1 Introduction 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
Recall that the local time Lμt(W) of the process W with the weight μ can be defined as the limit of the integral functionals
where μn, n≥1, is a sequence of absolutely continuous measures such that
for all continuous f with compact support, and (1) holds for μn, n≥1, uniformly. The limit Lμt(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 Lμt(εW). Namely, we prove the following theorem.
Theorem 1.
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
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 Lμt(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].
2 Proof of Theorem 1
2.1 Preliminaries
For a measure ν satisfying (1), denote by
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>0,
where s>0 is such that (7) holds. This inequality, combined with (6), leads to the following estimate.
Lemma 2.
For a nonzero measure ν satisfying (1), denote
For any λ≥1 and γ>0, there exists ελ,γ>0 such that
with
Proof.
If ε√s≤γ, then we have
Take
Then the inequality ε√s≤γ holds, provided that
Under this condition,
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 Lμt(εW) itself using the following simple inequality. Let μ=ν+κ and p,q>1 be such that 1/p+1/q=1. Then
and therefore by the Hölder inequality we get
We will also use another version of this upper bound, which has the form
We denote
We will prove Theorem 1 in several steps, in each of them extending the class of measures μ for which the required statement holds.
2.2 Step I: μ is a finite mixture of δ-measures
If μ=aδz is a weighted δ-measure at the point z, then we have
where
is the local time of a Wiener process at the point z. The distribution of L(z)t(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:
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,
Let j∗ be the number of the maximal value in {aj}, that is, Δ=aj∗. Then Lμt(εW)≥Δε−1L(zj∗)t(W), and it follows directly from (12) that
In what follows, we prove the corresponding upper bound
which, combined with this lower bound, proves (3).
Observe that, for γ>0 small enough,
Then by Lemma 2, for any λ≥1,
with
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
Fix some family of neighborhoods Oj of zj,j=1,…,k, such that the minimal distance between them equals ρ>0, and denote
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
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
Because Lμj(εW) is a time-homogeneous additive functional of the Markov process εW, we have
Then by (12), for any j1,…,jN∈{1,…,k},
Because we have a fixed number of sets Cj1,…,jN, this immediately yields
with
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
By the LDP for the Wiener process ([4], Chapter 3, §2),
where
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,
This means that, for any f∈closure(D),
which yields
If in this construction, N was chosen such that
then the latter inequality guarantees the analogue of (16) with D instead of C. This completes the proof of (14).
2.3 Step II: μ 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>0, we can find γ>0 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>1 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
On the other hand, we have N(ν,γ)<χ and then by Lemma 2
Hence, by (10),
Now we can finalize the argument. Fix Δ1>Δ:=maxx∈Rμ({x})2 and choose p,q>1 such that 1/p+1/q=1 and pΔ2<Δ21. Then there exists χ>0 small enough such that
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 is arbitrary, the same inequality holds for Δ.
2.4 Step III: μ is σ-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
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 Lqνt(εW) conditioned by Fτ is dominated by 2ec1M2tq2ε−2. This holds for ε≤εx,Rq,1, 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 εx,Rq,1 are strictly positive.
Now we take by τ the first time moment when |εWτ−x|=R. Observe that Lνt(εW) equals 0 on the set {τ>t} and it is well known that
Summarizing the previous statements, we get
which implies
where we denote a+=max(a,0). By (10) inequalities (17) and (19) yield
Now we finalize the argument in the same way as we did in the previous section. Fix Δ1>Δ and take p>1 such that pΔ2≤Δ21. Then take R large enough so that, for the corresponding q,
Under such a choice, the calculations made before yield (18) with Δ replaced by Δ1. Since Δ1>Δ is arbitrary, the same inequality holds for Δ.