Lyapunov Exponents for Hamiltonian Systems under Small Lévy Perturbations

Let $(\Omega,\mathscr{F},\{\mathscr{F}_{t}\}_{t\geqslant 0},\mathbb{P})$ be a filtered probability space, where $\mathscr{F}_{t}$ is a nondecreasing family of sub-$\sigma$-fields of $\mathscr{F}$ satisfying the usual conditions. An $\mathscr{F}_{t}$-adapted stochastic process $L_{t}=L(t)$ taking values in $\mathbb{R}^{d}$ with $L(0)=0$ $a.s.$ (almost surely) is called a Lévy motion if it is stochastically continuous, with independent increments and stationary increments.

An $d$-dimensional Lévy motion can be characterized by a drift vector $b\in\mathbb{R}^{d}$, an $d\times d$ non-negative-definite, symmetric matrix $Q$, and a Borel measure $\nu$ defined on ${\mathbb{R}^{d}}\backslash\{0\}$. We call $(b,Q,\nu)$ the generating triplet of the Lévy motion $L_{t}$ and write this Lévy motion as $L_{t}\sim(b,Q,\nu)$ for convenient. Moreover, we have the following Lévy-Itô decomposition for $L_{t}$:

$${L_{t}}=bt+B_{Q}(t)+\int_{\|z\|<c}z\tilde{N}(t,dz)+\int_{\|z\|\geq c}zN(t,dz),$$

(2.1)

where $N(dt,dz)$ is the Poisson random measure on $\mathbb{R}^{+}\times({\mathbb{R}^{d}}\backslash\{0\})$, $\tilde{N}(dt,dz)=N(dt,dz)-\nu(dz)dt$ is the compensated Poisson random measure, $\nu=\mathbb{E}N(1,\cdot)$ is the jump measure, $B_{Q}(t)$ is an independent $d$-dimensional Brownian motion with covariance matrix $Q$, and the last two terms describe the ‘small jumps’ and ‘big jumps’ of Lévy process, respectively. Here $\|\cdot\|$ denotes the Euclidean norm and $c$ is a positive constant. In the following, we denote $L_{c}(t)=\gamma t+B_{Q}(t)$ as the continuous part of $L_{t}$ and $L_{d}(t)=L_{t}-L_{c}(t)$ as the discontinuous part.

Let $H:\mathbb{R}^{2}\to\mathbb{R}$ be a smooth function with isolated critical points such that $H(x)\to\infty$ as $\|x\|\to\infty$, and $U_{1}=(\partial H/\partial x^{2},-\partial H/\partial x^{1})^{T}$ be the Hamiltonian vector field with respect to $H$. Given smooth vector fields $V_{1},\cdots,V_{d}$ and a $d$-dimensional Lévy motion $L_{t}=(L^{k}_{t})_{k=1,\cdots,d}$ with generating triplet $(0,I,\nu)$. For $0<\varepsilon<1$, we consider the following one degree-of-freedom Hamiltonian system with Lévy noise

$$dx_{t}=U_{1}(x_{t})dt+\varepsilon\sum_{k=1}^{d}V_{k}(x_{t})\diamond dL^{k}_{t},\;\;x(0)=x_{0},$$

(2.2)

or equivalently,

$$x_{t}=x_{0}+\int_{0}^{t}U_{1}(x_{s})ds+\varepsilon\sum_{k=1}^{d}\int_{0}^{t}V_{k}(x_{s-})\diamond dL^{k}_{s},$$

(2.3)

where “$\diamond$” indicates Marcus (canonical) integral [18, 15] defined by

	$\displaystyle\varepsilon\sum_{k=1}^{d}\int_{0}^{t}V_{k}(x_{s-})\diamond dL^{k}_{s}=$	$\displaystyle\varepsilon\sum_{k=1}^{d}\int_{0}^{t}V_{k}(x_{s-})\circ dL^{k}_{c,s}+\varepsilon\sum_{k=1}^{d}\int_{0}^{t}V_{k}(x_{s-})dL^{k}_{d,s}$
		$\displaystyle+\sum_{0\leqslant s\leqslant t}\left[\xi^{\varepsilon}(\Delta L_{s}(x_{s-}))-x_{s-}-\varepsilon\sum_{k=1}^{d}V_{k}(x_{s-})\Delta L^{k}_{s}\right]$

with $\int\cdots\circ dL^{k}_{c,s}$ denoting the Stratonovtich integral, $\int\cdots dL^{k}_{d,s}$ denoting the Itô integral and $\xi^{\varepsilon}$ being the value at $\tau=1$ of the solution of the following ODE: (for each $x\in\mathbb{R}^{2}$ and $z\in\mathbb{R}^{d}$)

$\displaystyle\frac{d\xi^{\varepsilon}(\tau z)(x)}{d\tau}=\varepsilon\sum_{k=1}^{d}z_{k}V_{k}(\xi^{\varepsilon}(\tau z)(x)),\;\;\xi^{\varepsilon}(0)(x)=x.$

(2.4)

Compared with Itô stochastic differentials, Marcus one has the advantage of leading to ordinary chain rule under a transformation (change of variable). This offers some slight reduction in the lengths of some of the calculations, but has no serious effect on the theorem. We also remark that (2.2) is referred as a stochastic Hamiltonian system preserving symplectic structure if $V_{k}$, $k=1,\cdots d$ are Hamiltonian vector fields [23].

Rewritting SDE (2.2) into Itô form, we have

	$\displaystyle dx_{t}=$	$\displaystyle U_{1}(x_{t})dt+\frac{1}{2}\varepsilon^{2}\sum_{k=1}^{d}(DV_{k}V_{k})(x_{t})dt+\varepsilon\sum_{k=1}^{d}V_{k}(x_{t})dB_{t}^{k}$
		$\displaystyle+\int_{\|z\|<c}\left[\xi^{\varepsilon}(z)(x_{t-})-x_{t-}-\varepsilon\sum_{k=1}^{d}z_{k}V_{k}(x_{t-})\right]\nu(dz)dt$
		$\displaystyle+\int_{\|z\|<c}\left[\xi^{\varepsilon}(z)(x_{t-})-x_{t-}\right]\tilde{N}(dt,dz)$
		$\displaystyle+\int_{\|z\|\geqslant c}\left[\xi^{\varepsilon}(z)(x_{t-})-x_{t-}\right]{N}(dt,dz),$		(2.5)

where $c$ is a positive constant. The term in (2.2) involving large jump is controlled by a function on ${\{\|z\|\geqslant c\}}$, and it would be absent if we take $c=\infty$. One standard way to handle this case is to use interlacing technique [4]. In this paper, we use this technique directly without making precise statements. For more details about interlacing, we would like to refer to Applebaum [4]. Hence, we can omit the large-jump terms and concentrate on the study of the equations driven by continuous noise interspersed with small jumps, then SDE (2.2) can be modified as

	$\displaystyle dx_{t}=$	$\displaystyle U_{1}(x_{t})dt+\frac{1}{2}\varepsilon^{2}\sum_{k=1}^{d}(DV_{k}V_{k})(x_{t})dt+\varepsilon\sum_{k=1}^{d}V_{k}(x_{t})dB_{t}^{k}$
		$\displaystyle+\int_{\|z\|<c}\left[\xi^{\varepsilon}(z)(x_{t-})-x_{t-}-\varepsilon\sum_{k=1}^{d}z_{k}V_{k}(x_{t-})\right]\nu(dz)dt$
		$\displaystyle+\int_{\|z\|<c}\left[\xi^{\varepsilon}(z)(x_{t-})-x_{t-}\right]\tilde{N}(dt,dz).$		(2.6)

If we assume that $U_{1}(x)$ and $\tilde{V}(x)\triangleq\sum_{k=1}^{d}(DV_{k}V_{k})(x)$ are locally Lipschitz continuous functions and satisfy ‘one sided linear growth’ condition in the following sense:

1. (A1)

   (Locally Lipschitz condition) For any $R>0$, there exists $K_{1}>0$ such that, for all $\|x_{1}\|$, $\|x_{2}\|\leqslant R$,

   $$\|U_{1}(x_{1})-U_{1}(x_{2})\|^{2}+\|\tilde{V}(x_{1})-\tilde{V}(x_{2})\|^{2}+\max_{1\leqslant k\leqslant d}\|V_{k}(x_{1})-V_{k}(x_{2})\|^{2}\leqslant K_{1}\|x_{1}-x_{2}\|^{2},$$

2. (A2)

   (One sided linear growth condition) There exists $K_{2}>0$ such that, for all $x\in\mathbb{R}^{2}$,

   $$\sum_{k=1}^{d}\|V_{k}(x)\|^{2}+2x\cdot(U_{1}+\tilde{V})(x)\leqslant K_{2}(1+\|x\|^{2}),$$

then there exists a unique solution to (2.2) and the solution process is adapted and cádlàg, based on Lemma 6.10.3 of [4] and Theorem 3.1 of [1]. Referring to Theorem 6.8.2 of [4], assumption (A2) also ensures that the solution process of (2.2) has a Lyapunov exponent.

We first claim that, when rewritten with respect to a suitable moving frame, the linearization of perturbed Hamiltonian system (3.1), that is, system (3.2), is a perturbed nilpotent linear system. For convenience, we write $U_{2}(x)=\nabla H(x)/\|\nabla H(x)\|^{2},$ and adopt the notation $U.f(x)=Df(x)(U(x))=\langle\nabla f(x),U(x)\rangle$, which is the action of a vector field $U$ as a first order differential operator acting on a function $f$.

Denote by $M$ the space $\mathbb{R}^{2}$ with all critical points of $H$ removed. Note that, for $x\in M$, $(U_{1}.H)(x)=0$, $(U_{2}.H)(x)=1$ and $\langle U_{1}(x),U_{2}(x)\rangle=0$. We rewrite smooth vector fields $V_{k}$, $k=1,\cdots,d$, into

$${V}_{k}(x)=a_{k}^{1}(x)U_{1}(x)+a_{k}^{2}(x)U_{2}(x),\;\;x\in M,$$

(3.3)

where $a_{k}^{1}$ and $a_{k}^{2}$ are smooth functions with respect to $x$. Then the original system (3.1) becomes

$$dx_{t}=U_{1}(x_{t})dt+\varepsilon\sum_{k=1}^{d}(a_{k}^{1}U_{1}+a_{k}^{2}U_{2})(x_{t})\diamond d\tilde{L}^{k}_{t}.$$

(3.4)

Furthermore, we have the following lemma:

Lemma 3.1 shows that we are indeed dealing with a small Lévy-type perturbation of a nilpotent system. Motivated by the remarkable work on small random perturbation of a nilpotent system in Pinsky-Wihstutz [19] and its applications in [9, 10], we define

$$T=\begin{bmatrix}\begin{matrix}\varepsilon^{\beta}&0\\ 0&1\end{matrix}\end{bmatrix},\;\;0<\beta<1,$$

(3.9)

for fixed $\varepsilon>0$. Since

$$\min(\varepsilon^{\beta},1)\|w\|\leqslant\|Tw\|\leqslant\max(\varepsilon^{\beta},1)\|w\|,$$

it follows that

$$\lim_{t\to\infty}\frac{1}{t}\log\|Tw_{t}\|=\lim_{t\to\infty}\frac{1}{t}\log\|w_{t}\|$$

and so the process $Tw_{t}$ has the same Lyapunov exponent as $w_{t}$. For simplicity of notation we continue to write $w_{t}$ in place of $Tw_{t}$.

Keeping in mind the equation for $x_{t}$ in (2.2) or equivalently (3.1), the process $w_{t}$ $(t<\tau_{\varepsilon})$ is now given by

$$dw_{t}=\varepsilon^{\beta}\Lambda(x_{t})w_{t}dt+\varepsilon\sum_{k=1}^{d}M_{k}^{\varepsilon}(x_{t})w_{t}\diamond d\tilde{L}^{k}_{t},$$

(3.10)

where $M_{k}^{\varepsilon}=TM_{k}T^{-1}=\begin{bmatrix}\begin{matrix}B_{k}&\varepsilon^{\beta}C_{k}\\ \varepsilon^{-\beta}D_{k}&E_{k}\end{matrix}\end{bmatrix}$ for $k=1,\cdots,d$.

Note that canonical differential satisfies the change of variable formula [15], we write

$$w_{t}=\|w_{t}\|\begin{bmatrix}\begin{matrix}\cos\theta_{t}\\ \sin\theta_{t}\end{matrix}\end{bmatrix},$$

and define $\rho_{t}=\log\|w_{t}\|$. Hence, equation (3.10) can be rewritten as a pair of equations:

	$\displaystyle d\theta_{t}=$	$\displaystyle-\varepsilon^{\beta}A(x_{t})\sin^{2}\theta_{t}+\sum_{k=1}^{d}\sigma_{k}^{1}(x_{t},\theta_{t})\diamond d\tilde{L}^{k}_{t},$		(3.11)
	$\displaystyle d\rho_{t}=$	$\displaystyle\varepsilon^{\beta}A(x_{t})\sin\theta_{t}\cos\theta_{t}+\sum_{k=1}^{d}\sigma_{k}^{2}(x_{t},\theta_{t})\diamond d\tilde{L}^{k}_{t},$		(3.12)

where, for $k=1,2,\cdots,d$,

	$\displaystyle\sigma_{k}^{1}(x,\theta)=$	$\displaystyle\varepsilon^{1-\beta}D_{k}(x)\cos^{2}\theta-\varepsilon(B_{k}-E_{k})(x)\sin\theta\cos\theta-\varepsilon^{1+\beta}C_{k}(x)\sin^{2}\theta$
	$\displaystyle\triangleq$	$\displaystyle\varepsilon^{1-\beta}Q_{1}^{k}(x,\theta)+\varepsilon Q_{2}^{k}(x,\theta)+\varepsilon^{1+\beta}Q_{3}^{k}(x,\theta),$		(3.13)
	$\displaystyle\sigma_{k}^{2}(x,\theta)=$	$\displaystyle\varepsilon^{1-\beta}D_{k}(x)\sin\theta\cos\theta+\varepsilon[B_{k}(x)\cos^{2}\theta+E_{k}(x)\sin^{2}\theta]+\varepsilon^{1+\beta}C_{k}(x)\sin\theta\cos\theta$
	$\displaystyle\triangleq$	$\displaystyle\varepsilon^{1-\beta}P_{1}^{k}(x,\theta)+\varepsilon P_{2}^{k}(x,\theta)+\varepsilon^{1+\beta}P_{3}^{k}(x,\theta).$		(3.14)

Notice that, under the change of variables, the vector fields $\sigma_{k}^{i}$, $k=1,\cdots,d$, $i=1,2$, should be understood as functions of $x$, $\theta$ and $\rho$, but they indeed do not depend on $\rho$. The fact that the equation for $\log\|w_{t}\|$ does not involve $\|w_{t}\|$ was first used by Khas’minskii [14] in the 1960s, and is an essential part of the derivation of the Furstenberg-Khas’minskii formula for the top Lyapunov exponent of a linearized system.

We next further convert Marcus equations (3.11)-(3.12) into the Itô form [4, Page 418]. It is important to note that the Marcus interpretation for SDE (3.11)-(3.12) depends also on the SDE (3.4) for $x_{t}$. More specifically, the Marcus interpretation is only for SDE describing a diffusion process [15], and the pair process $\{(\theta_{t},\rho_{t}):t\geqslant 0\}$ is not a diffusion process because its distribution depends also on $\{x_{t}:t\geqslant 0\}$. A jump in the Lévy process $L_{t}^{\alpha}$ causes jumps in both $x_{t}$ and $(\theta_{t},\rho_{t})$, we thus need to supplement ODE (2.4) with the following equations:

$\displaystyle\frac{d\zeta_{i}^{\varepsilon}(\tau z)(x,\theta)}{d\tau}=\sum_{k=1}^{d}z_{k}\sigma_{k}^{i}\big{(}\xi^{\varepsilon}(\tau z)(x),\zeta_{1}^{\varepsilon}(\tau z)(x,\theta)\big{)},\;\;i=1,2,$

(3.15)

with $\zeta^{\varepsilon}(0)(x,\theta)=(x,\theta)$. That is, we need to focus on the time one flow $\eta^{\varepsilon}(z)(x,\theta,\rho)=(\xi^{\varepsilon},\zeta_{1}^{\varepsilon},\rho+\zeta_{2}^{\varepsilon})$ started at $(x,\theta,\rho)$ along the vector field $\sum_{k=1}^{d}z_{k}\tilde{V}_{k}(x,\theta,\rho)$, where

$\displaystyle\tilde{V}_{k}(x,\theta,\rho)=\Big{(}\varepsilon V_{k}(x),\sigma_{k}^{1}(x,\theta),\sigma_{k}^{2}(x,\theta)\Big{)}.$

(3.16)

Therefore, we have the following Itô SDE:

	$\displaystyle d\theta_{t}=$	$\displaystyle-\varepsilon^{\beta}A(x_{t})\sin^{2}\theta_{t}dt+\frac{1}{2}\sum_{k=1}^{d}\tilde{\sigma}_{k}^{1}(x_{t},\theta_{t})dt+\sum_{k=1}^{d}\sigma_{k}^{1}(x_{t},\theta_{t})dB_{t}^{k}$
		$\displaystyle+\int_{\|z\|<c}\left[\zeta_{1}^{\varepsilon}(z)(x_{t-},\theta_{t-})-\theta_{t-}-\sum_{k=1}^{d}z_{k}\sigma_{k}^{1}(x_{t-},\theta_{t-})\right]\nu(dz)dt$
		$\displaystyle+\int_{\|z\|<c}\left[\zeta_{1}^{\varepsilon}(z)(x_{t-},\theta_{t-})-\theta_{t-}\right]\tilde{N}(dt,dz),$		(3.17)
	$\displaystyle d\rho_{t}=$	$\displaystyle\varepsilon^{\beta}A(x_{t})\sin\theta_{t}\cos\theta_{t}dt+\frac{1}{2}\sum_{k=1}^{d}\tilde{\sigma}_{k}^{2}(x_{t},\theta_{t})dt+\sum_{k=1}^{d}\sigma_{k}^{2}(x_{t},\theta_{t})dB_{t}^{k}$
		$\displaystyle+\int_{\|z\|<c}\left[\zeta_{2}^{\varepsilon}(z)(x_{t-},\theta_{t-})-\sum_{k=1}^{d}z_{k}\sigma_{k}^{2}(x_{t-},\theta_{t-})\right]\nu(dz)dt$
		$\displaystyle+\int_{\|z\|<c}\left[\zeta_{2}^{\varepsilon}(z)(x_{t-},\theta_{t-})\right]\tilde{N}(dt,dz)$		(3.18)

with Wong-Zakai correction terms

	$\displaystyle\tilde{\sigma}_{k}^{1}(x,\theta)=$	$\displaystyle\varepsilon D_{x}\sigma_{k}^{1}V_{k}+D_{\theta}\sigma_{k}^{1}\sigma_{k}^{1}$
	$\displaystyle=$	$\displaystyle\varepsilon^{2-2\beta}D_{\theta}Q_{1}^{k}Q_{1}^{k}+\varepsilon^{2-\beta}(D_{x}Q_{1}^{k}V_{k}+D_{\theta}Q_{1}^{k}P_{2}^{k}+D_{\theta}Q_{2}^{k}Q_{1}^{k})$
		$\displaystyle+\varepsilon^{2}(D_{x}Q_{2}^{k}V_{k}+D_{\theta}Q_{1}^{k}Q_{3}^{k}+D_{\theta}Q_{3}^{k}Q_{1}^{k})$
		$\displaystyle+\varepsilon^{2+\beta}(D_{x}Q_{3}^{k}V_{k}+D_{\theta}Q_{2}^{k}Q_{3}^{k}+D_{\theta}Q_{3}^{k}Q_{2}^{k})+\varepsilon^{2+2\beta}D_{\theta}Q_{3}^{k}Q_{3}^{k}$
	$\displaystyle\triangleq$	$\displaystyle\varepsilon^{2-2\beta}\tilde{Q}_{1}^{k}(x,\theta)+\varepsilon^{2-\beta}\tilde{Q}_{2}^{k}(x,\theta)+\varepsilon^{2}\tilde{Q}_{3}^{k}(x,\theta)+\varepsilon^{2+\beta}\tilde{Q}_{4}^{k}(x,\theta)+\varepsilon^{2+2\beta}\tilde{Q}_{5}^{k}(x,\theta),$
	$\displaystyle\tilde{\sigma}_{k}^{2}(x,\theta)=$	$\displaystyle\varepsilon D_{x}\sigma_{k}^{2}V_{k}+D_{\theta}\sigma_{k}^{2}\sigma_{k}^{1}$
	$\displaystyle=$	$\displaystyle\varepsilon^{2-2\beta}D_{\theta}P_{1}^{k}Q_{1}^{k}+\varepsilon^{2-\beta}(D_{x}P_{1}^{k}V_{k}+D_{\theta}P_{1}^{k}Q_{2}^{k}+D_{\theta}P_{2}^{k}Q_{1}^{k})$
		$\displaystyle+\varepsilon^{2}(D_{x}P_{2}^{k}V_{k}+D_{\theta}P_{1}^{k}Q_{3}^{k}+D_{\theta}P_{3}^{k}Q_{1}^{k})$
		$\displaystyle+\varepsilon^{2+\beta}(D_{x}P_{5}^{k}V_{k}+D_{\theta}P_{2}^{k}Q_{3}^{k}+D_{\theta}P_{3}^{k}Q_{2}^{k})+\varepsilon^{2+2\beta}D_{\theta}P_{3}^{k}Q_{3}^{k}$
	$\displaystyle\triangleq$	$\displaystyle\varepsilon^{2-2\beta}\tilde{P}_{1}^{k}(x,\theta)+\varepsilon^{2-\beta}\tilde{P}_{2}^{k}(x,\theta)+\varepsilon^{2}\tilde{P}_{3}^{k}(x,\theta)+\varepsilon^{2+\beta}\tilde{P}_{4}^{k}(x,\theta)+\varepsilon^{2+2\beta}\tilde{P}_{5}^{k}(x,\theta).$

Based on Pinsky-Wihstutz transformation (3.9), we now proceed to estimate the Lyapunov exponent for the linearized version of the perturbed Hamiltonian system (3.1). We first note that, for the process $\{(x_{t},\theta_{t}):t\geqslant 0\}$ given by (2.2) and (3.17), its generator $\mathcal{L}_{\varepsilon}$ can be written as

	$\displaystyle(\mathcal{L}_{\varepsilon}f)(x,\theta)=$	$\displaystyle\bigg{[}U_{1}(x)+\frac{1}{2}\varepsilon^{2}\sum_{k=1}^{d}(DV_{k}V_{k})(x)\bigg{]}\cdot\nabla_{x}f(x,\theta)$
		$\displaystyle+\bigg{[}-\varepsilon^{\beta}A(x)\sin^{2}\theta+\frac{1}{2}\sum_{k=1}^{d}\tilde{\sigma}_{k}^{1}(x,\theta)\bigg{]}\frac{\partial}{\partial\theta}f(x,\theta)$
		$\displaystyle+\frac{1}{2}\sum_{k=1}^{d}\bigg{(}\varepsilon^{2}\text{Tr}[V_{k}^{T}V_{k}\nabla_{x}\nabla_{x}]+2\varepsilon V_{k}\sigma_{k}^{1}(x,\theta)\cdot\nabla_{x}\frac{\partial}{\partial\theta}+({\sigma}_{k}^{1}(x,\theta))^{2}\frac{\partial^{2}}{\partial^{2}\theta}\bigg{)}f(x,\theta)$
		$\displaystyle+\int_{\|z\|<1}\Bigg{\{}f\big{(}\xi^{\varepsilon}(z)(x),\zeta_{1}^{\varepsilon}(z)(x,\theta)\big{)}-f(x,\theta)$
		$\displaystyle\;\;\;-\sum_{k=1}^{d}z_{k}\Big{[}\varepsilon V_{k}(x)\cdot\nabla_{x}f(x,\theta)+\sigma_{k}^{1}(x,\theta)\frac{\partial}{\partial\theta}f(x,\theta)\Big{]}\Bigg{\}}\nu(dz),$		(3.19)

for each $f\in C_{b}^{2}(M\times S^{1})$ (i.e., $f$ is a $C^{2}$ and bounded function). Referring to [3, 1] and keeping assumptions (A1)-(A2) in mind, we assume throughout the paper the following:

1. (A3)

   For each sufficiently small $\varepsilon>0$, the process $\{(x_{t},\theta_{t}):t\geqslant 0\}$ given by (2.2) and (3.17) is a positive recurrent diffusion process on $M\times S^{1}$ with a (unique) stationary probability $\mu^{\varepsilon}$. We write $\mu_{M}^{\epsilon}$ for the $M$ marginal of $\mu^{\varepsilon}$.

From the equation (3.18) for $\rho_{t}$, we observe that the leading drift term (with respect to $\varepsilon$) of this equation, formally, not only depends on the value of $\beta$, but also may contain contributions from the $\nu$-integral term. More specifically, it may come from the following terms: $\varepsilon^{\beta}A(x)\sin\theta\cos\theta$, $\frac{\varepsilon^{2-2\beta}}{2}\sum_{k=1}^{d}\tilde{P}_{1}^{k}(x,\theta)$ and $I_{\rho}^{\varepsilon}\triangleq\int_{\|z\|<c}\left[\zeta_{2}^{\varepsilon}(z)(x,\theta)-\sum_{k=1}^{d}z_{k}\sigma_{k}^{2}(x,\theta)\right]\nu(dz)$.

As mentioned in Remark 3.2, for pure Brownian case, it has been verified that $\beta=2/3$, so the leading drift term in pure Brownian case should be

		$\displaystyle\varepsilon^{2/3}\Big{(}A(x)\sin\theta\cos\theta+1/2\sum_{k=1}^{d}\tilde{P}_{1}^{k}(x,\theta)\Big{)}$
		$\displaystyle=\varepsilon^{2/3}\Big{(}A(x)\sin\theta\cos\theta+\sum_{k=1}^{d}D_{k}^{2}(x)(\frac{1}{2}\cos^{2}\theta-\sin^{2}\theta\cos^{2}\theta)\Big{)}.$		(3.20)

We next try to analyze the leading term of $I_{\rho}^{\epsilon}$ formally. By equations (2.4) and (3.15), and following calculations in the proof for Lemma 6.10.3 of [4], we conclude that

	$\displaystyle I_{\rho}^{\varepsilon}(x,\theta)=$	$\displaystyle\int_{\|z\|<c}\left[\int_{0}^{1}\sum_{k=1}^{d}z_{k}\sigma_{k}^{2}\big{(}\xi^{\varepsilon}(\tau z)(x),\zeta_{1}^{\varepsilon}(\tau z)(x,\theta)\big{)}d\tau-\sum_{k=1}^{d}z_{k}\sigma_{k}^{2}(x,\theta)\right]\nu(dz)$
	$\displaystyle=$	$\displaystyle\int_{\|z\|<c}\int_{0}^{1}\int_{0}^{a}\sum_{k,l=1}^{d}z_{k}z_{l}\big{(}\varepsilon D_{x}\sigma_{k}^{2}V_{l}+D_{\theta}\sigma_{k}^{2}\sigma_{l}^{1}\big{)}\big{(}\xi^{\varepsilon}(bz)(x),\zeta_{1}^{\varepsilon}(bz)(x,\theta)\big{)}dbda\;\nu(dz).$

Note that the expression for $\big{(}\varepsilon D_{x}\sigma_{k}^{2}V_{l}+D_{\theta}\sigma_{k}^{2}\sigma_{l}^{1}\big{)}(x,\theta)$ is quite similar to that of $\tilde{\sigma}_{k}^{2}(x,\theta)$. We adopt the following notation

$\displaystyle I_{\rho}^{\varepsilon}(x,\theta)\triangleq\int_{\|z\|<c}\left[\varepsilon^{2-2\beta}R_{0}^{\varepsilon}+\varepsilon^{2-\beta}R_{1}^{\varepsilon}+\varepsilon^{2}R_{2}^{\varepsilon}+\varepsilon^{2+\beta}R_{3}^{\varepsilon}+\varepsilon^{2+2\beta}R_{4}^{\varepsilon}\right](z)(x,\theta)\nu(dz),$

(3.21)

where, with $Int[f]\triangleq\int_{0}^{1}\int_{0}^{a}\sum_{k,l=1}^{d}z_{k}z_{l}f(\xi^{\varepsilon}(bz),\zeta_{1}^{\varepsilon}(bz))dbda$,

	$\displaystyle R_{0}^{\varepsilon}(z)$	$\displaystyle=Int[D_{\theta}P_{1}^{k}Q_{1}^{l}],\;\;\;$
	$\displaystyle R_{1}^{\varepsilon}(z)$	$\displaystyle=Int[D_{x}P_{1}^{k}V_{l}+D_{\theta}P_{1}^{k}Q_{2}^{l}+D_{\theta}P_{2}^{k}Q_{1}^{l}]$
	$\displaystyle R_{2}^{\varepsilon}(z)$	$\displaystyle=Int[D_{x}P_{2}^{k}V_{l}+D_{\theta}P_{1}^{k}Q_{3}^{l}+D_{\theta}P_{3}^{k}Q_{1}^{l}],\;\;\;$
	$\displaystyle R_{3}^{\varepsilon}(z)$	$\displaystyle=Int[D_{x}P_{5}^{k}V_{l}+D_{\theta}P_{2}^{k}Q_{3}^{l}+D_{\theta}P_{3}^{k}Q_{2}^{l}]$
	$\displaystyle R_{4}^{\varepsilon}(z)$	$\displaystyle=Int[D_{\theta}P_{3}^{k}Q_{3}^{l}].$

In particular,

$\displaystyle R_{0}^{\varepsilon}(z)=\int_{0}^{1}\int_{0}^{a}\left(\sum_{k=1}^{d}z_{k}D_{k}\big{(}\xi^{\varepsilon}(bz)\big{)}\right)^{2}\left[\frac{1}{2}\cos^{2}\big{(}\zeta_{1}^{\varepsilon}(bz)\big{)}-\sin^{2}\big{(}\zeta_{1}^{\varepsilon}(bz)\big{)}\cos^{2}\big{(}\zeta_{1}^{\varepsilon}(bz)\big{)}\right]dbda.$

(3.22)

Since $R_{i}^{\varepsilon}$, $i=0,1,\cdots,4$, are functions depending on the value of $\varepsilon$, the integral with respect to jump measure, $I_{\rho}$, cannot necessarily be expanded a finite linear combination of terms of size $\varepsilon^{\gamma}$ for various $\gamma$.