Small Deviations for Time-Changed Brownian Motions and Applications to Second-Order Chaos

We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.


Introduction
In this paper, we study small deviations for some time-changed Brownian motions, for the purpose of applications to certain elements of Wiener chaos. Large deviation estimates for Wiener chaos are well-studied (see for example [14]), largely due to the work of Borell (see for example [5] and [6]). However, small deviations in this setting are much less understood and are of interest for their myriad interactions with other concentration, comparison, and correlation inequalities as well as various limit laws for stochastic processes; see for example the surveys [16] and [18]. The present work gives strong small deviations results for certain elements of second-order homogeneous chaos. In particular, let (W, H, µ) be an abstract Wiener space, {W t } t≥0 denote Brownian motion on W, and ω be a continuous bilinear antisymmetric map on W. We will study processes {Z(t)} t≥0 of the form (A precise definition is given in Section 3.) In particular, we show that Z is equal in distribution to a Brownian motion running on an independent random clock for which strong small deviation probabilities are known, and thus the strong small deviations behavior of Z follows. From these results one may infer, for example, a functional law of iterated logarithm and hence a Chung-type law of iterated logarithm for Z. To the authors' knowledge, these are the first results for small deviations of elements of Wiener chaos in the infinite-dimensional context beyond the first-order Gaussian case.
1.1. Statement of main results. We first discuss the general small deviations result for time-changed Brownian motion we will be using. We will assume that the random clocks satisfy the following.
By the exponential Tauberian theorem (see Theorem 2.1), equation (1.2) is equivalent to Also via the exponential Tauberian theorem, equation (1.2) clearly holds when C is a subordinator satisfying lim ε↓0 ε α | log ε| β log P(C(t) ≤ ε) = −K(t) for any t > 0, or if C has independent increments which satisfy lim ε↓0 ε α | log ε| β log P(C(t) − C(s) ≤ ε) = −K(s, t) for all 0 ≤ s < t. However, it is not necessary for C to have independent or stationary increments for Assumption 1.1 to hold. One important source of examples for the present paper is the following theorem from [15] for weighted L p norms of a Brownian motion.
For example, ifρ is any non-negative continuous function on [0, ∞) and A particularly relevant example to our later applications is the simplest case where p = 2 andρ ≡ 1, for which Suppose that {Z(t)} t≥0 is a stochastic process given by Z(t) = B(C(t)), where C is as in Assumption 1.1 and B is a standard real-valued Brownian motion independent of C. Let M (t) := sup 0≤s≤t |Z(s)|. Then, for any m ∈ N, .
Such estimates have been previously studied for processes {Z t } t≥0 that are symmetric α-stable processes [8], fractional Brownian motions [12], certain stochastic integrals [13], m-fold integrated Brownian motions [27], and integrated α-stable processes [28]. In particular, the stochastic integrals studied in [13] are essentially finite-dimensional versions of the class of stochastic integral processes we study, and the proof that we give for Theorem 1.3 follows the outline of the proof of small ball estimates in that reference.
We apply Theorem 1.3 to stochastic integrals of the form (1.1) as follows.
where {e k } ∞ k=1 is any orthonormal basis of H contained in H * := {h ∈ H : h, · extends to a continuous linear functional on W} and {W k } ∞ k=1 are independent standard Brownian motions which are also independent of B. If we further suppose that ω(e k , ·) H * = O(k r ) for r > 1 or ω(e k , ·) H * = O(σ k ) for σ ∈ (0, 1), then, for any m ∈ N, 0 = t 0 < t 1 < · · · < t m , and where Applications of such estimates include using the small deviations in Theorem 1.4 to prove a Chung-type law of iterated logarithm as well as a functional law of iterated logarithm for the process Z. We record these results in Theorem 3.18 and 3.19.
1.2. Discussion. First-order small deviation estimates of the standard form log P sup 0≤s≤t |Z(s)| ≤ ε were studied in [23] for processes Z(t) = t 0 ω(W s , dW s ) with W an n-dimensional Brownian motion and ω : R n → R given by ω(x, y) = Ax · y for A a skew-symmetric n × n matrix. These estimates were then applied to prove an analogue of the classical limit result of Chung. (This was done earlier in [26] in the case n = 2 and , that is, for Z the stochastic Lévy area.) In [13], the authors improved these results by proving stronger asymptotic results like those in Theorem 1.3 for the same Z as in [23] and applying these results to prove a functional law of iterated logarithm. In the present paper, the proof of the small ball estimates established in Theorem 1.3 is a direct generalization of the techniques of [13]. However, Theorem 1.3 is sufficiently general to be of independent interest for other potential applications. Thus for that purpose, as well as for clarity and self-containment, we include the proof here. It is also clear from the proofs that, given only the weak asymptotic order for C, one could infer the weak asymptotic order for Z instead.
We also mention the reference [2], in which the authors study general iterated processes of the form X • Y where X is a two-sided self-similar process and Y is a continuous process independent of X. Since X is two-sided, it is not required that Y satisfy any monotonicity or positivity criteria. In this general setting, under the assumption that the strong first-order asymptotics are known for X and Y , the authors are able to prove a strong first-order small ball estimate (Theorem 4 of [2]). Theorem 1.3 is stated in the restricted setting that X is a Brownian motion; however, the the proof carries through for first-order estimates (m = 1) for general processes X satisfying approximately very general assumptions as in [2]. See Proposition 2.9 for more details.
The organization of the paper is as follows. In Section 2 we give the proof of Theorem 1.3. In Section 3, we apply Theorem 1.3 to prove small ball estimates for the relevant collection of stochastic integrals. In Section 3, we define precisely the processes of interest, and in Theorem 3.10 we prove that these processes have a representation as Brownian motions on an independent random clock. In Section 3.2, we determine the small ball asymptotics of the clock. Thus we are able to apply Theorem 1.3, and we additionally record a Chung-type law of iterated logarithm and functional law of iterated logarithm that follow from these estimates.
Acknowledgement. This paper is dedicated to the memory of Wenbo Li, who suggested the problems addressed in Section 3 of this paper, thus motivating the whole of this work.

Small ball estimates
In this section, we prove separately the upper and lower bounds of Theorem 1.3. The outline of the proof here follows Section 4 of [13]. First, we record a standard relation between asymptotics of the Laplace transform and small ball estimate of a positive random variable in the form of the exponential Tauberian theorem (see for example [3]). We give a special case of that theorem here.
Theorem 2.1. Suppose that X is a positive random variable. There exist α > 0, β ∈ R, and K ∈ (0, ∞) such that We will use this theorem repeatedly in the sequel along with the standard fact that, for any ε > 0, see for example [9]. Now the upper bound of Theorem 1.3 follows almost immediately from this and the upper bound for the random clock C via conditioning.

Proposition 2.3. Under the hypotheses of Theorem 1.3, we have that
Proof. We will show that Then applying equation (1.3) with d i = 1/b 2 i finishes the proof. So first we define Then we have that Since {Z(t)} t≥0 is a P C Gaussian centered process, we have by Anderson's inequality (see, for example Theorem 1.8.5 of [4]) that by the continuity of C and the stationary and scaling properties of Brownian motion. Thus By iterating the above computation m times we see that where the second inequality follows from the upper bound in (2.2). Taking the expectation of both sides yields (2.3).
We now move towards obtaining the lower bounds with the following lemma.
be given by Proof. Define Since sup By P C independent increments, and repeating this computation m times gives that Again we use the stationary and scaling properties of Brownian motion, as well as Sidak's Lemma (see for example, Corollary 4.6.2 of [4]), to show that Thus, taking expectations we have that Now the next three lemmas give the necessary estimates on the terms appearing in Lemma 2.4.
Proof. The lower bound in (2.2) implies that .
Given this strict inequality, Lemmas 2.5 and 2.7 imply that, for each Since δ > 0 was arbitrary, allowing δ ↓ 0 completes the proof.
As alluded to in the discussion from Section 1, a brief review of the proof shows that conditioning easily determines the strong first-order asymptotics of Z = X • C for general processes X satisfying their own small ball estimates. The following statement could also be inferred from the proofs of [2].
Proof. Under the assumptions on X, for any δ > 0, there exists ε 0 = ε 0 (δ) such that for any ε ∈ (0, ε 0 ) Thus, there exist c 1 , c 2 ∈ (0, ∞) depending only on ε 0 so that, for all ε > 0, Then continuity of C implies that Taking expectations and applying the asymptotics of C(t) gives Letting δ ↓ 0 proves the upper bound. The lower bound follows in a similar manner.
Remark 2.10. For example, if X is H-self-similar and there exists κ > 0 such that then (2.5) holds with ρ = θH. Note that this result can be more general than that for two-sided diffusions as in [2]. There it is necessary to require that θH = 1 (which is often satisfied with the supremum norm). There are basic processes in this setting for which this does not hold. For example, the process C defined in (1.4) is 2-selfsimilar, but by Theorem 1.2 satisfies a small ball estimate with α = 1. (And more generally, forρ ≡ 1 and general p ∈ [1, ∞), α = 2/p and H = (p + 2)/2.) Remark 2.11. Note also that we could have again allowed a slowly varying factor in the asymptotics of C, but we have omitted it for ease.

Remark 2.12.
It is in the iterative arguments for Theorem 1.3 that one uses, for example, the Gaussian properties of Brownian motion. It is clear that some of these estimates may be extended to other more general processes. For example, there is a known analogue of the Anderson inequality that holds for symmetric αstable processes (see for example Lemma 2.1 of [8]) that one could use to extend the proof of Proposition 2.3.

Applications to second order chaos
Here we apply the results of the previous section to prove small deviations estimates for stochastic integrals of the form where W is an infinite-dimensional Brownian motion and ω is an anti-symmetric continuous bilinear form. Small deviations have been studied for analogous integrals of finite-dimensional Brownian motions in [23] and [13].
First we define the integral processes we study. We will then prove that these processes are equal in distribution to a Brownian motion under an independent time-change, and we establish a small ball estimate for the relevant random clock. Then by applying the results of Section 2, we are able to prove small deviations result for Z. We fix the following notation for the sequel. for all s, t ≥ 0 and h, k ∈ H * . Let ω : W × W → R be a anti-symmetric continuous bilinear map.

Remark 3.2.
It is standard that continuity for a bilinear map ω on W × W implies that the restriction of ω to H × H is Hilbert-Schmidt, that is, is any orthonormal basis of H; see for example Proposition 3.14 of [10].
Recall that associated to any abstract Wiener space is a class of canonical projections. Suppose that P : H → H is a finite-rank orthogonal projection such that P H ⊂ H * . Let {e j } n j=1 be an orthonormal basis for P H. Then we may extend P to a (unique) continuous operator from W → H (still denoted by P ) by letting It is well-known that H * contains an orthonormal basis of H. Thus, we may always take a sequence P n ∈ Proj(W) so that P n | H ↑ I H .
Proposition 3.4. For P ∈ Proj(W) as in Notation 3.3, let {Z P t } t≥0 denote the continuous L 2 -martingale defined by In particular, if {P n } ∞ n=1 ⊂ Proj(W) is an increasing sequence of projections and Z n t := Z Pn t , then there exists an L 2 -martingale {Z t } t≥0 such that, for all p ∈ [1, ∞) and T > 0, and {Z t } t≥0 is independent of the sequence of projections. Thus, we will denote the limiting process by The quadratic variation of Z is given by Proof. First note that, for P as in (3.1), Let P, P ′ ∈ Proj(W), and let {h j } N j=1 be an orthonormal basis for P H + P ′ H. We then have that |ω((P − P ′ )e k , P e j )| 2 + |ω(P ′ e k , (P − P ′ )e j )| 2 .

(3.4)
Taking P = P n and P ′ = P m for m ≤ n gives Since the space of continuous L 2 -martingales on [0, T ] is complete in the norm N → E|N T | 2 , and, by Doob's maximal inequality, there exists c < ∞ such that it follows that there exists an L 2 -martingale {Z t } t≥0 such that (3.2) holds with p = 2. For p > 2, since Z is a chaos expansion of order 2, a theorem of Nelson (see Lemma 2 of [22] and pp. 216-217 of [21]) implies that, for each j ∈ N, there exists c j < ∞ such that , and again this combined with Doob's maximal inequality is sufficient to prove (3.2).
One may similarly use (3.4) to show that, for {e ′ j } ∞ j=1 ⊂ H * another orthonormal basis of H and P ′ n a corresponding sequence of orthogonal projections, that and thus Z is independent of choice of basis.
Since the quadratic variation of Z n is given by More general integrals of the form above are studied in [10] in the context of Brownian motions on infinite-dimensional Lie groups, and the above proposition should be compared with Proposition 4.1 of that reference.
We give the following basic example of the type of process Z we study here.
where w j = x j + iy j for each j. Then for a Brownian motion j=1 are independent standard real-valued Brownian motions, we have that is an infinite weighted sum of independent Lévy areas.
Remark 3.6. Since Z is a martingale with we know there exists a (not necessarily independent) real-valued Brownian motion B such that Z(t) = B( Z t ) by the Dubins-Schwarz representation. We will show in the next section that this representation in fact holds with B an independent Brownian motion.

A representation theorem.
In this section, we show that Z d = B( Z ) for an independent Brownian motion B. This representation is well-known for Z the standard stochastic Lévy area for two-dimensional Brownian motion (see for example Example 6.1 on page 470 of [11]), and was also proved for more general stochastic integrals of finite-dimensional Brownian motions in [13]. We summarize the latter result now; see Section 3 of [13] for a proof.
Lemma 3.7. Let W be a standard Brownian motion in R n and A be a real non-zero skew-symmetric n × n matrix with non-zero eigenvalues {±ia j } r j=1 (where 2r ≤ n). For t > 0, let where B and {X j , Y j } r j=1 are independent standard real-valued Brownian motions. Then the law of {L(t)} t≥0 is equivalent to the law of {L(t)} t≥0 .
Remark 3.8. In particular, this lemma implies that each of the finite-dimensional approximations Z n to Z has such a representation, in the following way. By Remark 3.2, the continuity assumption for ω implies that its restriction to the Cameron-Martin space is Hilbert-Schmidt, and thus the Riesz representation theorem implies the existence of an anti-symmetric Hilbert-Schmidt operator Q = Q ω : H → H such that ω(h, k) = Qh, k H , for all h, k ∈ H. Thus, and we may apply Lemma 3.7 to Z P , as P B is a Brownian motion on the finitedimensional space P H ⊂ H and A = P QP is a skew-symmetric linear operator on P H.
We will use this representation for the finite-dimensional approximations to show that an analogous statement is true for Z. First we record the following simple lemma.
Lemma 3.9. Let Q : H → H be a Hilbert-Schmidt operator, and let P n be an increasing sequence of orthogonal projections on H. Then, as n → ∞, P n QP n → Q in Hilbert-Schmidt norm.
is an orthonormal basis of P n H. We have The second term goes to zero since it is the tail of the convergent sum For the first term, we may use the dominated convergence theorem: since P n → I strongly we have (P n − I)Qe Theorem 3.10. Let Z(t) = t 0 ω(W s , dW s ) be as defined in Proposition 3.4, and let Q = Q ω be the linear operator on H such that ω(h, k) = Qh, k H for all h, k ∈ H as in Remark 3.8. Let {X j , Y j } ∞ j=1 be independent standard real-valued Brownian motions, {±iq j } ∞ j=1 be the eigenvalues of Q so that {q j } ∞ j=1 is ordered from largest to smallest, and define for t ≥ 0 (Note that C(t) is well-defined and finite almost surely for each t.) Then the law of {Z(t)} t≥0 is equivalent to the law of {Z(t)} t≥0 whereZ(t) = B(C(t)) for B a standard Brownian motion independent of {X j , Y j } ∞ j=1 .
Proof. Let {P n } ∞ n=1 ⊂ Proj(W ) be such that P n | H ↑ I H and Z n (t) = t 0 ω(P n W s , dP n W s ) = t 0 (P n QP n )P n W s , dP n W s H , as in Proposition 3.4, Then Lemma 3.7 implies that, for each n, the law of {Z n (t)} t≥0 is equal to the law of {Z n (t)} t≥0 wherẽ where {±iq nj } rn j=1 are the non-zero eigenvalues of P n QP n . For each n, we will assume the q nj are ordered in j from largest to smallest. Clearly, Proposition 3.4 and in particular (3.2) imply that Z n ⇒ Z and the collection {Z n } ∞ n=0 is tight. Equivalence in distribution then implies thatZ n ⇒ Z and {Z n } n≥0 is tight. Now we also have that, for each fixed t > 0, Note that Thus, for the second term, clearly, since this is the tail of a convergent sequence. For the first term, where the equality follows from the min-max theorem which implies that Qh h ≤ q j and the limit follows from Lemma 3.9 which implies that the Hilbert-Schmidt norms of Q n converge to the Hilbert-Schmidt norm of Q. Thus, for any (t 1 , . . . , t m ) ∈ (R + ) m , as n → ∞, and the finite-dimensional distributions ofZ n converge to those ofZ. Combining this with the tightness of {Z n } implies thatZ n ⇒Z. However, sincẽ Z n ⇒ Z also, it must be that {Z(t)} t≥0 and {Z(t)} t≥0 are equal in distribution.
Noting that, for {e k } ∞ k=1 an orthonormal basis of H, we see that indeed C(t) = Z t up to a reordering of terms. Given this last theorem, in order to prove small deviations for Z, it suffices to prove them forZ. The results of Section 2 lead us to find a small ball estimate for the process Z .

3.2.
Small deviations for Z t and applications. Note that we may write Recall that, if {ζ j } m j=1 are independent positive random variables satisfying small ball estimates with the same exponents α and β for coefficients {K j } m j=1 , then .
In particular, if {η j } m j=1 are positive i.i.d. random variables satisfying small ball estimates with K j = K for each j and ζ j = a j η j for some a j > 0, then we have that Equivalently, In the event this sum is actually infinite with a summable sequence of coefficients {a j } ∞ j=1 , analogous results hold under additional requirements on the coefficients. Small deviations of random variables of the form where {a j } ∈ ℓ 1 (R + ) and {ζ j } are non-negative i.i.d random variables, have been studied in [1,7,20,25,24,17]. In particular, we present the following two theorems without proof. The following is Theorem 3.1 of [7].
Theorem 3.11. Suppose that ζ is a non-negative random variable such that there exist α > 0 and a slowly varying function L so that as ε ↓ 0, and there exist κ, δ > 0 so that as ε ↓ 0.
The next theorem is Theorem 8 of [1] which gives small deviations estimates when {a j } is geometric. Theorem 3.12. Suppose that ζ is a non-negative random variable such that there exist α > 0 and K > 0 so that lim ε↓0 ε α log P(ζ ≤ ε) = −K.
Remark 3.14. When {a j } ∈ ℓ 1 (R + ) are such that a j ≤ã j for {ã j } a sequence satisfying the hypotheses of Theorems 3.11 or 3.12, then we may easily obtain a lower bound in terms of {ã j }. Since a j ≤ã j , S := ω(e k , ·) H * < ∞.
Then C = Z satisfies Assumption 1.1 with α = 1, β = 0, and That is, for any m ∈ N, 0 = t 0 < t 1 < · · · < t m , and {d i } m i=1 a decreasing sequence, Proof. We have that m i=1 d i ∆ i C = ∞ k=1 ω(e k , ·) 2 H * ξ k (t) where Thus, under the assumptions on ω, the desired result follows from Theorem 3.11 or 3.12.
Theorem 3.16. For any m ∈ N, 0 = t 0 < t 1 < · · · < t m , and 0 ≤ a 1 < b 1 ≤ a 2 < b 2 ≤ · · · ≤ a m < b m , Remark 3.17. Note that the weak asymptotics determined by Remarks 3.13 and 3.14 are sufficient to show that Theorem 1.3 has the correct order of asymptotics for general trace class ω.
As was done in [13], Theorem 3.16 may be used to prove a functional law of the iterated logarithm for Z. This immediately implies a Chung-like law of the iterated logarithm, or one may prove this directly from the first-order small deviations estimates proved in Theorem 3.16 as was done in [23]. The proofs follow exactly analogously to the finite-dimensional cases in [23] and [13], so we omit the proofs here.  From here it is possible to obtain various occupation measure results for the maximal process of Z, as was done in [8], [12], and [13].
Remark 3.20. Note that Theorems 3.16, 3.18, and 3.19 also include the finitedimensional stochastic integrals already studied in [23] and [13]. The differences in factors of 2 arises from the fact that the non-zero singular values of Q necessarily have multiplicity which is a factor of 2 and the sum in ω 1 counts all of these.