A Formal View on Level 2.5 Large Deviations and Fluctuation Relations

Abstract

We obtain the rate function for the level 2.5 of large deviations for pure jump and diffusion processes. This result is proved by two methods: tilting, for which a tilted process with an appropriate typical behavior is considered, and a spectral method, for which the scaled cumulant generating function is used. We also briefly discuss fluctuation relations, pointing out their connection with large deviations at the level 2.5.

Appendices

Appendix 1: Proof of (82)

We prove relation (82) from relation (81). Writing

$$\begin{aligned} \left( L_{0,V_{2}}+V_{1}\right) r\left[ V_{1},V_{2}\right] (x)=\lambda \left[ V_{1},V_{2}\right] r\left[ V_{1},V_{2}\right] (x), \end{aligned}$$

(90)

we obtain

$$\begin{aligned} V_{1}-\lambda \left[ V_{1},V_{2}\right] =-\left( r\left[ V_{1},V_{2}\right] (x)\right) ^{-1}L_{0,V_{2}}\left( r\left[ V_{1},V_{2}\right] \right) (x). \end{aligned}$$

(91)

With this last equation (81) becomes

$$\begin{aligned} I\left[ \rho ,j\right]= & {} \sup _{V_{1},V_{2}}\left( \int _{\mathcal {E}}dx\rho (x)\left( V_{1}(x)-\lambda \left[ V_{1},V_{2}\right] \right) +j(x) \cdot V_{2}(x)\right) \nonumber \\= & {} \sup _{V_{1},V_{2}}\left( \int _{\mathcal {E}}dx\rho (x)\left( -\left( r\left[ V_{1},V_{2}\right] (x)\right) ^{-1}L_{0,V_{2}}\left( r\left[ V_{1},V_{2}\right] \right) (x)\right) +j(x) \cdot V_{2}(x)\right) \nonumber \\= & {} \sup _{V'_{1},V_{2}}\left( \int _{\mathcal {E}}dx\rho (x)\left( -\exp \left( -V'_{1}(x)\right) L_{0,V_{2}}\left[ \exp \left( V'_{1}\right) \right] (x)\right) +j(x) \cdot V_{2}(x)\right) ,\nonumber \\ \end{aligned}$$

(92)

where $V_1'= \ln r(V_1,V_2)$.

Appendix 2: Proof of (83)

The goal here is to prove relation (83) from (82). From a direct calculation we obtain

$$\begin{aligned} \exp \left( -V'_{1}\right) L_{0,V_{2}}\left( \exp V'_{1}\right) =L_{0,V_{2}+\nabla V'_{1}}[1]. \end{aligned}$$

(93)

Relation (82) then becomes

$$\begin{aligned} I\left[ \rho ,j\right]&= \sup _{V'_{1},V_{2}}\left( \int _{\mathcal {E}}dx\left( j(x) \cdot V_{2}(x)-\rho (x)L_{0,V_{2}+\nabla V'_{1}}[1](x)\right) \right) \nonumber \\&= \sup _{V'_{1},V'_{2}}\left( \int _{\mathcal {E}}dx\left( -j(x) \cdot \nabla V'_{1}+j(x) \cdot V'_{2}(x)-\rho (x)L_{0,V'_{2}}[1](x)\right) \right) \nonumber \\&= -\inf _{V'_{1}}\left( \int _{\mathcal {E}}dxj(x) \cdot \nabla V'_{1}\right) +\sup _{V_{2}'}\left( \int _{\mathcal {E}}dx\left( j(x) \cdot V'_{2}(x)-\rho (x)L_{0,V'_{2}}[1](x)\right) \right) . \end{aligned}$$

(94)

We obtain the final relation (83) with $L_{0,V'_{2}}[1] = \widehat{A_{0}} \cdot V'_{2}+V'_{2} \cdot \frac{D}{2} \cdot V'_{2}+\nabla \cdot \left( \frac{D}{2} \cdot V'_{2}\right)$ and the algebraic manipulation

$$\begin{aligned}&\int _{\mathcal {E}}dx\left( j(x) \cdot V'_{2}(x)-\rho (x)L_{0,V'_{2}}[1](x)\right) \nonumber \\&\quad =\int _{\mathcal {E}}dx\left( j(x) \cdot V'_{2}(x)-\rho (x)\left( \widehat{A_{0}} \cdot V'_{2}+V'_{2} \cdot \frac{D}{2} \cdot V'_{2}+\nabla \cdot \left( \frac{D}{2} \cdot V'_{2}\right) \right) \right) \nonumber \\&\quad = \int _{\mathcal {E}}dxj(x) \cdot V'_{2}(x)-\left[ \rho (x)V'_{2} \cdot \frac{D}{2} \cdot V'_{2}+V'_{2} \cdot \left( \widehat{A_{0}}\rho (x)-\frac{D}{2}\cdot \nabla \rho +j\right) \right] \nonumber \\&\quad = \int _{\mathcal {E}}dx\left[ -\rho (x)V'_{2} \cdot \frac{D}{2} \cdot V'_{2}+V'_{2} \cdot \left( j-J_{\rho }\right) \right] \nonumber \\&\quad = -\int _{\mathcal {E}}dx\left[ \left( V'_{2}-\left( \rho D\right) ^{-1}\left( j-J_{\rho }\right) \right) \frac{\rho D}{2}\left( V'_{2}-\left( \rho D\right) ^{-1}\left( j-J_{\rho }\right) \right) \right. \nonumber \\&\quad \quad \left. -\left( j-J_{\rho }\right) \frac{\left( \rho D\right) ^{-1}}{2}\left( j-J_{\rho }\right) \right] , \end{aligned}$$

(95)

which included formal integration by parts.