Large Deviations in Fast–Slow Systems

Abstract

The incidence of rare events in fast–slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior—such fluctuations are rare on short time-scales but become ubiquitous eventually. Classical results prove that this LDP involves an Hamilton–Jacobi equation whose Hamiltonian is related to the leading eigenvalue of the generator of the fast process, and is typically non-quadratic in the momenta—in other words, the LDP for the slow variables in fast–slow systems is different in general from that of any stochastic differential equation (SDE) one would write for the slow variables alone. It is shown here that the eigenvalue problem for the Hamiltonian can be reduced to a simpler algebraic equation for this Hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables, and the fast variables enter quadratically the equation for the slow variables. These results are illustrated via examples, inspired by kinetic theories of turbulent flows and plasma, in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an SDE.

Appendices

Appendix 1: Derivation of the Limiting Equation (5) from the LLN

Here we derive the limiting equation (5) of the LLN by formally taking the limit as $\alpha \rightarrow 0$ on the backward Kolmogorov equation (20). To this end expand u as $u=u_0+\alpha u_1 +O(\alpha ^2)$, insert this ansatz in (20), and collect term of increasing power in $\alpha$. This gives the hierarchy

$$\begin{aligned} L_1 u_0= & {} 0\nonumber \\ L_1 u_1= & {} \partial _t u_0 - L_0 u_0\nonumber \\&\vdots \end{aligned}$$

(101)

The first implies that $u_0$ is a only a function of x and not of y, or equivalently

$$\begin{aligned} Pu_0 = u_0 \end{aligned}$$

(102)

Since $L_1$ is not invertible ($PL_1 = 0$), the second equation requires a solvability condition, which reads

$$\begin{aligned} 0 = \partial _t Pu_0 - P L_0 u_0 = \partial _t u_0 - P L_0P u_0 \end{aligned}$$

(103)

It is easy to see that $PL_0P = F(x) \cdot \partial _x$, i.e. (103) is the backward Kolmogorov equation of the limiting ODE (5).

Appendix 2: Derivation of the CLT Equation (10)

To derive the linear SDE (10) of the CLT, notice that, using (5), (1) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\tilde{\xi }}} = \frac{1}{\sqrt{\alpha }}\tilde{f}(\bar{X}, Y) + \partial _x f(\bar{X},Y) \tilde{\xi }+ O(\sqrt{\alpha })\\ \displaystyle dY = \frac{1}{\alpha }b(\bar{X},Y) dt + \frac{1}{\sqrt{\alpha }} \partial _x b(\bar{X}, Y) \tilde{\xi }dt+ \frac{1}{\sqrt{\alpha }} \sigma (\bar{X}, Y) dW(t) + O(1) \end{array}\right. } \end{aligned}$$

(104)

This means that the joint process $(\bar{X},\tilde{\xi }, Y)$ is Markov with generator $L' = L_0'+ \alpha ^{-1/2} L_{\frac{1}{2}} + \alpha ^{-1} L_1 + O(\alpha ^{3/2})$ where $L_1$ is defined in (18) and

(105)

Letting

$$\begin{aligned} v(t,\bar{x},\xi ,y) = \mathbb {E}^{\bar{x},\xi ,y} g(\bar{X}(t),\tilde{\xi }(t)) \end{aligned}$$

(106)

this function satisfies the backward Kolmogorov equation

$$\begin{aligned} \partial _t v = L'_0 v + \frac{1}{\sqrt{\alpha }} L_{\frac{1}{2}} v + \frac{1}{\alpha }L_1 v + \text {higher order terms}, \qquad v(0) = g \end{aligned}$$

(107)

Formally expand v as $v=v_0 +\sqrt{\alpha } v_{\frac{1}{2}}+\alpha v_1 +O(\alpha ^{3/2})$, insert this ansatz in (107), and collect term of increasing power in $\alpha$:

(108)

The first equation implies that $v_0 = P v_0$, i.e. $v_0$ is a function of $\bar{x}$ and $\xi$ only. The solvability condition for the second equation is automatically satisfied since $P\tilde{f} = 0$ implies that $P L_{\frac{1}{2}} P =0$. Therefore, the solution to this equation is

$$\begin{aligned} v_{\frac{1}{2}} = - L_1^{-1} L_{\frac{1}{2}}P v_0 \end{aligned}$$

(109)

where $L_1^{-1}$ denotes the pseudo-inverse of $L_1$. Alternatively, this solution can also be expressed as

$$\begin{aligned} v_{\frac{1}{2}} = \int _0^\infty d\tau \, e^{\tau L_1} L_{\frac{1}{2}} Pv_0 \end{aligned}$$

(110)

Using this expression in the solvability condition for the third equation in (108) finally gives the evolution equation for $v_0$:

$$\begin{aligned} \partial _t v_0 = PL'_0P v_0 + P L_{\frac{1}{2}}\int _0^\infty d\tau \, e^{\tau L_1} L_{\frac{1}{2}} P v_0 \end{aligned}$$

(111)

The first term at the right hand side is explicitly

$$\begin{aligned} PL'_0Pv_0 = \bar{F}(\bar{x}) \cdot \partial _{\bar{x}} v_0 + B_1(\bar{x}) \xi \cdot \partial _\xi v_0 \end{aligned}$$

(112)

where $B_1(x)$ is

$$\begin{aligned} B_1(x) = \int _{\mathbb {R}^n} \partial _x f(x,y) \mu _x(dy) \end{aligned}$$

(113)

The second term at the right hand side of (111) is

$$\begin{aligned} P L_{\frac{1}{2}}\int _0^\infty d\tau \, e^{\tau L_1} L_{\frac{1}{2}} P v_0 = B_2(\bar{x}) \xi \cdot \partial _\xi v_0 + A(\bar{x}) : \partial _\xi \partial _\xi v_0 \end{aligned}$$

(114)

where A(x) is the matrix defined in (7) and

$$\begin{aligned} B_2(x) = \int _0^\infty d\tau \int _{\mathbb {R}^n} \left( \partial _y \mathbb {E}^y f(x,\tilde{Y}^x(\tau ))\right) \partial _x b(x,y) \mu _x(dy) \end{aligned}$$

(115)

Inserting (112) and (114) in (111) shows that this equation is indeed the backward Kolmogorov equation of the joint process governed by (5) and (10).