1 Introduction and the main result
Let $\{W_{t},t\ge 0\}$ be a real-valued Wiener process, and μ be a σ-finite measure on $\mathbb{R}$ such that
Recall that the local time ${L_{t}^{\mu }}(W)$ of the process W with the weight μ can be defined as the limit of the integral functionals
where $\mu _{n}$, $n\ge 1$, is a sequence of absolutely continuous measures such that
for all continuous f with compact support, and (1) holds for $\mu _{n}$, $n\ge 1$, uniformly. The limit ${L_{t}^{\mu }}(W)$ 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 $\varepsilon W$ instead of W for any positive ε. In what follows, we will treat $\varepsilon W$ as a Markov process whose initial value may vary, and with a slight abuse of notation, we denote by $\mathbf{P}_{x}$ the law of $\varepsilon W$ with $\varepsilon W_{0}=x$ and by $\mathbf{E}_{x}$ the expectation w.r.t. this law.
(2)
\[{L_{t}^{\mu _{n}}}(W):={\int _{0}^{t}}k_{n}(W_{s})\hspace{0.1667em}ds,\hspace{1em}k_{n}(x):=\frac{\mu _{n}(dx)}{dx},\hspace{2.5pt}n\ge 1,\]In this note, we study the asymptotic behavior as $\varepsilon \to 0$ of the exponential moments of the family of weighted local times ${L_{t}^{\mu }}(\varepsilon W)$. Namely, we prove the following theorem.
Theorem 1.
For arbitrary finite measure μ on $\mathbb{R}$,
For arbitrary σ-finite measure μ on $\mathbb{R}$ that satisfies (1),
(3)
\[\underset{\varepsilon \to 0}{\lim }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}=\frac{t}{2}\underset{y\in \mathbb{R}}{\sup }\mu {\big(\{y\}\big)}^{2}.\]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,\sigma $. In [7], a Wentzel–Freidlin-type large deviation principle (LDP) was established in the case $a\equiv 0$ under mild assumptions on the diffusion coefficient σ. In [8], this result was extended to the particular class of SDEs such that the function $a/{\sigma }^{2}$ has a bounded derivative. This limitation had appeared because of formula (7) in [8] for the rate transform of the family ${X}^{\varepsilon }$. This formula contains an integral functional with kernel ${(a/{\sigma }^{2})}^{\prime }$ of a certain diffusion process obtained from $\varepsilon W$ by the time change procedure. If $a/{\sigma }^{2}$ 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 $\mu ={(a/{\sigma }^{2})}^{\prime }$. 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 ${L_{t}^{\mu }}(\varepsilon W)$ are negligible at the logarithmic scale with rate function ${\varepsilon }^{2}$. 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 $a/{\sigma }^{2}$ is a continuous function of bounded variation. The problem how to describe in a more general situation the influence of the jumps of $a/{\sigma }^{2}$ 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,\sigma $ obtained in [1, 2, 6].
(5)
\[d{X_{t}^{\varepsilon }}=a\big({X_{t}^{\varepsilon }}\big)\hspace{0.1667em}dt+\varepsilon \sigma \big({X_{t}^{\varepsilon }}\big)\hspace{0.1667em}dW_{t}\]2 Proof of Theorem 1
2.1 Preliminaries
For a measure ν satisfying (1), denote by
the characteristic of the local time ${L}^{\nu }(\varepsilon W)$ considered as a W-functional of $\varepsilon W$; see [3], Chapter 6.
(6)
\[{f_{t}^{\nu ,\varepsilon }}(x)=\mathbf{E}_{x}{L_{t}^{\nu }}(\varepsilon W)={\int _{0}^{t}}\int _{\mathbb{R}}\frac{1}{\sqrt{2\pi s{\varepsilon }^{2}}}{e}^{-\frac{{(y-x)}^{2}}{2s{\varepsilon }^{2}}}\nu (dy)\hspace{0.1667em}ds,\hspace{1em}t\ge 0,\hspace{2.5pt}x\in \mathbb{R},\]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.
(8)
\[\underset{x\in \mathbb{R}}{\sup }\mathbf{E}_{x}{e}^{{L_{t}^{\nu }}(\varepsilon W)}\le {2}^{1+t/s}=2{e}^{(\log 2)(t/s)},\]Lemma 2.
For a nonzero measure ν satisfying (1), denote
\[N(\nu ,\gamma )=\underset{x\in \mathbb{R}}{\sup }\nu \big([x-\gamma ,x+\gamma ]\big),\hspace{1em}\gamma >0.\]
For any $\lambda \ge 1$ and $\gamma >0$, there exists $\varepsilon _{\lambda ,\gamma }>0$ such that
with
Proof.
If $\varepsilon \sqrt{s}\le \gamma $, then we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle {f_{s}^{\nu ,\varepsilon }}(x)& \displaystyle =\sum \limits_{k\in \mathbb{Z}}{\int _{0}^{s}}\int _{|y-x-2k\gamma |\le \gamma }\frac{1}{\sqrt{2\pi v{\varepsilon }^{2}}}{e}^{-\frac{{(y-x)}^{2}}{2v{\varepsilon }^{2}}}\nu (dy)\hspace{0.1667em}dv\\{} & \displaystyle \le \sqrt{c_{0}}N(\nu ,\gamma )\sqrt{\frac{s}{{\varepsilon }^{2}}}.\end{array}\]
Take
Then the inequality $\varepsilon \sqrt{s}\le \gamma $ holds, provided that
\[\varepsilon \le {\big(\gamma {\big(2N(\nu ,\gamma )\big)}^{2}c_{0}{\lambda }^{2}\big)}^{1/3}=:\varepsilon _{\lambda ,\gamma }.\]
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 ${L_{t}^{\mu }}(\varepsilon W)$ itself using the following simple inequality. Let $\mu =\nu +\kappa $ and $p,q>1$ be such that $1/p+1/q=1$. Then
\[{L_{t}^{\mu }}(\varepsilon W)={L_{t}^{\nu }}(\varepsilon W)+{L_{t}^{\kappa }}(\varepsilon W)=(1/p){L_{t}^{p\nu }}(\varepsilon W)+(1/q){L_{t}^{q\kappa }}(\varepsilon W),\]
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 $\mu =a\delta _{z}$ is a weighted δ-measure at the point z, then we have
where
Note that in this formula the supremum is attained at the point $x=z$.
\[{L_{t}^{(z)}}(W)=\underset{\eta \to 0}{\lim }\frac{1}{2\eta }{\int _{0}^{t}}1_{|W_{s}-z|\le \eta }\hspace{0.1667em}ds\]
is the local time of a Wiener process at the point z. The distribution of ${L_{t}^{(z)}}(W)$ 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 $\mu =a\delta _{z}$ is straightforward, and we have the following:
(12)
\[\underset{\varepsilon \to 0}{\lim }{\varepsilon }^{2}\underset{x}{\sup }\log \mathbf{E}_{x}{e}^{a{\varepsilon }^{-1}{L_{t}^{(z)}}(W)}=\frac{t{a}^{2}}{2}.\]In this section, we will extend this result to the case where μ is a finite mixture of δ-measures, that is,
Let $j_{\ast }$ be the number of the maximal value in $\{a_{j}\}$, that is, $\varDelta =a_{j_{\ast }}$. Then ${L_{t}^{\mu }}(\varepsilon W)\ge \varDelta {\varepsilon }^{-1}{L_{t}^{(z_{j_{\ast }})}}(W)$, and it follows directly from (12) that
In what follows, we prove the corresponding upper bound
which, combined with this lower bound, proves (3).
(13)
\[\underset{\varepsilon \to 0}{\liminf }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\ge \frac{t{\varDelta }^{2}}{2}.\](14)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\le \frac{t{\varDelta }^{2}}{2},\]Observe that, for $\gamma >0$ small enough,
Then by Lemma 2, for any $\lambda \ge 1$,
with
(15)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{\lambda {L_{t}^{\mu }}(\varepsilon W)}\le c_{1}{\lambda }^{2}t{\varDelta }^{2}\]
\[c_{1}=(4\log 2)c_{0}=\frac{8\log 2}{\pi }{\Bigg(1+2\sum \limits_{k=1}^{\infty }{e}^{-\frac{{(2k-1)}^{2}}{2}}\Bigg)}^{2}.\]
In particular, taking $\lambda =1$, 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 $\varepsilon W$, the Markov property, and the “individual” identities (12).Denote $\mu _{j}=a_{j}\delta _{z_{j}},j=1,\dots ,k$. Then
Fix some family of neighborhoods $O_{j}$ of $z_{j},j=1,\dots ,k$, such that the minimal distance between them equals $\rho >0$, and denote
For some $N\ge 1$ whose particular value will be specified later, consider the partition $t_{n}=t(n/N)$, $n=0,\dots ,N$, of the segment $[0,t]$ and denote
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 $D=C(0,t)\setminus C$. Using (11) with $p=2$, $A=\{\varepsilon W\in D\}$, and (15) with $\lambda =2$, we get
\[B_{n,j}=\big\{f\in C(0,t):f_{s}\in {O}^{j},s\in [t_{n-1},t_{n}]\big\},\hspace{1em}j\in \{1,\dots ,k\},\hspace{2.5pt}n\in \{1,\dots ,N\},\]
\[C_{j_{1},\dots ,j_{N}}=\bigcap \limits_{n=1}^{N}B_{n,j_{n}},\hspace{1em}j_{1},\dots ,j_{N}\in \{1,\dots ,k\}.\]
Observe that if the process $\varepsilon W$ does not visit $O_{j}$ on the time segment $[u,v]$, then ${L}^{\mu _{j}}(\varepsilon W)$ on this segment stays constant. This means that, on the set $\{\varepsilon W\in C_{j_{1},\dots ,j_{N}}\}$, we have
\[{L_{t}^{\mu }}(\varepsilon W)=\sum \limits_{n=1}^{N}\big({L_{t_{n}}^{\mu _{j_{n}}}}(\varepsilon W)-{L_{t_{n-1}}^{\mu _{j_{n}}}}(\varepsilon W)\big).\]
Because ${L}^{\mu _{j}}(\varepsilon W)$ is a time-homogeneous additive functional of the Markov process $\varepsilon W$, we have
\[E_{x}\big[{e}^{{L_{t_{n}}^{\mu _{j_{n}}}}(\varepsilon W)-{L_{t_{n-1}}^{\mu _{j_{n}}}}(\varepsilon W)}\big|\mathcal{F}_{t_{n-1}}\big]=E_{y}{e}^{{L_{t/N}^{\mu _{j_{n}}}}(\varepsilon W)}\Big|_{y=\varepsilon W_{t_{n-1}}}.\]
Then by (12), for any $j_{1},\dots ,j_{N}\in \{1,\dots ,k\}$,
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}1_{\varepsilon W\in C_{j_{1},\dots ,j_{N}}}\le \frac{t}{2N}\sum \limits_{n=1}^{N}{(a_{j_{n}})}^{2}\le \frac{t{\varDelta }^{2}}{2}.\]
Because we have a fixed number of sets $C_{j_{1},\dots ,j_{N}}$, this immediately yields
(16)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}1_{\varepsilon W\in C}\le \frac{t{\varDelta }^{2}}{2}\]
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}1_{\varepsilon W\in D}\le 2c_{1}t{\varDelta }^{2}+\frac{1}{2}\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{P}_{x}(\varepsilon W\in D).\]
By the LDP for the Wiener process ([4], Chapter 3, §2),
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{P}_{x}(\varepsilon W\in D)=-\underset{f\in \mathrm{closure}(D)}{\inf }I(f),\]
where
\[I(f)=\left\{\begin{array}{l@{\hskip10.0pt}l}(1/2){\textstyle\int _{0}^{t}}{({f^{\prime }_{s}})}^{2}\hspace{0.1667em}ds,& f\hspace{2.5pt}\text{is absolutely continuous on}\hspace{2.5pt}[0,t];\\{} +\infty & \text{otherwise.}\end{array}\right.\]
For any trajectory $f\in D$, 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 $f\in \mathrm{closure}(D)$ exhibits an oscillation $\ge \rho $ on this time segment. On the other hand, for an absolutely continuous f,
\[|f_{u}-f_{v}|=\Bigg|{\int _{u}^{v}}{f^{\prime }_{s}}\hspace{0.1667em}ds\Bigg|\le |u-v{|}^{1/2}{\Bigg({\int _{0}^{t}}{\big({f^{\prime }_{s}}\big)}^{2}\hspace{0.1667em}ds\Bigg)}^{1/2}.\]
This means that, for any $f\in \mathrm{closure}(D)$,
which yields
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}1_{\varepsilon W\in D}\le 2c_{1}t{\varDelta }^{2}-\frac{{\rho }^{2}N}{2t}.\]
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 $\chi >0$, we can find $\gamma >0$ and decompose $\mu =\mu _{0}+\nu $ in such a way that $\mu _{0}$ is a finite mixture of δ-measures and $N(\nu ,\gamma )<\chi $. Let $p,q>1$ be such that $1/p+1/q=1$. The measure $p\mu _{0}$ has the maximal weight of an atom equal to $p\varDelta $. Since we have already proved the required statement for finite mixtures of δ-measures, we have
On the other hand, we have $N(\nu ,\gamma )<\chi $ and then by Lemma 2
(17)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log {\big(\mathbf{E}_{x}{e}^{{L_{t}^{p\mu _{0}}}(\varepsilon W)}\big)}^{1/p}\le \frac{t}{2}p{\varDelta }^{2}.\]
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log {\big(\mathbf{E}_{x}{e}^{{L_{t}^{q\nu }}(\varepsilon W)}\big)}^{1/q}\le c_{1}qt{\chi }^{2}.\]
Hence, by (10),
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\le \frac{t}{2}p{\varDelta }^{2}+c_{1}qt{\chi }^{2}.\]
Now we can finalize the argument. Fix $\varDelta _{1}>\varDelta :=\max _{x\in \mathbb{R}}\mu {(\{x\})}^{2}$ and choose $p,q>1$ such that $1/p+1/q=1$ and $p{\varDelta }^{2}<{\varDelta _{1}^{2}}$. Then there exists $\chi >0$ small enough such that
Taking the decomposition $\mu =\mu _{0}+\nu $ 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 $\varDelta _{1}$. Since $\varDelta _{1}>\varDelta $ 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 $\mu =a\delta _{z}$, 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 $\mu =\mu _{0}+\nu $ with finite $\mu _{0}$ 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).
(18)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\le \frac{t{\varDelta }^{2}}{2},\hspace{1em}x\in \mathbb{R}.\]Namely, for a given x, we define $\mu _{0},\nu $ 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 $M=\sup _{x\in \mathbb{R}}\mu ([x-1,x+1])$ and observe that $N(\nu ,1)\le M$. Then by Lemma 2 with $\gamma =1$ and the strong Markov property, for any stopping time τ, the exponential moment of ${L_{t}^{q\nu }}(\varepsilon W)$ conditioned by $\mathcal{F}_{\tau }$ is dominated by $2{e}^{c_{1}{M}^{2}t{q}^{2}{\varepsilon }^{-2}}$. This holds for $\varepsilon \le {\varepsilon _{q,1}^{x,R}}$, 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 ${\varepsilon _{q,1}^{x,R}}$ are strictly positive.
Now we take by τ the first time moment when $|\varepsilon W_{\tau }-x|=R$. Observe that ${L_{t}^{\nu }}(\varepsilon W)$ equals 0 on the set $\{\tau >t\}$ and it is well known that
where we denote $a_{+}=\max (a,0)$. By (10) inequalities (17) and (19) yield
\[\mathbf{P}_{x}(\tau <t)\le 4\mathbf{P}_{x}(\varepsilon W_{t}>R)\le C{e}^{-t{R}^{2}{\varepsilon }^{-2}/2}.\]
Summarizing the previous statements, we get
\[\mathbf{E}_{x}{e}^{{L_{t}^{q\nu }}(\varepsilon W)}\le 1+2C{e}^{t{\varepsilon }^{-2}(c_{1}{M}^{2}{q}^{2}-{R}^{2}/2)},\hspace{1em}\varepsilon \le {\varepsilon _{\lambda ,1}^{x,R}},\]
which implies
(19)
\[\underset{\varepsilon \to 0}{\limsup }{\varepsilon }^{2}\log {\big(\mathbf{E}_{x}{e}^{{L_{t}^{q\nu }}(\varepsilon W)}\big)}^{1/q}\le t\big(c_{1}{M}^{2}q-{R}^{2}/(2q)\big)_{+},\]Now we finalize the argument in the same way as we did in the previous section. Fix $\varDelta _{1}>\varDelta $ and take $p>1$ such that $p{\varDelta }^{2}\le {\varDelta _{1}^{2}}$. 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 $\varDelta _{1}$. Since $\varDelta _{1}>\varDelta $ is arbitrary, the same inequality holds for Δ.
3 Example
Let
Then μ satisfies (1) and $\varDelta =1$. However, it is an easy observation that when the initial value x is taken in the form $x_{k}={k}^{2}$, the respective exponential moments satisfy
\[\mathbf{E}_{x_{k}}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\to \mathbf{E}_{0}{e}^{{L_{t}^{\nu }}(\varepsilon W)},\hspace{1em}k\to \infty ,\]
with $\nu =2\delta _{0}$. Then
\[\underset{\varepsilon \to 0}{\liminf }{\varepsilon }^{2}\underset{x\in \mathbb{R}}{\sup }\log \mathbf{E}_{x}{e}^{{L_{t}^{\mu }}(\varepsilon W)}\ge \underset{\varepsilon \to 0}{\liminf }{\varepsilon }^{2}\log \mathbf{E}_{0}{e}^{{L_{t}^{\nu }}(\varepsilon W)}=2t>\frac{t}{2},\]
and therefore (3) fails.
