Stochastic Limit-Cycle Oscillations of a Nonlinear System Under Random Perturbations

Abstract

Dynamical systems with $\varepsilon$ small random perturbations appear in both continuous mechanical motions and discrete stochastic chemical kinetics. The present work provides a detailed analysis of the central limit theorem (CLT), with a time-inhomogeneous Gaussian process, near a deterministic limit cycle in $\mathbb {R}^n$. Based on respectively the theory of random perturbations of dynamical systems and the WKB approximation that codes the large deviations principle (LDP), results are developed in parallel from both standpoints of stochastic trajectories and transition probability density and their relations are elucidated. We show rigorously the correspondence between the local Gaussian fluctuations and the curvature of the large deviation rate function near its infimum, connecting the CLT and the LDP of diffusion processes. We study uniform asymptotic behavior of stochastic limit cycles through the interchange of limits of time $t\rightarrow \infty$ and $\varepsilon \rightarrow 0$. Three further characterizations of stochastic limit cycle oscillators are obtained: (i) An approximation of the probability flux near the cycle; (ii) Two special features of the vector field for the cyclic motion; (iii) A local entropy balance equation along the cycle with clear physical meanings. Lastly and different from the standard treatment, the origin of the $\varepsilon$ in the theory is justified by a novel scaling hypothesis via constructing a sequence of stochastic differential equations.

Appendix

1.1 Proofs of Lemmas 2.2 and 2.3

1.1.1 Proof of Lemma 2.2

Proof

For the special case of $\mathbf{x}\in \mathbb {R}$, $\varTheta =3$, $\varLambda =15$, and $\varvec{\varXi }=[h''(\mathbf{x}^*)]^{-1}$. Then Eq. (2.25)

$$\begin{aligned} \eta (\mathbf{x}^*) = \frac{f''(\mathbf{x}^*)}{2h''(\mathbf{x})}-\left[ \frac{f'(\mathbf{x}^*)h'''(\mathbf{x}^*)}{2[h''(\mathbf{x}^*)]^2}+ \frac{f(\mathbf{x}^*)h''''(\mathbf{x}^*)}{8[h''(\mathbf{x}^*)]^2}\right] + \frac{5f(\mathbf{x}^*)[h'''(\mathbf{x}^*)]^2}{24[h'''(\mathbf{x}^*)]^3}. \end{aligned}$$

This result can be found on P. 273 of [53], Eq. (6.4.35).

For the general case,

$$\begin{aligned}&\int _{\mathbb {R}^n} f(\mathbf{x})e^{-\frac{h(\mathbf{x})}{\varepsilon }}\mathrm{d}\mathbf{x}\nonumber \\&\quad = \int _{\mathbb {R}^n} \left[ f(\mathbf{x}^*) + (\mathbf{x}-\mathbf{x}^*)\cdot \nabla f(\mathbf{x}^*) +\frac{(\mathbf{x}-\mathbf{x}^*)^T\nabla \nabla f(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*)}{2} + \dots \right] \nonumber \\&\qquad \times \exp \left[ -\frac{h(\mathbf{x}^*)}{\varepsilon }-\frac{(\mathbf{x}-\mathbf{x}^*)_ih''_{ij}(\mathbf{x}^*) (\mathbf{x}-\mathbf{x}^*)_j}{2\varepsilon } -\frac{h'''_{ijk}(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k}{6\varepsilon } \right. \nonumber \\&\qquad - \left. \frac{h''''_{ijk\ell }(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k(\mathbf{x}-\mathbf{x}^*)_{\ell }}{24\varepsilon } + \cdots \right] \mathrm{d}\mathbf{x}\nonumber \\&\quad = e^{-\frac{h(\mathbf{x}^*)}{\varepsilon }} \int _{\mathbb {R}^N} \left[ f(\mathbf{x}^*) +(\mathbf{x}-\mathbf{x}^*)\cdot \nabla f(\mathbf{x}^*) + \frac{(\mathbf{x}-\mathbf{x}^*)^T\nabla \nabla f(\mathbf{x}^*) (\mathbf{x}-\mathbf{x}^*)}{2} + \dots \right] \nonumber \\&\qquad \times \left[ 1 -\frac{h'''_{ijk}(\mathbf{x}^*)}{6\varepsilon }(\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k - \frac{h''''_{ijk\ell }(\mathbf{x}^*)}{24\varepsilon }(\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j \right. \nonumber \\&\qquad \times \left. (\mathbf{x}-\mathbf{x}^*)_k(\mathbf{x}-\mathbf{x}^*)_{\ell } + \frac{\big [h'''_{ijk}(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*)_i(\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k\big ]^2}{72\varepsilon ^2} +\cdots \right] \nonumber \\&\qquad \times e^{-\frac{ (\mathbf{x}-\mathbf{x}^*)^T\nabla \nabla h(\mathbf{x}^*) (\mathbf{x}-\mathbf{x}^*)}{2\varepsilon } } \mathrm{d}\mathbf{x}\nonumber \\&\quad = e^{-\frac{h(\mathbf{x}^*)}{\varepsilon }} \int _{\mathbb {R}^n} e^{-\frac{(\mathbf{x}-\mathbf{x}^*)^T\nabla \nabla h(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*) }{2\varepsilon } } \left[ f(\mathbf{x}^*) + (\mathbf{x}-\mathbf{x}^*)\cdot \nabla f(\mathbf{x}^*) \right. \nonumber \\&\qquad + \frac{ (\mathbf{x}-\mathbf{x}^*)^T\nabla \nabla f(\mathbf{x}^*) (\mathbf{x}-\mathbf{x}^*)}{2} - \frac{ f(\mathbf{x}^*)h'''_{ijk}(\mathbf{x}^*)}{6\varepsilon }(\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k \nonumber \\&\qquad - \left( \frac{f'_i(\mathbf{x}^*)h'''_{jk\ell }(\mathbf{x}^*)}{6\varepsilon } + \frac{f(\mathbf{x}^*)h''''_{ijk\ell }(\mathbf{x}^*)}{24\varepsilon } \right) (\mathbf{x}-\mathbf{x}^*)_i (\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k(\mathbf{x}-\mathbf{x}^*)_{\ell } \nonumber \\&\qquad + \left. \frac{f(\mathbf{x}^*)[h'''_{ijk}(\mathbf{x}^*)(\mathbf{x}-\mathbf{x}^*)_i(\mathbf{x}-\mathbf{x}^*)_j(\mathbf{x}-\mathbf{x}^*)_k ]^2}{72\varepsilon ^2} + \cdots \right] \mathrm{d}\mathbf{x}\nonumber \\&\quad = \sqrt{\frac{(2\pi \varepsilon )^N}{\det \big [\nabla \nabla h(x^*)\big ]}} e^{-\frac{h(\mathbf{x}^*)}{\varepsilon }} \left\{ f(\mathbf{x}^*) + \frac{\varepsilon f''_{ij}(\mathbf{x}^*)}{2}\varXi _{ij}(\mathbf{x}^*) \right. \nonumber \\&\qquad - \varepsilon \left[ \frac{f'_i(\mathbf{x}^*)h'''_{jk\ell }(\mathbf{x}^*)}{6} + \frac{f(\mathbf{x}^*)h''''_{ijk\ell }(x^*)}{24} \right] \varXi ^{\frac{1}{2}}_{i\mu } \varXi ^{\frac{1}{2}}_{j\nu }\varXi ^{\frac{1}{2}}_{k\rho } \varXi ^{\frac{1}{2}}_{\ell \kappa }\varTheta _{\mu \nu \rho \kappa } \nonumber \\&\qquad + \left. \varepsilon \left( \frac{f(\mathbf{x}^*)[h'''_{ijk}(\mathbf{x}^*)]^2}{72} \right) \varXi ^{-\frac{1}{2}}_{i\mu } \varXi ^{-\frac{1}{2}}_{i\mu '}\varXi ^{-\frac{1}{2}}_{j\nu }\varXi ^{-\frac{1}{2}}_{j\nu '}\varXi ^{-\frac{1}{2}}_{k\rho }\varXi ^{-\frac{1}{2}}_{k\rho '} \varLambda _{\mu \mu '\nu \nu '\rho \rho '} + \cdots \right\} . \end{aligned}$$

(6.1)

A multivariate normal distribution with covariance matrix $\varvec{\varXi }$, which is positive definite thus $\varvec{\varXi }=\varvec{\varXi }^{\frac{1}{2}}\varvec{\varXi }^{\frac{T}{2}}$ [70], has

$$\begin{aligned}&\frac{1}{\big [ \big (2\pi \varepsilon \big )^N \det \big (\varvec{\varXi }\big )\big ]^{\frac{1}{2}} } \int _{\mathbb {R}^n} f''_{ij}(\mathbf{0})x_ix_j \exp \left[ -\frac{1}{2\varepsilon }\mathbf{x}^T \varvec{\varXi }^{-1} \mathbf{x}\right] \mathrm{d}\mathbf{x}\\&\quad = \frac{\varepsilon \varXi ^{\frac{1}{2}}_{i\nu } \varXi ^{\frac{1}{2}}_{j\mu } f''_{ij}(\mathbf{0})}{ (2\pi )^{N/2} } \int _{\mathbb {R}^n}y_{\nu }y_{\mu } \exp \left[ -\frac{\mathbf{y}^T\mathbf{y}}{2} \right] \mathrm{d}\mathbf{y}= \varepsilon f''_{ij}(\mathbf{0})\varXi _{ij}, \end{aligned}$$

the Frobenius product of the Hessian matrix and covariant matrix $\varvec{\varXi }$,

$$\begin{aligned}&\Big [ \big (2\pi \varepsilon \big )^n \det \big (\varvec{\varXi }\big )\Big ]^{-\frac{1}{2}} \int _{\mathbb {R}^n} f'''_{ijk}(\mathbf{0})x_ix_jx_k \exp \left[ -\frac{1}{2\varepsilon }\mathbf{x}^T \varvec{\varXi }\mathbf{x}\right] \mathrm{d}\mathbf{x}= 0, \\&\Big [ \big (2\pi \varepsilon \big )^n \det \big (\varvec{\varXi }\big )\Big ]^{-\frac{1}{2}} \int _{\mathbb {R}^n} f''''_{ijk\ell }(\mathbf{0})x_ix_jx_kx_{\ell } \exp \left[ -\frac{1}{2\varepsilon }\mathbf{x}^T \varvec{\varXi }\mathbf{x}\right] \mathrm{d}\mathbf{x}\\&\quad = \varepsilon ^2 f''''_{ijk\ell }(\mathbf{0}) \varXi ^{\frac{1}{2}}_{i\mu }\varXi ^{\frac{1}{2}}_{j\nu } \varXi ^{\frac{1}{2}}_{k\rho }\varXi ^{\frac{1}{2}}_{\ell \kappa } \varTheta _{\mu \nu \rho \kappa },\\&\Big [ \big (2\pi \varepsilon \big )^n \det \big (\varvec{\Sigma }\big )\Big ]^{-\frac{1}{2}} \int _{\mathbb {R}^n} f'''_{ijk}(\mathbf{0})x_i^2x_j^2x_k^2 \exp \left[ -\frac{1}{2\varepsilon }\mathbf{x}^T \varvec{\varXi }\mathbf{x}\right] \mathrm{d}\mathbf{x}\\= & {} \varepsilon ^3\big (2\pi \big )^{-\frac{n}{2}} f'''_{ijk}(\mathbf{0}) \varXi ^{-\frac{1}{2}}_{i\mu } \varXi ^{-\frac{1}{2}}_{i\mu '}\varXi ^{-\frac{1}{2}}_{j\nu }\varXi ^{-\frac{1}{2}}_{j\nu '}\varXi ^{-\frac{1}{2}}_{k\rho }\varXi ^{-\frac{1}{2}}_{k\rho '} \int _{\mathbb {R}^n} y_{\mu }y_{\mu '}y_{\nu }y_{\nu '}y_{\rho } y_{\rho '} \exp \left[ -\frac{\mathbf{y}^T\mathbf{y}}{2}\ \right] \mathrm{d}\mathbf{x}\\= & {} \varepsilon ^3 f'''_{ijk}(\mathbf{0}) \varXi ^{-\frac{1}{2}}_{i\mu } \varXi ^{-\frac{1}{2}}_{i\mu '}\varXi ^{-\frac{1}{2}}_{j\nu }\varXi ^{-\frac{1}{2}}_{j\nu '}\varXi ^{-\frac{1}{2}}_{k\rho }\varXi ^{-\frac{1}{2}}_{k\rho '} \varLambda _{\mu \mu '\nu \nu '\rho \rho '}. \end{aligned}$$

Applying (2.23) to both numerator and denominator of the lhs of (2.24),

$$\begin{aligned}&\frac{ f(\mathbf{x}^*)+\varepsilon \left\{ \begin{array}{c} \frac{f''_{ij}(\mathbf{x}^*)\varXi _{ij}}{2}-\left[ \frac{f'_i(\mathbf{x}^*)h'''_{jk\ell }(\mathbf{x}^*)}{6}+ \frac{f_{\alpha }(\mathbf{x}^*)h''''_{ijk\ell }(\mathbf{x}^*)}{24}\right] \varXi ^{\frac{1}{2}}_{i\mu } \varXi ^{\frac{1}{2}}_{j\nu }\varXi ^{\frac{1}{2}}_{k\rho } \varXi ^{\frac{1}{2}}_{\ell \kappa }\varTheta _{\mu \nu \rho \kappa } \\ + \frac{f(\mathbf{x}^*)[h'''_{ijk}(\mathbf{x}^*)]^2}{72} \varXi ^{-\frac{1}{2}}_{i\mu } \varXi ^{-\frac{1}{2}}_{i\mu '}\varXi ^{-\frac{1}{2}}_{j\nu }\varXi ^{-\frac{1}{2}}_{j\nu '}\varXi ^{-\frac{1}{2}}_{k\rho }\varXi ^{-\frac{1}{2}}_{k\rho '} \varLambda _{\mu \mu '\nu \nu '\rho \rho '} \end{array} \right\} + O(\varepsilon ^2)}{ 1+\varepsilon \left\{ \begin{array}{c} -\left[ \frac{h''''_{ijk\ell }(\mathbf{x}^*)}{24}\right] \varXi ^{\frac{1}{2}}_{i\mu } \varXi ^{\frac{1}{2}}_{j\nu }\varXi ^{\frac{1}{2}}_{k\rho } \varXi ^{\frac{1}{2}}_{\ell \kappa }\varTheta _{\mu \nu \rho \kappa } \\ + \frac{[h'''_{ijk}(\mathbf{x}^*)]^2}{72} \varXi ^{-\frac{1}{2}}_{i\mu } \varXi ^{-\frac{1}{2}}_{i\mu '}\varXi ^{-\frac{1}{2}}_{j\nu }\varXi ^{-\frac{1}{2}}_{j\nu '}\varXi ^{-\frac{1}{2}}_{k\rho }\varXi ^{-\frac{1}{2}}_{k\rho '} \varLambda _{\mu \mu '\nu \nu '\rho \rho '} \end{array} \right\} + O(\varepsilon ^2)}\\&\quad = f(\mathbf{x}^*) + \varepsilon \left[ \frac{f''_{ij}(\mathbf{x}^*)\varXi _{ij}}{2}- \frac{f'_i(\mathbf{x}^*)h'''_{jk\ell }(\mathbf{x}^*)\varXi ^{\frac{1}{2}}_{i\mu } \varXi ^{\frac{1}{2}}_{j\nu }\varXi ^{\frac{1}{2}}_{k\rho } \varXi ^{\frac{1}{2}}_{\ell \kappa }\varTheta _{\mu \nu \rho \kappa }}{6} \right] + O(\varepsilon ^2). \end{aligned}$$

1.1.2 Proof of Lemma 2.3

We only provide the proof for the case $\mathbf{x}\in \mathbb {R}^1$, which is denoted by x. For higher dimensions, results are the same by using the notations from Lemma 2.2.

Proof

Let the global minimum of $[h(x)-\varepsilon \ln g(x)]$ be at $\tilde{x}^*=x^*+\varDelta x(\varepsilon )$. Clearly, $\varDelta x\rightarrow 0$ as $\varepsilon \rightarrow 0$. In fact,

$$\begin{aligned}&\left[ h'(x)-\varepsilon \left( \frac{g'(x)}{g(x)}\right) \right] _{x=x^*+\varDelta x} = 0,\\&\quad h'(x^*)+h''(x^*)\varDelta x -\varepsilon \left( \frac{g'(x^*)}{g(x^*)}\right) = 0,\\&\quad \varDelta x = \varepsilon \left( \frac{g'(x^*)}{g(x^*)h''(x^*)}\right) +O\big (\varepsilon ^2\big ). \end{aligned}$$

Now we apply the Eq. (2.24) in Lemma 2.2:

$$\begin{aligned}&\frac{\displaystyle \int _{-\infty }^{\infty } f(x)g(x)e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } g(x) e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x } = \frac{\displaystyle \int _{-\infty }^{\infty } f(x)e^{-\frac{h(x)-\varepsilon \ln g(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } e^{-\frac{h(x)-\varepsilon \ln g(x)}{\varepsilon }}\mathrm{d}x } \nonumber \\&\quad = f( \tilde{x}^*) + \varepsilon \left[ \frac{f''(\tilde{x}^*)}{2h''(\tilde{x}^*)}- \frac{f'(\tilde{x}^*)h'''(\tilde{x}^*)}{2[h''(\tilde{x}^*)]^2} \right] + O(\varepsilon ^2), \nonumber \\&\quad = f(x^*) +\varepsilon \underbrace{ \left( \frac{ f'(x^*)g'(x^*)}{g(x^*)h''(x^*)} \right. }_{ f'(x^*)\varDelta x\text { due to } g(x) } + \underbrace{ \frac{f''(x^*)}{2h''(x^*)} }_{ \text { due to } f(x) } - \underbrace{ \left. \frac{f'(x^*)h'''(x^*)}{2[h''(x^*)]^2} \right) }_{ \text { due to non-quadratic } h(x) } + O(\varepsilon ^2). \end{aligned}$$

(6.2)

For the terms on the order of $\varepsilon$, replacing $\tilde{x}^*$ by $x^*$ only affects the order $\varepsilon ^2$ term.

1.1.3 Proof of Lemma 6.1

Here we provide an additional Lemma, which is not used in the present work, but it is useful for related fields.

Lemma 6.1

$$\begin{aligned} \frac{\displaystyle \int _{-\infty }^{\infty } f^2(x)g(x)e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } g(x) e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x } - \left[ \frac{\displaystyle \int _{-\infty }^{\infty } f(x)g(x)e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } g(x) e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x } \right] ^2 = \varepsilon \left( \frac{f'^2(x^*)}{h''(x^*)} \right) + O\big (\varepsilon ^2\big ). \end{aligned}$$

(6.3)

Proof

$$\begin{aligned}&\frac{\displaystyle \int _{-\infty }^{\infty } f^2(x)g(x)e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } g(x) e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x } - \left[ \frac{\displaystyle \int _{-\infty }^{\infty } f(x)g(x)e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x}{ \displaystyle \int _{-\infty }^{\infty } g(x) e^{-\frac{h(x)}{\varepsilon }}\mathrm{d}x } \right] ^2 \nonumber \\&\quad = \varepsilon \left( \frac{ 2f(x^*)f'(x^*)g'(x^*)}{g(x^*)h''(x^*)}+ \frac{2f'^2(x^*)+2f(x^*)f''(x^*) }{2h''(x^*)}-\frac{2f(x^*)f'(x^*)h'''(x^*)}{2[h''(x^*)]^2} \right) \nonumber \\&\qquad - 2\varepsilon f(x^*) \left( \frac{ f'(x^*)g'(x^*)}{g(x^*)h''(x^*)}+ \frac{ f''(x^*) }{2h''(x^*)}-\frac{f'(x^*)h'''(x^*)}{2[h''(x^*)]^2} \right) + O\big (\varepsilon ^2\big ) \nonumber \\&\quad = \varepsilon \left( \frac{f'^2(x^*)}{h''(x^*)} \right) + O\big (\varepsilon ^2\big ). \end{aligned}$$

(6.4)

1.2 Proof of Theorem 3.3

Proof

Given $\mathbf {x_1} \in \varGamma$, let $\mathcal {P}_1$ be a plane containing $\mathbf {x_1}$ and perpendicular to the vector $\varvec{{\varvec{\gamma }}}(\mathbf {x_1})$. Given another point $\mathbf {x}_2 \in \varGamma$, let $\mathcal {P}_2$ be a plane perpendicular to the vector $\varvec{{\varvec{\gamma }}}(\mathbf {x}_2)$. Let $\mathcal {S}_1 \subset \mathcal {P}_1$ and $\mathcal {S}_2 \subset \mathcal {P}_2$ be compact sets such that

$$\begin{aligned} \max _{\mathbf {x}, \mathbf {y} \in \mathcal {S}_1} ||\mathbf {x} - \mathbf {y} ||= \max _{\mathbf {x}, \mathbf {y} \in \mathcal {S}_2} ||\mathbf {x} - \mathbf {y} ||= \delta >0. \end{aligned}$$

(6.5)

We then can define a tube $\varPhi (\delta )$ with two side boundaries $\mathcal {S}_1$ and $\mathcal {S}_2$ and $\varGamma \subset \varPhi (\delta )$.

Recall that $\varvec{\gamma }_\varepsilon ( \mathbf {x}) = \pi _\varepsilon (\mathbf {x})^{-1} \mathbf {J}[ \pi _\varepsilon (\mathbf {x}) ]$. By the stationary Fokker-Planck equation, we have that

$$\begin{aligned} \nabla \cdot ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x})) = 0, \quad \text {for all} \ \mathbf {x} \in \mathbb {R}^n. \end{aligned}$$

(6.6)

Furthermore, by the Gauss’s theorem,

$$\begin{aligned} \int _{\mathcal {S}} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S} = \int _{\mathcal {V}} \big ( \nabla \cdot (\varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \big ) \mathrm{d}\mathcal {V} = 0, \end{aligned}$$

(6.7)

where $\mathcal {S}$ in the surface of $\varPhi (\delta )$, $\mathbf{n}$ is the outward normal vector to $\mathcal {S}$, and $\mathcal {V}$ is the volume of $\varPhi (\delta )$.

For the left hand side of Eq. (6.7), it can be written as a sum of three terms

$$\begin{aligned} \int _{\mathcal {S}} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S} =&\int _{\mathcal {S}_1} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_1 + \int _{\mathcal {S}_2} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_2 \nonumber \\&+ \int _{\mathcal {S}_3} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_3 , \end{aligned}$$

(6.8)

in which $\mathcal {S}_3$ is the lateral surface of $\varPhi (\delta )$ and $\mathcal {S}_3 \cap \varGamma = \emptyset$.

For every $\mathbf {y} \in \varGamma$, we denote $\mathcal {S}_\mathbf{y}:=\varPhi (\delta ) \cap \mathcal {P}_\mathbf{y}$, $\mathcal {P}_\mathbf{y}$ is the plane perpendicular to ${\varvec{\gamma }}(\mathbf{y})$. By Lemma 2.2,

$$\begin{aligned} \frac{ \int _{\mathcal {S}_\mathbf {y}} f(\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf {y}}{\int _{\mathcal {S}_\mathbf {y}} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf {y}} = f(\mathbf {y}) + O(\varepsilon ), \end{aligned}$$

(6.9)

for any continuous and bounded function $f: \mathbb {R}^n \rightarrow \mathbb {R}$. Furthermore, by the definition of function v in Lemma 3.3, we can approximate the ratio of two integrals

$$\begin{aligned} \frac{ \int _{\mathcal {S}_\mathbf {y}} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf {y} }{\int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } = \frac{\sqrt{2\pi \varepsilon v(\mathbf {y}) } e^{ \frac{-\varphi (\mathbf{y}) }{\varepsilon }} }{ \sqrt{2\pi \varepsilon v(\mathbf{x}) } e^{ \frac{-\varphi (\mathbf{x}_1) }{\varepsilon }} } + O(\varepsilon ) = \frac{ \sqrt{v(\mathbf {y})} }{ \sqrt{v(\mathbf {x}_1)} } + O(\varepsilon ), \end{aligned}$$

(6.10)

in which we use Laplace’s method in the first equality and $\varphi \equiv 0$ on $\varGamma$ in the second equality. To choose $f(\mathbf {x}) = \omega ({\mathbf {x}})$ for Eq. (6.9), combined with the result of (6.10), we can obtain

$$\begin{aligned} \frac{ \int _{\mathcal {S}_\mathbf {y}} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf {y} }{\int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } = \frac{ \omega (\mathbf {y})\sqrt{v(\mathbf {y})} }{ \sqrt{v(\mathbf {x}_1)} } + O(\varepsilon ). \end{aligned}$$

(6.11)

Since the function $e^{ -\frac{\varphi (\mathbf {x}) }{\varepsilon }}$ is concentrated near $\varGamma$, we can further approximate the normalization factor $\int _{\mathbb {R}^n} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathbf {x}$ as follows

$$\begin{aligned} \frac{\int _{\mathbb {R}^n} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathbf {x}}{\int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } = \frac{\int _{\varGamma } \int _{\mathcal {S}_\mathbf {y}} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf {y} \mathrm{d}\mathbf {\mathbf {y}} }{\int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } + O(\varepsilon ). \end{aligned}$$

(6.12)

By (6.11) and (6.12), we have that

$$\begin{aligned} \frac{\int _{\mathbb {R}^n} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathbf {x}}{\int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } = \frac{\int _{\varGamma }\omega (\mathbf {y}) \sqrt{v(\mathbf {y})} \mathrm{d}\mathbf {\mathbf {y}} }{\sqrt{v(\mathbf {x}_1)}} + O(\varepsilon ). \end{aligned}$$

(6.13)

By the WKB expansion of $\pi _\varepsilon$ in Eq. (3.6), with Eq. (6.13), the first term on the right hand side of Eq. (6.8) can be written as

$$\begin{aligned} \int _{\mathcal {S}_1} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_1&= \frac{\int _{\mathcal {S}_1} \left( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \cdot \mathbf {n} \right) \omega (\mathbf {x}) e^{ \frac{- \varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 }{ \int _{\mathbb {R}^n} \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathbf {x} } \nonumber \\&= \left( \frac{\sqrt{v(\mathbf {x}_1)}}{ \int _{\varGamma } \omega (\mathbf {y}) \sqrt{v(\mathbf {y})} \mathrm{d}\mathbf {y}} \right) \left( \frac{ \int _{\mathcal {S}_1} \left( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \cdot \mathbf {n} \right) \omega (\mathbf {x}) e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 }{ \int _{\mathcal {S}_1} e^{ \frac{-\varphi (\mathbf {x}) }{\varepsilon }} \mathrm{d}\mathcal {S}_1 } \right) + O(\varepsilon ) . \end{aligned}$$

(6.14)

Note that $\varvec{\gamma }_\varepsilon (\mathbf {x}) \rightarrow \varvec{\gamma }(\mathbf {x})$. Without loss of generality, we assume that $\varvec{\gamma }(\mathbf {x}_1)$ is inflow and $\varvec{\gamma }(\mathbf {x}_2)$ is outflow of $\varPhi (\delta )$. To choose $f(\mathbf {x}) = \left( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \cdot \mathbf {n} \right) \omega (\mathbf {x})$ for Eq. (6.9), combined with Eq. (6.14), we then obtain

$$\begin{aligned} \int _{\mathcal {S}_1} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_1 \rightarrow - C \sqrt{v(\mathbf {x}_1)} \omega (\mathbf {x}_1) ||\varvec{\gamma }(\mathbf {x}_1) ||\quad \text {as} \quad \varepsilon \rightarrow 0, \end{aligned}$$

(6.15)

in which the constant $C = 1/ \int _{\varGamma } \omega (\mathbf {y}) \sqrt{v(\mathbf {y})} \mathrm{d}\mathbf {y}$. By the same approach, the second term on the right hand side of Eq. (6.8) has a convergence

$$\begin{aligned} \int _{\mathcal {S}_2} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_2 \rightarrow C \sqrt{v(\mathbf {x}_2)} \omega (\mathbf {x}_2) ||\varvec{\gamma }(\mathbf {x}_2) ||\quad \text {as} \quad \varepsilon \rightarrow 0. \end{aligned}$$

(6.16)

Since $\mathcal {S}_3 \cap \varGamma = \emptyset$, the third term

$$\begin{aligned} \int _{\mathcal {S}_3} \big ( \varvec{\gamma }_\varepsilon ( \mathbf {x}) \pi _\varepsilon (\mathbf {x}) \cdot \mathbf {n} \big ) \mathrm{d}\mathcal {S}_3 \rightarrow 0 \quad \text {as} \quad \varepsilon \rightarrow 0. \end{aligned}$$

(6.17)

To apply the results (6.15), (6.16), and (6.17) to the equations (6.7) and (6.8), we can show that

$$\begin{aligned} \Big | C \sqrt{v(\mathbf {x}_1)} \omega (\mathbf {x}_1) ||\varvec{\gamma }(\mathbf {x}_1) ||- C \sqrt{v(\mathbf {x}_2)} \omega (\mathbf {x}_2) ||\varvec{\gamma }(\mathbf {x}_2) ||\Big | = 0. \end{aligned}$$

(6.18)

Since Eq. (6.18) holds for every pair of two points on $\varGamma$, $\sqrt{v(\mathbf {x})} \omega (\mathbf {x}) ||\varvec{\gamma }(\mathbf {x}) ||$ is constant on $\varGamma$.

For $\mathbf{x}\in \varGamma$, the marginal density can be approximated by

$$\begin{aligned} g_\varepsilon (\mathbf {x}) = \frac{ \int _{\mathbb {R}^n \backslash \varGamma } \omega (\mathbf {y}) e^{ \frac{-\varphi (\mathbf {y}) }{\varepsilon }} \mathrm{d}\mathbf {y} }{ \int _{\mathbb {R}^n} \omega (\mathbf {y}) e^{ \frac{-\varphi (\mathbf {y}) }{\varepsilon }} \mathrm{d}\mathbf {y} } = \frac{\int _{\mathcal {S}_\mathbf{x}} \omega (\mathbf {y}) e^{ \frac{- \varphi (\mathbf {y}) }{\varepsilon }} \mathrm{d}\mathcal {S}_\mathbf{x}}{ \int _{\mathbb {R}^n} \omega (\mathbf {y}) e^{ \frac{-\varphi (\mathbf {y}) }{\varepsilon }} \mathrm{d}\mathbf {y} } + O(\varepsilon ) = \frac{ \omega (\mathbf {x}) \sqrt{v(\mathbf {x})}}{ \int _{\varGamma } \omega (\mathbf {y}) \sqrt{v(\mathbf {y})} \mathrm{d}\mathbf {y}} + O(\varepsilon ), \end{aligned}$$

(6.19)

which follows the steps from Eqs. (6.9) to (6.14) with choosing $\varvec{\gamma }_\varepsilon ( \mathbf {x}) \cdot \mathbf {n} = 1$. Furthermore, since $\sqrt{v(\mathbf{x})} \omega (\mathbf {x}) ||\gamma (\mathbf {x}) ||$ is constant on $\varGamma$, with the result (6.19), there exists a constant K such that

$$\begin{aligned} g_\varepsilon (\mathbf {x})||\varvec{\gamma }(\mathbf {x}) ||= K + O(\varepsilon ). \end{aligned}$$

(6.20)