that, there exists A=A(m0)>0 such that

We discuss as follows

we get

When l is large enough, for constantwe can also prove thatThus we have

where c0is a constant.We get

consequently

we have

Case IISet ρ = 1.By applying Lemma 2.1, similarly to the corresponding that of(3.3) in [2].

3 Proof of Theorem 1.2

Proof of Theorem 1.2 is completed by the following lemmas.

Lemma 3.1If condition(iii)also holds,then there exists an non-increasingfor any f ∈K with 2I(f)<1, we have

ProofOwing tothere exists a subsequence {un;n ≥ 1} such that

where c0is a constant.If d is chosen in a suitable way, then

by Borel-Cantelli’s lemma

Lemma 3.2If conditions (i), (ii) and (iii) hold, then for any f ∈K with 2I(f) < 1,we have

ProofLet φt,u(s) = βu(w(t+aus)? w(t)), unis defined as in Lemma 3.1.Sincewe have

Moreover

We can conclude Lemma 3.2 from (3.1)–(3.3) and Lemma 3.1.