Martingale Structure for General Thermodynamic Functionals of Diffusion Processes Under Second-Order Averaging

Abstract

Novel hidden thermodynamic structures have recently been uncovered during the investigation of nonequilibrium thermodynamics for multiscale stochastic processes. Here we reveal the martingale structure for a general thermodynamic functional of inhomogeneous singularly perturbed diffusion processes under second-order averaging, where a general thermodynamic functional is defined as the logarithmic Radon–Nykodim derivative between the laws of the original process and a comparable process (forward case) or its time reversal (backward case). In the forward case, we prove that the regular and anomalous parts of a thermodynamic functional are orthogonal martingales. In the backward case, while the regular part may not be a martingale, we prove that the anomalous part is still a martingale. With the aid of the martingale structure, we prove the integral fluctuation theorem satisfied by the regular and anomalous parts of a general thermodynamic functional. Further extensions and applications to stochastic thermodynamics are also discussed, including the martingale structure and fluctuation theorems for the regular and anomalous parts of entropy production and housekeeping heat in the absence or presence of odd variables.

Appendix A: An Class of Processes Satisfying Assumption 4.7

Clearly, Assumption 4.7 is automatically satisfied if the original process is defined on a compact set. However, in many cases, to guarantee this assumption to hold, we only need to require that the fast component is defined on a compact set. To give such an example, we consider the case where the fast process $Y^\epsilon$ is defined on a compact subset $K\subset {\mathbb {R}}^m$ and the slow process $X^\epsilon$ is a one-dimensional diffusion defined on the entire space. Moreover, we assume that $X^\epsilon _0 = 0$ and the drift and diffusion coefficients of the slow process satisfy $f = 0$, $\sigma = 1$, and

$$\begin{aligned} -\alpha x \le b(x,y,t)\le 0,\quad x\in {\mathbb {R}},\;y\in K,\;t\in [0,T], \end{aligned}$$

where $\alpha >0$ is a constant. Under these conditions, it follows from the Ikeda–Watanabe comparison theorem [81, p. 437, Theorem 1.1] that for any $\epsilon >0$, the slow process can be controlled by

$$\begin{aligned} U_t \le X^\epsilon _t \le W_t,\;\;\;\;t\in [0,T], \end{aligned}$$

(21)

where $U_t$ is an Ornstein-Uhlenbeck process satisfying

$$\begin{aligned} dU_t = -\alpha U_t+dW_t,\;\;\;U_0 = 0. \end{aligned}$$

Since $W_t$ is a continuous martingale, it follows from the Burkholder–Davis–Gundy inequality that for any $p>0$, there exists a constant $D_p>0$ independent of T such that

$$\begin{aligned} {\mathbb {E}}\big [\sup _{t\le T}|W_t|^p\big ] \le D_pT^{p/2}. \end{aligned}$$

Recently, it has been proved that the Ornstein-Uhlenbeck process satisfies the following $L^p$ maximal inequality [82, 83]: for any $p>0$, there exists a constant $C_p>0$ independent of $\alpha$, $\beta$, and T such that

$$\begin{aligned} {\mathbb {E}}\big [\sup _{t\le T}|U_t|^p\big ] \le \frac{C_p}{\alpha ^{p/2}}\log ^{p/2}(1+\alpha T). \end{aligned}$$

For any $M>0$, it then follows from Chebychev’s inequality that

$$\begin{aligned}&{\mathbb {P}}\big (\sup _{t\le T}|W_t|>M\big ) \le \frac{1}{M^p}{\mathbb {E}}\big [\sup _{t\le T}|W_t|^p\big ] \le \frac{D_p}{M^p}T^{p/2},\\&{\mathbb {P}}\big (\sup _{t\le T}|U_t|>M\big ) \le \frac{1}{M^p}{\mathbb {E}}\big [\sup _{t\le T}|U_t|^p\big ] \le \frac{C_p}{M^p\alpha ^{p/2}}\log ^{p/2}(1+\alpha T). \end{aligned}$$

Thus for any $\delta >0$, when M is sufficiently large, it follows from (21) that

$$\begin{aligned} {\mathbb {P}}\big (\inf _{\epsilon>0}\tau ^\epsilon _{B_0(3M)}\ge T\big )\ge & {} {\mathbb {P}}\big (\sup _{t\le T}|X^\epsilon _t|\le 2M\;\text {for all}\;\epsilon>0\big )\\\ge & {} {\mathbb {P}}\big (\sup _{t\le T}|W_t|+\sup _{t\le T}|U_t|\le 2M\big )\\= & {} 1-{\mathbb {P}}\big (\sup _{t\le T}|W_t|+\sup _{t\le T}|U_t|>2M\big )\\\ge & {} 1-{\mathbb {P}}\big (\sup _{t\le T}|W_t|>M\big ) -{\mathbb {P}}\big (\sup _{t\le T}|U_t|>M\big ) \ge 1-\delta , \end{aligned}$$

where $B_0(3M)\subset {\mathbb {R}}^{m+1}$ is the closed ball centered at zero with radius 3M. Therefore, Assumption 4.7 is satisfied.