We prove a large deviation type estimate for the asymptotic behavior of a weighted local time of εW as ε→0.

Local timeexponential momentlarge deviations principle60J5560F1060H10Introduction and the main result

Let {Wt,t≥0} be a real-valued Wiener process, and μ be a σ-finite measure on R such that
supx∈Rμ([x−1,x+1])<∞.
Recall that the local timeLtμ(W)of the process W with the weight μ can be defined as the limit of the integral functionals
Ltμn(W):=∫0tkn(Ws)ds,kn(x):=μn(dx)dx,n≥1,
where μn, n≥1, is a sequence of absolutely continuous measures such that
∫Rf(x)μn(dx)→∫Rf(x)μ(dx)
for all continuous f with compact support, and (1) holds for μn, n≥1, uniformly. The limit Ltμ(W) exists in the mean square sense due to the general results from the theory of W-functionals; see [3], Chapter 6. This definition also applies to εW instead of W for any positive ε. In what follows, we will treat εW as a Markov process whose initial value may vary, and with a slight abuse of notation, we denote by Px the law of εW with εW0=x and by Ex the expectation w.r.t. this law.

In this note, we study the asymptotic behavior as ε→0 of the exponential moments of the family of weighted local times Ltμ(εW). Namely, we prove the following theorem.

We note that in this statement the measure μ can be changed to a signed measure; in this case, in the right-hand side, only the atoms of the positive part of μ should appear. We also note that, in the σ-finite case, the uniform statement (3) may fail; one example of such a type is given in Section 3.

Let us briefly discuss the problem that was our initial motivation for the study of such exponential moments. Consider the one-dimensional SDE
dXtε=a(Xtε)dt+εσ(Xtε)dWt
with discontinuous coefficients a,σ. In [7], a Wentzel–Freidlin-type large deviation principle (LDP) was established in the case a≡0 under mild assumptions on the diffusion coefficient σ. In [8], this result was extended to the particular class of SDEs such that the function a/σ2 has a bounded derivative. This limitation had appeared because of formula (7) in [8] for the rate transform of the family Xε. This formula contains an integral functional with kernel (a/σ2)′ of a certain diffusion process obtained from εW by the time change procedure. If a/σ2 is not smooth but is a function of a bounded variation, this integral function still can be interpreted as a weighted local time with weight μ=(a/σ2)′. Thus, Theorem 1 can be used in order to study the LDP for the SDE (5) with discontinuous coefficients. One of such particular results can be derived immediately. Namely, if μ is a continuous measure, then by Theorem 1 the exponential moments of Ltμ(εW) are negligible at the logarithmic scale with rate function ε2. This, after simple rearrangements, allows us to neglect the corresponding term in (7) of [8] and to obtain the statement of Theorem 2.1 of [8] under the weaker condition that a/σ2 is a continuous function of bounded variation. The problem how to describe in a more general situation the influence of the jumps of a/σ2 on the LDP for the solution to (5) still remains open and is the subject of our ongoing research. We just remark that due to Theorem 1 the respective integral term is no longer negligible, which well corresponds to the LDP results for piecewise smooth coefficients a,σ obtained in [1, 2, 6].

Proof of Theorem <xref rid="j_vmsta49_stat_001">1</xref>Preliminaries

For a measure ν satisfying (1), denote by
ftν,ε(x)=ExLtν(εW)=∫0t∫R12πsε2e−(y−x)22sε2ν(dy)ds,t≥0,x∈R,
the characteristic of the local time Lν(εW) considered as a W-functional of εW; see [3], Chapter 6.

The following statement is a version of Khas’minskii’s lemma; see [9], Section 1.2.

Using the Markov property, as a simple corollary, we obtain, for arbitrary t>00$]]>,
supx∈RExeLtν(εW)≤21+t/s=2e(log2)(t/s),
where s>00$]]> is such that (7) holds. This inequality, combined with (6), leads to the following estimate.

For a nonzero measure ν satisfying (1), denoteN(ν,γ)=supx∈Rν([x−γ,x+γ]),γ>0.0.\]]]>For anyλ≥1andγ>00$]]>, there existsελ,γ>00$]]>such thatsupx∈RExeλLtν(εW)≤2e(4log2)c0N(ν,γ)2tλ2ε−2,ε∈(0,ελ,γ),withc0=2π(1+2∑k=1∞e−(2k−1)22)2.

If εs≤γ, then we have
fsν,ε(x)=∑k∈Z∫0s∫|y−x−2kγ|≤γ12πvε2e−(y−x)22vε2ν(dy)dv≤c0N(ν,γ)sε2.
Take
s=(2N(ν,γ))−2(c0)−1λ−2ε2.
Then the inequality εs≤γ holds, provided that
ε≤(γ(2N(ν,γ))2c0λ2)1/3=:ελ,γ.
Under this condition,
fsλν,ε(x)=λfsν,ε(x)≤12.
Now the required inequality follows immediately from (8). □

In what follows, we will repeatedly decompose μ into sums of two components and analyze separately the exponential moments of the local times that correspond to these components. We will combine these estimates and obtain an estimate for Ltμ(εW) itself using the following simple inequality. Let μ=ν+κ and p,q>11$]]> be such that 1/p+1/q=1. Then
Ltμ(εW)=Ltν(εW)+Ltκ(εW)=(1/p)Ltpν(εW)+(1/q)Ltqκ(εW),
and therefore by the Hölder inequality we get
EeLtμ(εW)≤(EeLtpν(εW))1/p(EeLtqκ(εW))1/q.
We will also use another version of this upper bound, which has the form
EeLtμ(εW)1A≤(EeLtpμ(εW))1/p(P(A))1/q,A∈F.

We denote
Δ=supx∈Rμ({x}).
We will prove Theorem 1 in several steps, in each of them extending the class of measures μ for which the required statement holds.

Step I: <italic>μ</italic> is a finite mixture of <italic>δ</italic>-measures

If μ=aδz is a weighted δ-measure at the point z, then we have
Ltμ(εW)=aε−1Lt(z)(W),
where
Lt(z)(W)=limη→012η∫0t1|Ws−z|≤ηds
is the local time of a Wiener process at the point z. The distribution of Lt(z)(W) is well known; see, e.g., [5], Chapter 2.2 and expression (6) in Chapter 2.1. Hence, the required statement in the particular case μ=aδz is straightforward, and we have the following:
limε→0ε2supxlogExeaε−1Lt(z)(W)=ta22.
Note that in this formula the supremum is attained at the point x=z.

In this section, we will extend this result to the case where μ is a finite mixture of δ-measures, that is,
μ=∑j=1kajδzj.
Let j∗ be the number of the maximal value in {aj}, that is, Δ=aj∗. Then Ltμ(εW)≥Δε−1Lt(zj∗)(W), and it follows directly from (12) that
lim infε→0ε2supx∈RlogExeLtμ(εW)≥tΔ22.
In what follows, we prove the corresponding upper bound
lim supε→0ε2supx∈RlogExeLtμ(εW)≤tΔ22,
which, combined with this lower bound, proves (3).

Observe that, for γ>00$]]> small enough,
N(μ,γ)=Δ.
Then by Lemma 2, for any λ≥1,
lim supε→0ε2supx∈RlogExeλLtμ(εW)≤c1λ2tΔ2
with
c1=(4log2)c0=8log2π(1+2∑k=1∞e−(2k−1)22)2.
In particular, taking λ=1, we obtain an upper bound of the form (14), but with a worse constant c1 instead of required 1/2. We will improve this bound by using the large deviations estimates for εW, the Markov property, and the “individual” identities (12).

Denote μj=ajδzj,j=1,…,k. Then
Ltμ(εW)=∑j=1kLtμj(εW).
Fix some family of neighborhoods Oj of zj,j=1,…,k, such that the minimal distance between them equals ρ>00$]]>, and denote
Oj=R∖⋃i≠jOi.
For some N≥1 whose particular value will be specified later, consider the partition tn=t(n/N), n=0,…,N, of the segment [0,t] and denote
Bn,j={f∈C(0,t):fs∈Oj,s∈[tn−1,tn]},j∈{1,…,k},n∈{1,…,N},Cj1,…,jN=⋂n=1NBn,jn,j1,…,jN∈{1,…,k}.
Observe that if the process εW does not visit Oj on the time segment [u,v], then Lμj(εW) on this segment stays constant. This means that, on the set {εW∈Cj1,…,jN}, we have
Ltμ(εW)=∑n=1N(Ltnμjn(εW)−Ltn−1μjn(εW)).
Because Lμj(εW) is a time-homogeneous additive functional of the Markov process εW, we have
Ex[eLtnμjn(εW)−Ltn−1μjn(εW)|Ftn−1]=EyeLt/Nμjn(εW)|y=εWtn−1.
Then by (12), for any j1,…,jN∈{1,…,k},
lim supε→0ε2supx∈RlogExeLtμ(εW)1εW∈Cj1,…,jN≤t2N∑n=1N(ajn)2≤tΔ22.
Because we have a fixed number of sets Cj1,…,jN, this immediately yields
lim supε→0ε2supx∈RlogExeLtμ(εW)1εW∈C≤tΔ22
with
C=⋃j1,…,jN∈{1,…,k}Cj1,…,jN.
Hence, to get the required upper bound (14), it suffices to prove an analogue of (16) with the set C replaced by its complement D=C(0,t)∖C. Using (11) with p=2, A={εW∈D}, and (15) with λ=2, we get
lim supε→0ε2supx∈RlogExeLtμ(εW)1εW∈D≤2c1tΔ2+12lim supε→0ε2supx∈RlogPx(εW∈D).

By the LDP for the Wiener process ([4], Chapter 3, §2),
lim supε→0ε2supx∈RlogPx(εW∈D)=−inff∈closure(D)I(f),
where
I(f)=(1/2)∫0t(fs′)2ds,fis absolutely continuous on[0,t];+∞otherwise.
For any trajectory f∈D, there exists n such that f visits at least two sets Oj on the time segment [tn−1,tn]. Therefore, any trajectory f∈closure(D) exhibits an oscillation ≥ρ on this time segment. On the other hand, for an absolutely continuous f,
|fu−fv|=|∫uvfs′ds|≤|u−v|1/2(∫0t(fs′)2ds)1/2.
This means that, for any f∈closure(D),
I(f)≥ρ2N2t,
which yields
lim supε→0ε2supx∈RlogExeLtμ(εW)1εW∈D≤2c1tΔ2−ρ2N2t.
If in this construction, N was chosen such that
N≥(4c1−1)ρ−2t2Δ2,
then the latter inequality guarantees the analogue of (16) with D instead of C. This completes the proof of (14).

Step II: <italic>μ</italic> is finite

Exactly the same argument as that used in Section 2.2 provides the lower bound (13). In this section, we prove the upper bound (14) for a finite measure μ and thus complete the proof of the first assertion of the theorem. For finite μ and any χ>00$]]>, we can find γ>00$]]> and decompose μ=μ0+ν in such a way that μ0 is a finite mixture of δ-measures and N(ν,γ)<χ. Let p,q>11$]]> be such that 1/p+1/q=1. The measure pμ0 has the maximal weight of an atom equal to pΔ. Since we have already proved the required statement for finite mixtures of δ-measures, we have
lim supε→0ε2supx∈Rlog(ExeLtpμ0(εW))1/p≤t2pΔ2.
On the other hand, we have N(ν,γ)<χ and then by Lemma 2lim supε→0ε2supx∈Rlog(ExeLtqν(εW))1/q≤c1qtχ2.
Hence, by (10),
lim supε→0ε2supx∈RlogExeLtμ(εW)≤t2pΔ2+c1qtχ2.
Now we can finalize the argument. Fix Δ1>Δ:=maxx∈Rμ({x})2\varDelta :=\max _{x\in \mathbb{R}}\mu {(\{x\})}^{2}$]]> and choose p,q>11$]]> such that 1/p+1/q=1 and pΔ2<Δ12. Then there exists χ>00$]]> small enough such that
pΔ2+2c1qtχ2<Δ12.
Taking the decomposition μ=μ0+ν that corresponds to this value of χ and applying the previous calculations, we obtain an analogue of the upper bound (14) with Δ replaced by Δ1. Since Δ1>Δ\varDelta $]]> is arbitrary, the same inequality holds for Δ.

Step III: <italic>μ</italic> is <italic>σ</italic>-finite

In this section, we prove the second assertion of the theorem. As before, the lower bound can be obtained directly from the case μ=aδz, and hence we concentrate ourselves on the proof of the upper bound
lim supε→0ε2logExeLtμ(εW)≤tΔ22,x∈R.
We will use an argument similar to that from the previous section and decompose μ into a sum μ=μ0+ν with finite μ0 and ν, which is negligible in a sense. However, such a decomposition relies on the initial value x, and this is the reason why we obtain an individual upper bound (18) instead of the uniform one (14).

Namely, for a given x, we define μ0,ν by restricting μ to [x−R,x+R] and its complement, respectively. Without loss of generality, we assume that for each R, the corresponding ν is nonzero. Since we have already proved the required statement for finite measures, we get (17).

Next, denote M=supx∈Rμ([x−1,x+1]) and observe that N(ν,1)≤M. Then by Lemma 2 with γ=1 and the strong Markov property, for any stopping time τ, the exponential moment of Ltqν(εW) conditioned by Fτ is dominated by 2ec1M2tq2ε−2. This holds for ε≤εq,1x,R, where we put the indices x,R in order to emphasize that this constant depends on ν, which, in turn, depends on x,R. Since we have assumed that, for any x,R, the respective ν is nonzero, the constants εq,1x,R are strictly positive.

Now we take by τ the first time moment when |εWτ−x|=R. Observe that Ltν(εW) equals 0 on the set {τ>t}t\}$]]> and it is well known that
Px(τ<t)≤4Px(εWt>R)≤Ce−tR2ε−2/2.R)\le C{e}^{-t{R}^{2}{\varepsilon }^{-2}/2}.\]]]>
Summarizing the previous statements, we get
ExeLtqν(εW)≤1+2Cetε−2(c1M2q2−R2/2),ε≤ελ,1x,R,
which implies
lim supε→0ε2log(ExeLtqν(εW))1/q≤t(c1M2q−R2/(2q))+,
where we denote a+=max(a,0). By (10) inequalities (17) and (19) yield
lim supε→0ε2logExeLtμ0(εW)≤t2pΔ2+t(c1M2q−R2/(2q))+.

Now we finalize the argument in the same way as we did in the previous section. Fix Δ1>Δ\varDelta $]]> and take p>11$]]> such that pΔ2≤Δ12. Then take R large enough so that, for the corresponding q,
c1M2q−R2/(2q)≤0.
Under such a choice, the calculations made before yield (18) with Δ replaced by Δ1. Since Δ1>Δ\varDelta $]]> is arbitrary, the same inequality holds for Δ.

Example

Let
μ=∑k=1∞(δk2+δk2+2−k).
Then μ satisfies (1) and Δ=1. However, it is an easy observation that when the initial value x is taken in the form xk=k2, the respective exponential moments satisfy
ExkeLtμ(εW)→E0eLtν(εW),k→∞,
with ν=2δ0. Then
lim infε→0ε2supx∈RlogExeLtμ(εW)≥lim infε→0ε2logE0eLtν(εW)=2t>t2,\frac{t}{2},\]]]>
and therefore (3) fails.

Acknowledgments

The first author gratefully acknowledges the DFG Grant Schi 419/8-1 and the joint DFFD-RFFI project No. 09-01-14.

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