Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume

Abstract

When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker–Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron–Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian–Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton–Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron–Frobenius theorem concerning the positivity of the (left and right) dominant eigenvectors. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.

Appendices

Appendix

This appendix deals with the biasing and rectification of Markov diffusion processes. We first review the case of a single process in the linear operator framework, and then we study the case of $N$ independent processes within the Lagrangian–Hamiltonian formalism.

Appendix A: Single diffusion process

We consider a diffusion process described by the following Langevin equation

$$\begin{aligned} \dot{x}_t = b(x_t) + \sigma (x_t) \circ \xi _t, \end{aligned}$$

(A.1)

where $x_t$ is a random variable and $\xi _t$ a Gaussian white noise of mean $\langle \xi _t \rangle = 0$ and variance $\langle \xi _t \xi _{t'} \rangle \,=\,\delta (t-t')$. The stochastic integrals are defined according to the mid-point Stratonovich convention, referred to by a circle $\circ$. The drift b and the diffusion $\sigma$ are functions of $x_t$ and do not depend explicitly on time. The probability density $\varrho (x,t)$ satisfies the Fokker–Planck equation

$$\begin{aligned} \frac{\partial \varrho (x,t)}{\partial t} = - \nabla J(x,t), \end{aligned}$$

(A.2)

with $\nabla$ the derivative with respect to x and

$$\begin{aligned} J^{\varrho }(x,t) \equiv \hat{b}(x) \varrho (x,t) - \frac{1}{2} \sigma (x)^2 \nabla \varrho (x,t), \end{aligned}$$

(A.3)

where we introduced the modified drift $\hat{b}(x) \equiv b(x) - \frac{1}{2} \sigma (x) \nabla \sigma (x)$. The Fokker–Planck equation appears to be a continuity equation that conserves the normalization of the probability density, i.e. $\int \text {d}x \varrho (x,t) =1 , \forall t$. From now on, unspecified integrations are implicitly on x, e.g. $\int \varrho =1$. One can rewrite the Fokker–Planck equation as

$$\begin{aligned} \frac{\partial \varrho }{\partial t} = \mathfrak {L}\varrho , \end{aligned}$$

(A.4)

where $\mathfrak {L}$ is the Fokker–Planck operator in the Stratonovich convention, defined by its action on a function $\varphi$:

$$\begin{aligned} \mathfrak {L}\varphi (x) \equiv - \nabla \left[ \hat{b}(x) \varphi (x) \right] + \frac{1}{2} \nabla \left[ \sigma (x)^2 \nabla \varphi (x) \right] \end{aligned}$$

(A.5)

The adjoint Fokker–Planck operator $\mathfrak {L}^\dagger$ is given by

$$\begin{aligned} \mathfrak {L}^\dagger \varphi (x) = \hat{b}(x) \nabla \varphi (x) + \frac{1}{2} \nabla \left[ \sigma (x)^2 \nabla \varphi (x) \right] , \end{aligned}$$

(A.6)

both operators being related by

$$\begin{aligned} \int \left( \mathfrak {L}\varphi \right) \psi = \int \left( \mathfrak {L}^\dagger \psi \right) \varphi , \end{aligned}$$

(A.7)

for any functions $\varphi$, $\psi$. We denote a path by $[x_t]$, with $x_t$ the state of the system at time t. The path probability of $[x_t]$ between the initial time 0 and the final time t within the Stratonovich convention reads [97]

$$\begin{aligned} \mathbb {P}_{b,\sigma ,\varrho (0)}[x_t] = \varrho (x_0,0) \exp \left\{ - \int _0^t \text {d}t' \left[ \left( \dot{x}_{t'} - \hat{b}(x_{t'}) \right) ^2 + \frac{1}{2} \nabla b(x_{t'}) \right] \right\} , \end{aligned}$$

(A.8)

Many dynamical and thermodynamic observables such as heat, matter currents, work, entropy production, etc. can be written as linear combinations of the empirical occupancy $\tilde{\rho }_t$ and the empirical current $\tilde{j}_t$. The function $\tilde{\rho }_t(x)$ counts the rate of occupancy of the position x along the trajectory $[x_t]$:

$$\begin{aligned} \tilde{\rho }_t(x) = \frac{1}{t} \int _{0}^{t} \text {d}\tau \delta (x_\tau -x), \end{aligned}$$

(A.9)

while the function $\tilde{j}_t(x)$ informs on the time-averaged local velocity at x [56]:

$$\begin{aligned} \tilde{j}_t(x) = \frac{1}{t} \int _0^t \text {d}\tau \delta (x_\tau -x) \circ \dot{x}_\tau , \end{aligned}$$

(A.10)

where the circle $\circ$ refers to the Stratonovich convention. We would like to condition our original Markov process by filtering the ensemble of paths to select those leading to a chosen value of ${\tilde{\varvec{A}}}_t(x) = ( \tilde{j}_t(x), \tilde{\rho }_t(x))$, for each state x. This defines the conditioned process for which we aim to find an equivalent Markov process in the long-time limit, namely the driven process [67]. This process is described by the microcanonical path probability [56]

$$\begin{aligned} \mathbb {P}^{\text {micro}}_\mathbf{a ,\varrho (0)}[x_t]= \mathbb {P}_{\Lambda ,\varrho (0)}\left[ x_t ~\Big |~ {\tilde{\varvec{A}}}_t = \mathbf{a} \right] . \end{aligned}$$

(A.11)

To explicit the generator of the driven process, let us introduce the generating function

$$\begin{aligned} G_{\varvec{\gamma }}(x,t) \equiv \left\langle \text {e}^{t \varvec{\gamma }\cdot {\tilde{\varvec{A}}}_{t} } \delta (x_t - x) \right\rangle _{\varrho (0)}, \end{aligned}$$

(A.12)

where $\langle \cdots \rangle _{\varrho (0)}$ is the path average based on (A.8), $\varvec{\gamma }(x) \equiv \left( \gamma _1(x) , \gamma _2(x) \right)$ is the conjugate variable of ${\tilde{\varvec{A}}}_t(x)$ and where the dot stands for the scalar product $\varvec{\gamma }\cdot {\tilde{\varvec{A}}}_{t} = \int \text {d}x \left[ \gamma _1(x) \tilde{j}_t(x) + \gamma _2(x) \tilde{\rho }_t(x) \right]$. From now on, we drop in the notation the x-dependency of the functions for clarity. The generating function (A.12) evolves according to

$$\begin{aligned} \dot{G}_{\varvec{\gamma }} = \Lambda _{\varvec{\gamma }} G_{\varvec{\gamma }}, \end{aligned}$$

(A.13)

where the biased Fokker–Planck operator $\Lambda _{\varvec{\gamma }}$ is given by [56]:

$$\begin{aligned} \Lambda _{\varvec{\gamma }} \varphi \equiv (- \nabla + \gamma _1) ( \hat{b} \varphi ) + \frac{1}{2} (- \nabla + \gamma _1) \left[ \sigma ^2 (-\nabla + \gamma _1) \varphi \right] + \gamma _2 \varphi , \end{aligned}$$

(A.14)

One can compute its adjoint operator

$$\begin{aligned} \Lambda _{\varvec{\gamma }}^\dagger \varphi = \hat{b} (\nabla + \gamma _1) \varphi + \frac{1}{2} (\nabla + \gamma _1) \left[ \sigma ^2 (\nabla + \gamma _1) \varphi \right] + \gamma _2 \varphi . \end{aligned}$$

(A.15)

The biased Fokker–Planck operator $\Lambda _{\varvec{\gamma }}$ generates a Markov process that is not norm-conserving since $\int \Lambda _{\varvec{\gamma }} \varrho \ne 0$. As with Markov jump processes, we can build the operator of the driven process $\mathcal {L}$ by taking the Doob transform of the biased operator $\Lambda _{\varvec{\gamma }}$ associated with its dominant left eigenfunction:

$$\begin{aligned} \mathcal {L}\varphi \equiv l \Lambda _{\varvec{\gamma }} (l^{-1} \varphi ) - l (\Lambda _{\varvec{\gamma }}^\dagger l^{-1}) \varphi , \end{aligned}$$

(A.16)

with $l \equiv l(x)$ being the left eigenfunction of $\Lambda _{\varvec{\gamma }}$ for the highest eigenvalue ${\bar{\Gamma }}$

$$\begin{aligned} \Lambda _{\varvec{\gamma }}^\dagger l = {\bar{\Gamma }}l. \end{aligned}$$

(A.17)

Since the Krein–Rutman theorem ensures the positivity of l, we introduce a new function $u \equiv u(x)$ such that $l \equiv \text {e}^u$. It follows from (A.17):

$$\begin{aligned} {\bar{\Gamma }}= \int \text {e}^{-u} (\Lambda _{\varvec{\gamma }}^\dagger \text {e}^u) \rho . \end{aligned}$$

(A.18)

Computing explicitly Eq. (A.16) using Eqs. (A.14, A.15) and $l = \text {e}^u$, we finally find that the driven Fokker–Planck operator is given by

$$\begin{aligned} \mathcal {L}\varphi = - \nabla \left[ \hat{B}_\gamma \varphi - \frac{1}{2} \sigma ^2 \nabla \varphi \right] , \end{aligned}$$

(A.19)

where we introduced the rectified drift

$$\begin{aligned} \hat{B}_{\varvec{\gamma }} \equiv \hat{b} + \sigma ^2 (\nabla u + \gamma _1). \end{aligned}$$

(A.20)

The driven process is thus a diffusive process obeying the same stochastic equation (A.1) as the original process but with a new drift (A.20). Note that the dependence of u on $\varvec{\gamma }$ is made implicit.

Appendix B: $N$ Independent Diffusion Processes

We consider $N$ independent and identical systems, each one modeled by a time-homogeneous Markov diffusion process of time-independent drift b and diffusion coefficient $\sigma$. We denote by $\nu \in \{1, 2, \dots N\}$ the $\nu ^\text {th}$ system and by $x^\nu _t$ the stochastic process of the system $\nu$ which evolves according to the Langevin equation

$$\begin{aligned} \dot{x}^\nu _t = b(x_t^\nu ) + \sigma (x_t^\nu ) \xi ^\nu _t, \end{aligned}$$

(B.1)

We are interested in the empirical occupation density:

$$\begin{aligned} \rho (x,t) = \frac{1}{N} \sum _{\nu =1}^N \delta (x^\nu _t-x) , \end{aligned}$$

(B.2)

and the empirical current:

$$\begin{aligned} j(x,t) = \frac{1}{N} \sum _{\nu =1}^N \delta (x^\nu _t-x) \circ \dot{x}^\nu _t, \end{aligned}$$

(B.3)

playing respectively the role of the variables ${\varvec{z}}$ and $\varvec{\lambda }$ in our general framework of section 3. The empirical occupation density gives the density of systems being at a state in $[x,x+\text {d}x[$ at time t, and the empirical current measures the density of systems performing a displacement between x and $x+\text {d}x$ within the time interval $[t,t+\text {d}t[$. Both variables are related by

$$\begin{aligned} \dot{\rho }(x,t) = - \nabla j(x,t), \end{aligned}$$

(B.4)

with $- \nabla$ playing the role of $\mathcal {D}$. Notice that these observables are related to the empirical occupancy $\tilde{\rho }_t^\nu$ (A.9) and the empirical transition current $\tilde{j}_t^\nu$ (A.10) for a single process through

$$\begin{aligned} \begin{aligned} \frac{1}{t} \int _0^t \text {d}\tau \, \rho (\tau )&= \frac{1}{N} \sum _{\nu = 1}^N \tilde{\rho }_t^\nu , \\ \frac{1}{t} \int _0^t \text {d}\tau \, j(\tau )&= \frac{1}{N} \sum _{\nu = 1}^N \tilde{j}_t^\nu , \end{aligned} \end{aligned}$$

(B.5)

where the superscript $\nu$ indicates that the empirical occupancy or current are those for the trajectory of the $\nu ^{\text {th}}$ system.

1.1 Stochastic equation for the empirical occupation density

We aim to give a coarse-grained description of the global system by deriving the stochastic equation for $\rho$. To do so, we compute the quantity: $\Delta x^\nu ~\equiv ~x^\nu _{t+\Delta t}~-~x^\nu _t$. Using the Langevin equation (A.1) for $x^\nu _t$, we get:

$$\begin{aligned} \Delta x^\nu = \int _t^{t+\Delta t} \text {d}\tau \left[ b(x^\nu _\tau ) + \sigma (x_\tau ^\nu ) \circ \xi ^\nu _\tau \right] . \end{aligned}$$

(B.6)

For an infinitesimal $\Delta t$, we have in the Stratonovich convention [97]:

$$\begin{aligned} \int _t^{t+\Delta t} \text {d}\tau \, b(x^\nu _\tau )&\simeq b (x^\nu _t + \frac{\Delta x^\nu }{2} ) \Delta t, \end{aligned}$$

(B.7)

$$\begin{aligned} \int _t^{t+\Delta t} \text {d}\tau \, \left[ \sigma (x_\tau ^\nu ) \xi ^\nu _\tau \right]&\simeq \sigma (x^\nu _t + \frac{\Delta x^\nu }{2} ) \int _t^{t+\Delta t} \text {d}\tau \, \xi ^\nu _\tau . \end{aligned}$$

(B.8)

Expanding up to order $\Delta t$ and using the identity $(\int _t^{t+\Delta t} \xi _t \text {d}t)^2 = \Delta t$ when $\Delta t \rightarrow 0$ [13, 98], it follows

$$\begin{aligned} \Delta x^\nu&\simeq b(x^\nu _t) \Delta t + \sigma (x^\nu _t) \int _t^{t+\Delta t} \xi ^\nu _{\tau } \text {d}\tau + \frac{1}{2} \sigma (x^\nu _t) \nabla \sigma (x^\nu _t) \Delta t, \end{aligned}$$

(B.9)

$$\begin{aligned} (\Delta x^\nu )^2&\simeq \sigma (x^\nu _t)^2 \Delta t. \end{aligned}$$

(B.10)

We now compute $\rho (x,t+\Delta t) = \frac{1}{N} \sum _{\nu =1}^N \delta (x^\nu _{t+\Delta t}-x)$. Let $\varphi$ be a test function, then

$$\begin{aligned} \int \text {d}x \varphi (x) \rho (x,t+\Delta t)&= \frac{1}{N} \sum _{\nu = 1}^N \varphi (x_{t+\Delta t}^\nu ) \end{aligned}$$

(B.11)

$$\begin{aligned}&= \frac{1}{N} \sum _{\nu = 1}^N \varphi (x_{t}^\nu + \Delta x^\nu ) \end{aligned}$$

(B.12)

$$\begin{aligned}&\simeq \frac{1}{N} \sum _{\nu = 1}^N \varphi (x_t^\nu ) + \frac{1}{N} \sum _{\nu = 1}^N \Delta x^\nu \varphi '(x_t^\nu ) + \frac{1}{N} \sum _{\nu = 1}^N \frac{1}{2} (\Delta {x^\nu })^2 \varphi ''(x_t^\nu ), \end{aligned}$$

(B.13)

where we used Taylor’s formula around $x^\nu _t$ up to second order in $\Delta x^\nu$ in the last equation. Using Eqs. (B.9, B.10) and the fact that $\frac{1}{N} \sum _{\nu =1}^N \varphi (x_t^\nu ) = \int \text {d}x \varphi (x) \rho (x,t)$, Eq. (B.13) gives

$$\begin{aligned}&\int \text {d}x \varphi (x) \dot{\rho }(x,t) \nonumber \\&\quad = \int \text {d}x \varphi (x)\left\{ - \nabla \left[ \hat{b}(x) \rho (x,t) - \frac{1}{2} \sigma (x)^2 \nabla \rho (x,t) + \sigma (x) \sqrt{\frac{\rho (x,t)}{N}} \eta (x,t) \right] \right\} , \nonumber \\ \end{aligned}$$

(B.14)

with $\dot{\rho }(x,t) = \lim _{\Delta t \rightarrow 0} \frac{\rho (x,t+\Delta t) - \rho (x,t)}{\Delta t}$ and where we introduced

$$\begin{aligned} \eta (x,t) \equiv \frac{1}{\sqrt{N\, \rho (x,t)} } \sum _{\nu =1}^N\delta (x-x_t^\nu ) \bar{\xi }^\nu _t, \end{aligned}$$

(B.15)

with $\bar{\xi }^\nu _t \equiv \lim _{\Delta t \rightarrow 0} \frac{1}{\Delta t} \int _t^{t+\Delta t} \text {d}\tau \xi _\tau ^\nu$. The stochastic process $\eta$ is a Gaussian white noise in time and space [99, 100] with mean and variance

$$\begin{aligned} \langle \eta (x,t) \rangle= & {} 0, \end{aligned}$$

(B.16)

$$\begin{aligned} \langle \eta (x,t) \eta (x',t') \rangle= & {} \delta (x-x') \delta (t-t'). \end{aligned}$$

(B.17)

Since (B.14) is valid for any function $\varphi$, we obtain the stochastic equation for the density $\rho$:

$$\begin{aligned} \dot{\rho }(x,t) = - \nabla \left[ \hat{b}(x) \rho (x,t) - \frac{1}{2} \sigma (x)^2 \nabla \rho (x,t) + \sigma (x) \sqrt{\frac{\rho (x,t)}{N}} \eta (x,t) \right] . \end{aligned}$$

(B.18)

Eq. (B.18) is known as the Dean equation.

1.2 Derivation of the Lagrangians and Hamiltonians

In order to obtain the Lagrangian $\mathscr {L}(j,\rho )$, we compute the conditional probability $P_{\delta t}(j \,|\, \rho )$ using Eqs. (B.4, B.18):

$$\begin{aligned} P_{\delta t}(j \,|\, \rho ) = \prod _x \left\langle \delta \left[ j - \hat{b} \rho + \frac{1}{2} \sigma ^2 \nabla \rho - \sigma \sqrt{\frac{\rho }{N}} \eta \right] \right\rangle _{\eta }. \end{aligned}$$

(B.19)

The continuous product $\prod _x$ runs over the states $x_\ell \equiv \ell \delta x$ with $\ell$ an integer and $\delta x$ an infinitesimal space step, and $\langle \cdots \rangle _\eta$ is the average over the noise $\eta$:

$$\begin{aligned} \left\langle \mathcal {O}(\eta ) \right\rangle _\eta \equiv \frac{1}{\mathsf {N}} \int \mathcal {O}(\eta ) \text {e}^{-\frac{1}{2} \delta t \delta x \eta ^2} \text {d}\eta , \end{aligned}$$

(B.20)

with $\mathsf {N}$ the normalization factor and $O(\eta )$ an arbitrary function of $\eta$. We have dropped the (x, t)-dependency in all functions for clarity. It follows

$$\begin{aligned} P_{\delta t}(j \,|\, \rho )&= \frac{1}{\mathsf {N}} \int \text {d}\eta \prod _{x} \text {e}^{-\frac{1}{2} \delta t \delta x \eta ^2} \delta \left[ j - \hat{b} \rho + \frac{1}{2} \sigma ^2 \nabla \rho - \sigma \sqrt{\frac{\rho }{N}} \eta \right] \nonumber \\&= \frac{1}{\mathsf {N}} \int \text {d}\eta \prod _{x} e^{-\frac{1}{2} \delta t \delta x \eta ^2} \frac{\sqrt{N}}{\sigma \sqrt{\rho }} \delta \left[ \eta - \frac{ j - \hat{b} \rho + \frac{1}{2} \sigma ^2 \nabla \rho }{\frac{1}{\sqrt{N}} \sigma \sqrt{\rho }} \right] \nonumber \\&= \frac{1}{\mathsf {N}} \exp { \left[ - \delta t \int \frac{N}{2\sigma ^2 \rho } \left( j - \hat{b} \rho + \frac{1}{2} \sigma ^2 \nabla \rho \right) ^2 + \frac{1}{2} \int \ln \left( \frac{N}{\rho \sigma ^2}\right) \right] }, \end{aligned}$$

(B.21)

where we used in the second equality the relation $\delta (\varphi (y)) = |\varphi '(y_0) |^{-1} \delta (y-y_0)$, for any smooth test function $\varphi$ and any root $y_0$ of $\varphi (y)=0$. In the limit of large $N$, the last term in the exponential is asymptotically dominated by $N$ and we obtain

$$\begin{aligned} P_{\delta t}(j \,|\, \rho ) \underset{N \rightarrow \infty }{\asymp }\text {e}^{- N\delta t \mathscr {L}(j,\rho )}, \end{aligned}$$

(B.22)

where the Lagrangian is given by

$$\begin{aligned} \mathscr {L}(j,\rho ) = \int \frac{1}{2\sigma ^2 \rho } \left( j - J^\rho \right) ^2, \end{aligned}$$

(B.23)

with $J^\rho = {\hat{b}} \rho - \frac{1}{2} \sigma ^2 \nabla \rho$. Computing the Legendre transform of $\mathscr {L}$ with respect to j yields the detailed Hamiltonian

$$\begin{aligned} \mathscr {H}(f,\rho ) = \int f \left[ \frac{1}{2} \sigma ^2 f \rho + J^\rho \right] . \end{aligned}$$

(B.24)

We are interested in the observable

$$\begin{aligned} \varvec{A}_{t}(x) \equiv \frac{N}{t} \left( \begin{array}{c} \int _0^t \text {d}\tau j(x,\tau ) \\ \int _0^t \text {d}\tau \rho (x,\tau ) \end{array} \right) . \end{aligned}$$

(B.25)

Using the results of Sect. 3.2, the dynamical fluctuations of $\varvec{A}$ are encoded in the biased Lagrangian and Hamiltonian

$$\begin{aligned} \mathscr {L}_{\varvec{\gamma }}(j,\rho )&= \mathscr {L}(j,\rho ) - \gamma _1 \cdot j - \gamma _2 \cdot \rho , \end{aligned}$$

(B.26)

$$\begin{aligned} \mathscr {H}_{\varvec{\gamma }}(f,\rho )&= \int (f+\gamma _1) \left[ \frac{1}{2} \sigma ^2 (f+\gamma _1) \rho + J^\rho \right] + \gamma _2 \cdot \rho = \mathscr {H}(f+\gamma _1,\rho ) + \gamma _2 \rho , \end{aligned}$$

(B.27)

where $\gamma _1(x)$ (resp. $\gamma _2(x)$) is conjugated to the first (resp. second) component of $\varvec{A}(x)$.

1.3 SCGF and HJ equation

In order to derive the rectified Hamiltonian, we first need to translate the spectral properties of the biased generator from the linear operator formalism to the Hamiltonian formalism. Because of the independence of the $N$ processes, it suffices to look at a single process. Indeed, we can relate the SCGF $\Gamma$ of the global system to the SCGF ${\bar{\Gamma }}$ of the single process by:

$$\begin{aligned} \Gamma&= \lim _{t \rightarrow \infty } \frac{1}{t} \ln \left\langle \text {e}^{t \varvec{\gamma }\cdot \varvec{A}_t} \right\rangle _{x_0} \end{aligned}$$

(B.28)

$$\begin{aligned}&= \lim _{t \rightarrow \infty } \frac{1}{t} \ln \left\langle \text {e}^{N\int _0^t \text {d}\tau \gamma _1 \cdot j(\tau ) + N\int _0^t \text {d}\tau \gamma _2 \cdot \rho (\tau )} \right\rangle _{x_0} \end{aligned}$$

(B.29)

$$\begin{aligned}&= \lim _{t \rightarrow \infty } \frac{1}{t} \ln \left\langle \text {e}^{t \sum _{\nu = 1}^N \left( \gamma _1 \cdot \tilde{j}_t^\nu + \gamma _2 \cdot \tilde{\rho }_t^\nu \right) } \right\rangle _{x_0} \end{aligned}$$

(B.30)

$$\begin{aligned}&= N\lim _{t \rightarrow \infty } \frac{1}{t} \ln \left\langle \text {e}^{t \left( \gamma _1 \cdot \tilde{j}_t^\nu + \gamma _2 \cdot \tilde{\rho }_t^\nu \right) } \right\rangle _{x_0} \end{aligned}$$

(B.31)

$$\begin{aligned}&= N{\bar{\Gamma }}, \end{aligned}$$

(B.32)

where we used Eq. (B.25) in Eqs. (B.29), (B.5) in Eqs. (B.30) and the fact that the $N$ processes are independent and identically distributed in Eq. (B.31). Computing explicitly the right-hand-side of Eq. (A.18), we find

$$\begin{aligned} {\bar{\Gamma }}= \mathscr {H}_{\varvec{\gamma }}(f=\nabla u, \rho ), \end{aligned}$$

(B.33)

with $\nabla = (-\nabla )^\dag$. As expected, the function u appearing in the left eigenfunction of the single-process biased Fokker–Planck operator is the solution of the HJ equation and we write $u = \partial _{\rho } W_{\text {s}}$.

1.4 Rectified Hamiltonian

The rectified Hamiltonian follows from Eq. (5.6):

$$\begin{aligned} \mathscr {H}^r(f,\rho ;\varvec{\gamma }) = \mathscr {H}_{\varvec{\gamma }}(f+\nabla u, \rho ) - \mathscr {H}_{\varvec{\gamma }}(\nabla u, \rho ), \end{aligned}$$

(B.34)

leading after explicit computation to

$$\begin{aligned} \mathscr {H}^{r}(f,\rho ;\varvec{\gamma }) = \int f \left[ \frac{1}{2} \sigma ^2 f \rho + J^{{\text {r}},\rho }_{\varvec{\gamma }} \right] , \end{aligned}$$

(B.35)

with $J^{{\text {r}},\rho }_{\varvec{\gamma }} \equiv \hat{B}_{\varvec{\gamma }} - \frac{1}{2} \sigma ^2 \nabla \rho$, where the rectified drift $\hat{B}_{\varvec{\gamma }}$ is defined in Eq. (A.20). Unsurprisingly, the rectified Hamiltonian corresponds to an unbiased Hamiltonian associated with the drift $\hat{B}_{\varvec{\gamma }}$ of the driven process obtained from a Doob transform as seen in Eq. (A.20). This illustrates the fact that the rectification of biased Hamiltonians is equivalent to the rectification of biased generators in the linear operator formalism using the Doob transform.