Asymptotics of exponential moments of a weighted local time of a Brownian motion with small variance

We prove a large deviation type estimate for the asymptotic behavior of a weighted local time of $\varepsilon W$ as $\varepsilon\to0$.


Introduction and the main result
Let {W t , t ≥ 0} be a real-valued Wiener process, and µ be a σ-finite measure on R such that sup Recall that the local time L µ t (W ) of the process W with the weight µ can be defined as the limit of the integral functionals where µ n , n ≥ 1, is a sequence of absolutely continuous measures such that for all continuous f with compact support, and (1) holds for µ n , n ≥ 1, uniformly. The limit L µ t (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 P x the law of εW with εW 0 = x and by E x 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 L µ t (εW ). Namely, we prove the following theorem.
For arbitrary σ-finite measure µ on R that satisfies (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/σ 2 has a bounded derivative. This limitation had appeared because of formula (7) in [8] for the rate transform of the family X ε . This formula contains an integral functional with kernel (a/σ 2 ) ′ of a certain diffusion process obtained from εW by the time change procedure. If a/σ 2 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/σ 2 ) ′ . 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 L µ t (εW ) are negligible at the logarithmic scale with rate function ε 2 . 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/σ 2 is a continuous function of bounded variation. The problem how to describe in a more general situation the influence of the jumps of a/σ 2 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].

Preliminaries
For a measure ν satisfying (1), denote by the characteristic of the local time L ν (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.

Lemma 1. Suppose that
Then sup 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.
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 L µ t (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 ∆ = sup x∈R µ {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: µ is a finite mixture of δ-measures
If µ = aδ z is a weighted δ-measure at the point z, then we have 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 = aδ z 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 {a j }, that is, ∆ = a j * . Then L µ t (εW ) ≥ ∆ε −1 L (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, In particular, taking λ = 1, we obtain an upper bound of the form (14), but with a worse constant c 1 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 µ j = a j δ zj , j = 1, . . . , k. Then Fix some family of neighborhoods O j of z j , 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 Observe that if the process εW does not visit O j on the time segment [u, v], then L µj (εW ) on this segment stays constant. This means that, on the set {εW ∈ C j1,...,jN }, we have Because L µj (εW ) is a time-homogeneous additive functional of the Markov process εW , we have Then by (12), for any j 1 , . . . , j N ∈ {1, . . . , k}, Because we have a fixed number of sets C j1,...,jN , this immediately yields with C = j1,...,jN ∈{1,...,k} C j1,...,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 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 O j on the time segment [t n−1 , t n ]. Therefore, any trajectory f ∈ closure(D) exhibits an oscillation ≥ ρ on this time segment. On the other hand, for an absolutely continuous f , We will use an argument similar to that from the previous section and decompose µ into a sum µ = µ 0 + ν with finite µ 0 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 µ 0 , 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 = sup x∈R µ([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 L qν t (εW ) conditioned by F τ is dominated by 2e c1M 2 tq 2 ε −2 . This holds for ε ≤ ε x,R q,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 ε x,R q,1 are strictly positive. Now we take by τ the first time moment when |εW τ − x| = R. Observe that L ν t (εW ) equals 0 on the set {τ > t} and it is well known that Summarizing the previous statements, we get where we denote a + = max(a, 0). By (10) inequalities (17) and (19) yield lim sup ε→0 ε 2 log E x e L µ 0 t (εW ) ≤ t 2 p∆ 2 + t c 1 M 2 q − R 2 /(2q) + .
Now we finalize the argument in the same way as we did in the previous section. Fix ∆ 1 > ∆ and take p > 1 such that p∆ 2 ≤ ∆ 2 1 . Then take R large enough so that, for the corresponding q, Under such a choice, the calculations made before yield (18) with ∆ replaced by ∆ 1 .
Since ∆ 1 > ∆ is arbitrary, the same inequality holds for ∆.