arXiv subjects

Cecile Monthus

Publications and source records attributed to Cecile Monthus.

At least 19 recordsLinked to original sources

arXiv2026

Markov chains at the onset of non-reversibility

For a one-dimensional path graph and a lifted path graph constructed from a duplication of each of its sites, we study how a reversible Markov chain can be perturbed and gradually driven into non-reversibility. The reversible Markov chain has a transition matrix that is diagonalizable and features real-valued eigenvalues and eigenvectors. The left and right eigenvectors form a biorthogonal system. We discuss in concrete examples how the transition matrix of a non-reversible Markov chain may be diagonalizable or non-diagonalizable, and it may have real eigenvalues and complex-conjugate pairs. For a number of steady states (flat, square-wave, wedge, V-shape), we compute eigenvalue spectra on both graphs and discuss the speedup that can be achieved through lifting. We develop a Green's matrix formalism, which we use to compute Kemeny times and mean first-passage times, and which provides valuable information and allows us to interpret the results for the characteristic times.

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arXiv2026

Using the slowest observable in one-dimensional Markov processes to construct quasi-exactly-solvable generators with two explicit levels

The construction of Quasi-Exactly-Solvable quantum Hamiltonians where only the first two eigenstates $\Phi_0(x)$ and $\Phi_1(x)$ of energies $E_0$ and $E_1$ are explicit is revisited from the point of view of one-dimensional Markov processes satisfying detailed-balance, whose generators are related to quantum Hamiltonians via similarity transformations. Here, the lowest energy vanishes $E_0=0$ and is associated with the conservation of probability and with the steady state $P_*(x)$, while $E_1>0$ is the rate that governs the exponential relaxation towards the steady-state, and is associated with the slowest observable $L_1(x)$ that corresponds to the ratio $ \frac{\Phi_1(x) }{\Phi_0(x)}$ of the two quantum eigenstates. Our main conclusion is that the Markov perspective leads to interesting re-interpretations and that the construction of quasi-exactly-solvable models with $N=2$ explicit levels is more intuitive and technically simpler when one takes the slowest observable $L_1(x)$ as the central object from which all the other properties can be reconstructed. This general approach is then applied to Fokker-Planck generators in continuous space and to Markov jump generators on the lattice.

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arXiv2025

Classification of diffusion processes in dimension $d$ via the Carleman approach with applications to models involving additive, multiplicative or square-root noises

The Carleman approach is well-known in the field of deterministic classical dynamics as a method to replace a finite number $d$ of non-linear differential equations by an infinite-dimensional linear system. Here this approach is applied to a system of $d$ stochastic differential equations for $[x_1(t),..,x_d(t)]$ when the forces and the diffusion-matrix elements are polynomials, in order to write the linear system governing the dynamics of the averaged values ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) ... x_d^{n_d}(t) )$ labelled by the $d$ integers $(n_1,..,n_d)$. The natural decomposition of the Carleman matrix into blocks associated to the global degree $n=n_1+n_2+..+n_d$ is useful to identify the models that have the simplest spectral decompositions in the bi-orthogonal basis of right and left eigenvectors. This analysis is then applied to models with a single noise per coordinate, that can be either additive or multiplicative or square-root, or with two types of noises per coordinate, with many examples in dimensions $d=1,2$. In $d=1$, the Carleman matrix governing the dynamics of the moments ${\mathbb E} ( x^{n}(t) )$ is diagonal for the Geometric Brownian motion, while it is lower-triangular for the family of Pearson diffusions containing the Ornstein-Uhlenbeck and the Square-Root processes, as well as the Kesten, the Fisher-Snedecor and the Student processes that converge towards steady states with power-law-tails. In dimension $d=2$, the Carleman matrix governing the dynamics of the correlations ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) )$ has a natural decomposition into blocks associated to the global degree $n=n_1+n_2$, and we discuss the simplest models where the Carleman matrix is either block-diagonal or block-lower-triangular or block-upper-triangular.

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arXiv2025

Markov dualities via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors

The notion of Markov duality between two Markov processes that can live in two different configurations spaces $(x,{\tilde x})$ is revisited via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors. In this formulation, the two generators should have the same eigenvalues $(-E)$ that may be complex, while the duality function $\Omega(x,{\tilde x})$ can be considered as a mapping between the right and the left eigenvectors of the two models. We describe how this spectral perspective is useful to better understand two well-known dualities between processes defined in the same configuration space: the Time-Reversal duality corresponds to an exchange between the right and the left eigenvectors that involves the steady state, while in the Siegmund duality, the left eigenvectors correspond to integrals of the dual right eigenvectors. We then focus on the famous Moment-Duality between the Wright-Fisher diffusion on the interval $x \in [0,1] $ and the Kingman Markov jump process on the semi-infinite lattice $n \in {\mathbb N}$ in order to analyze the relations between their eigenvectors living in two different configuration spaces. Finally, we discuss how the spectral perspective can be used to construct new dualities and we give an example for the case of non-degenerate real eigenvalues, where one can always construct a dual Directed Jump process on the semi-infinite lattice $n \in {\mathbb N}$, whose transitions rates are the opposite-eigenvalues.

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arXiv2025

Supersymmetric properties of one-dimensional Markov generators with the links to Markov-dualities and to shape-invariance-exact-solvability

For diffusion process involving the force $F(x)$ and the diffusion coefficient $D(x)$, the continuity equation $\partial_t P_t(x)=- \partial_xJ_t(x)$ gives the dynamics of the probability $P_t( x)$ in terms of the current $J_t( x)=F(x)P_t(x)-D(x)\partial_x P_t(x)={\cal J}P_t( x)$ obtained from $P_t( x) $ via the application of the first-order differential current-operator ${\cal J}$. So the dynamics of the probability $P_t( x)$ is governed by the factorized Fokker-Planck generator ${\cal F}=-\partial_x{\cal J}$, while the dynamics of the current $J_t( x)$ is governed by its supersymmetric partner ${\hat {\cal F} }= - {\cal J}\partial_x$, so that their right and left eigenvectors are directly related using the two intertwining relations ${\cal J}{\cal F}=-{\cal J}\partial_x{\cal J}={\hat {\cal F}}{\cal J}$ and ${\cal F}\partial_x=-\partial_x{\cal J}\partial_x=\partial_x{\hat {\cal F} }$. We also describe the link with the factorization of the adjoint $ {\cal F}^{\dagger}=\frac{d}{dm(x)}\frac{d}{ds(x)} $ in terms of the scale function $s(x)$ and speed measure $m(x)$. We then analyze how the supersymmetric partner ${\hat {\cal F} } = - {\cal J} \partial_x$ can be re-interpreted in two ways: (1) as the adjoint ${\mathring {\cal F}}^{\dagger} ={\mathring {\cal J} }^{\dagger} \partial_x$ of the Fokker-Planck generator ${\mathring {\cal F}}=- \partial_x{\mathring {\cal J} }$ associated to the dual force ${\mathring F}(x)=-F(x)-D'(x)$, that unifies various known Markov dualities; (2) as the non-conserved Fokker-Planck generator ${\tilde {\cal F}}_{nc} = -\partial_x{\tilde {\cal J}}-{\tilde K }(x)$ involving the force ${\tilde F}(x)=F(x) +D'(x)$ and the killing rate ${\tilde K }(x)=-F'(x)-D''(x)$, with application to shape-invariance-solvability. Finally, we describe how all these ideas can be also applied to Markov jump processes with nearest-neighbors transition rates $w(x \pm 1,x)$.

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arXiv2025

Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators

The statistical properties of non-linear observables of the fractal Gaussian field $\phi(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the L\'evy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$.

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arXiv2025

Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes

Within the Wilson RG of 'incomplete integration' as a function of the effective RG-time $t$, the non-linear differential RG-flow for the energy $E_t[\phi(.)]$ translates for the probability distribution $P_t[\phi(.)] \sim e^{- E_t[\phi(.)]} $ into the linear Fokker-Planck RG-flow associated to independent non-identical Ornstein-Uhlenbeck processes for the Fourier modes. The corresponding Langevin stochastic differential equations for the real-space field $\phi_t(\vec x)$ have been recently interpreted by Carosso as genuine infinitesimal coarsening-transformations that are the analog of spin-blocking, and whose irreversible character is essential to overcome the paradox of the naive description of the Wegner-Morris Continuity-Equation for the RG-flow as a meaningless infinitesimal change of variables in the partition function integral. This interpretation suggests to consider new RG-schemes, in particular the Carosso RG where the Langevin SDE corresponds to the stochastic heat equation also known as the Edwards-Wilkinson dynamics. After a pedestrian self-contained introduction to this stochastic formulation of RG-flows, we focus on the case where the field theory is defined on the large volume $L^d$ with periodic boundary conditions, in order to distinguish between extensive and intensives observables while keeping the translation-invariance. Since the empirical magnetization $m_e \equiv \frac{1}{L^d} \int_{L^d} d^d \vec x \ \phi(\vec x) $ is an intensive variable corresponding to the zero-momentum Fourier coefficient of the field, its probability distribution $p_L(m_e)$ can be obtained from the gradual integration over all the other Fourier coefficients associated to non-vanishing-momenta via an appropriate adaptation of the Carosso stochastic RG, in order to obtain the large deviation properties with respect to the volume $L^d$.

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arXiv2024

Markov spin models for image generation : explicit large deviations with respect to the number of pixels

For the discrete-time or the continuous-time Markov spin models for image generation when each pixel $n=1,..,N$ can take only two values $S_n=\pm 1$, the finite-time forward propagator depends on the initial and on the final configurations of the $N$ spins only via a single global variable, namely the extensive overlap that counts the number of spins that have the same value or not in the two configurations. The joint probability distribution of the overlap and of the magnetization during the forward noising dynamics can be written for any finite number $N$ of pixels and in the limit $N \to + \infty$ to extract the large deviations properties. The consequences for the backward reconstructive dynamics are then analyzed for various initial conditions, namely (i) a single image (ii) a mixture of two images (iii) when the initial condition corresponds to the Curie-Weiss mean-field ferromagnetic model in the microcanonical ensemble, as a simple analog of the manifold-hypothesis concerning continuous generative diffusion models.

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arXiv2024

Convergence properties of Markov models for image generation with applications to spin-flip dynamics and to diffusion processes

In the field of Markov models for image generation, the main idea is to learn how non-trivial images are gradually destroyed by a trivial forward Markov dynamics over the large time window $[0,t]$ converging towards pure noise for $t \to + \infty$, and to implement the non-trivial backward time-dependent Markov dynamics over the same time window $[0,t]$ starting from pure noise at $t$ in order to generate new images at time $0$. The goal of the present paper is to analyze the convergence properties of this reconstructive backward dynamics as a function of the time $t$ using the spectral properties of the trivial continuous-time forward dynamics for the $N$ pixels $n=1,..,N$. The general framework is applied to two cases : (i) when each pixel $n$ has only two states $S_n=\pm 1$ with Markov jumps between them; (ii) when each pixel $n$ is characterized by a continuous variable $x_n$ that diffuses on an interval $]x_-,x_+[$ that can be either finite or infinite.

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arXiv2024

Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region

The Pelikan random trajectories $x_t \in [0,1[$ are generated by choosing the chaotic doubling map $x_{t+1}=2 x_t [mod 1]$ with probability $p$ and the non-chaotic half-contracting map $x_{t+1}=\frac{x_t}{2}$ with probability $(1-p)$. We compute various dynamical observables as a function of the parameter $p$ via two perspectives. In the first perspective, we focus on the closed dynamics within the subspace of probability densities that remain constant on the binary-intervals $x \in [ 2^{-n-1}, 2^{-n}[$ partitioning the interval $x \in [0,1[$ : the dynamics for the weights $\pi_t(n)$ of these intervals corresponds to a biased random walk on the half-infinite lattice $n \in \{0,1,2,..+\infty\}$ with resetting occurring with probability $p$ from the origin $n=0$ towards any site $n$ drawn with the distribution $2^{-n-1}$. In the second perspective, we study the Pelikan dynamics for any initial condition $x_0$ via the binary decomposition $x_t = \sum_{l=1}^{+\infty} \frac{\sigma_l (t)}{2^l} $, where the dynamics for the half-infinite lattice $l=1,2,..$ of the binary variables $\sigma_l(t) \in \{0,1\}$ can be reformulated in terms of two global variables : $z_t$ corresponds to a biased random walk on the half-infinite lattice $z \in \{0,1,2,..+\infty\}$ that may remain at the origin $z=0$ with probability $p$, while $F_t \in \{0,1,2,..t\}$ counts the number of time-steps $\tau \in [0,t-1]$ where $z_{\tau+1}=0=z_{\tau}$ and represents the number of the binary coefficients of the initial condition that have been erased. We discuss typical and large deviations properties in the chaotic region $\frac{1}{2}<p<1 $ as well as at the intermittent critical point $p_c=\frac{1}{2}$ towards the non-chaotic region $0<p<\frac{1}{2}$.

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arXiv2024

A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains

The large deviations at various levels that are explicit for Markov jump processes satisfying detailed-balance are revisited in terms of the supersymmetric quantum Hamiltonian $H$ that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density ${\hat p}(C) $ of the configurations $C$ seen during a trajectory over the large time-window $[0,T]$ and rewrite the explicit Donsker-Varadhan rate function as the matrix element $I^{[2]}[{\hat p}(.) ] = \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{\hat p} \rangle $ involving the square-root ket $\vert \sqrt{\hat p} \rangle $. [The analog formula is also discussed for reversible diffusion processes as a comparison.] We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density ${\hat p}(C) $ and the empirical local activities ${\hat a}(C,C') $ characterizing the density of jumps between two configurations $(C,C')$. Finally, the explicit rate function for the joint probability of the empirical density ${\hat p}(C) $ and of the empirical total activity ${\hat A} $ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements $ \langle \sqrt{ {\hat p} } \vert H \vert \sqrt{\hat p} \rangle$ and $\langle \sqrt{ {\hat p} } \vert H^{off} \vert \sqrt{\hat p} \rangle $, where $H^{off} $ represents the off-diagonal part of the supersymmetric Hamiltonian $H$. This general formalism is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian $H$ correspond to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites.

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arXiv2024

Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions

Continuity equations associated to continuous-time Markov processes can be considered as Euclidean Schr\"odinger equations, where the non-hermitian quantum Hamiltonian $\bold{H}={\bold{div}}{\bold J}$ is naturally factorized into the product of the divergence operator ${\bold {div}}$ and the current operator ${\bold J}$. For non-equilibrium Markov jump processes in a space of $N$ configurations with $M$ links and $C=M-(N-1)\geq 1$ independent cycles, this factorization of the $N \times N$ Hamiltonian ${\bold H}={\bold I}^{\dagger}{\bold J}$ involves the incidence matrix ${\bold I}$ and the current matrix ${\bold J}$ of size $M \times N$, so that the supersymmetric partner ${\hat{\bold H}}= {\bold J}{\bold I}^{\dagger}$ governing the dynamics of the currents living on the $M$ links is of size $M \times M$. To better understand the relations between the spectral decompositions of these two Hamiltonians $\bold{H}={\bold I}^{\dagger}{\bold J}$ and ${\hat {\bold H}} ={\bold J}{\bold I}^{\dagger}$ with respect to their bi-orthogonal basis of right and left eigenvectors that characterize the relaxation dynamics towards the steady state and the steady currents, it is useful to analyze the properties of the Singular Value Decompositions of the two rectangular matrices ${\bold I}$ and ${\bold J} $ of size $M \times N$ and the interpretations in terms of discrete Helmholtz decompositions. This general framework concerning Markov jump processes can be adapted to non-equilibrium diffusion processes governed by Fokker-Planck equations in dimension $d$, where the number $N$ of configurations, the number $M$ of links and the number $C=M-(N-1)$ of independent cycles become infinite, while the two matrices ${\bold I}$ and ${\bold J}$ become first-order differential operators acting on scalar functions to produce vector fields.

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arXiv2024

Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension $d$ involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing $b$ ; (ii) discrete-time random walks in continuous space with a small time-step $τ$. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits $b \to 0$ and $τ\to 0$ respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits $b \to 0$ and $τ\to 0$ for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

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arXiv2023

Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics

The large deviations properties of trajectory observables for chaotic non-invertible deterministic maps as studied recently by N. R. Smith, Phys. Rev. E 106, L042202 (2022) and by R. Gutierrez, A. Canella-Ortiz, C. Perez-Espigares, arXiv:2304.13754 are revisited in order to analyze in detail the similarities and the differences with the case of stochastic Markov chains. To be concrete, we focus on the simplest example displaying the two essential properties of local-stretching and global-folding, namely the doubling map $ x_{t+1} = 2 x_t [\text{mod} 1] $ on the real-space interval $x \in [0,1[$ that can be also analyzed via the decomposition $x= \sum_{l=1}^{+\infty} \frac{σ_l}{2^l} $ into binary coefficients $σ_l=0,1$. The large deviations properties of trajectory observables can be studied either via deformations of the forward deterministic dynamics or via deformations of the backward stochastic dynamics. Our main conclusions concerning the construction of the corresponding Doob canonical conditioned processes are: (i) non-trivial conditioned dynamics can be constructed only in the backward stochastic perspective where the reweighting of existing transitions is possible, and not in the forward deterministic perspective ; (ii) the corresponding conditioned steady state is not smooth on the real-space interval $x \in [0,1[$ and can be better characterized in the binary space $σ_{l=1,2,..,+\infty}$. As a consequence, the backward stochastic dynamics in the binary space is also the most appropriate framework to write the explicit large deviations at level 2 for the probability of the empirical density of long backward trajectories.

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arXiv2023

Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient

For diffusion processes in dimension $d>1$, the statistics of trajectory observables over the time-window $[0,T]$ can be studied via the Feynman-Kac deformations of the Fokker-Planck generator, that can be interpreted as euclidean non-hermitian electromagnetic quantum Hamiltonians. It is then interesting to compare the four regimes corresponding to the time $T$ either finite or large and to the diffusion coefficient $D$ either finite or small. (1) For finite $T$ and finite $D$, one needs to consider the full time-dependent quantum problem that involves the full spectrum of the Hamiltonian. (2) For large time $T \to + \infty$ and finite $D$, one only needs to consider the ground-state properties of the quantum Hamiltonian to obtain the generating function of rescaled cumulants and to construct the corresponding canonical conditioned processes. (3) For finite $T$ and $D \to 0$, one only needs to consider the dominant classical trajectory and its action satisfying the Hamilton-Jacobi equation, as in the semi-classical WKB approximation of quantum mechanics. (4) In the double limit $T \to + \infty$ and $D \to 0$, the simplifications in the large deviations in $\frac{T}{D}$ of trajectory observables can be analyzed via the two orders of limits, i.e. either from the limit $D \to 0$ of the ground-state properties of the quantum Hamiltonians of (2), or from the limit of long classical trajectories $T \to +\infty$ in the semi-classical WKB approximation of (3). This general framework is illustrated in dimension $d=2$ with rotational invariance.

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arXiv2023

Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?

In the field of large deviations for stochastic dynamics, the canonical conditioning of a given Markov process with respect to a given time-local trajectory observable over a large time-window has attracted a lot of interest recently. In the present paper, we analyze the following inverse problem: when two Markov generators are given, is it possible to connect them via some canonical conditioning and to construct the corresponding time-local trajectory observable? We focus on continuous-time Markov processes and obtain the following necessary and sufficient conditions: (i) for continuous-time Markov jump processes, the two generators should involve the same possible elementary jumps in configuration space, i.e. only the values of the corresponding rates can differ; (ii) for diffusion processes, the two Fokker-Planck generators should involve the same diffusion coefficients, i.e. only the two forces can differ. In both settings, we then construct explicitly the various time-local trajectory observables that can be used to connect the two given generators via canonical conditioning. This general framework is illustrated with various applications involving a single particle or many-body spin models. In particular, we describe several examples to show how non-equilibrium Markov processes with non-vanishing steady currents can be interpreted as the canonical conditionings of detailed-balance processes with respect to explicit time-local trajectory observables.

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arXiv2023

Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force

The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force are characterized by explicit dynamics of their integer moments and by explicit relaxation spectral properties towards their steady state. Besides the Ornstein-Uhlenbeck process with a Gaussian steady state, the other representative examples of the Pearson family are the Square-Root or the Cox-Ingersoll-Ross process converging towards the Gamma-distribution, the Jacobi process converging towards the Beta-distribution, the reciprocal-Gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the Inverse-Gamma-distribution, the Fisher-Snedecor process, and the Student process, so that the last three steady states display heavy-tails. The goal of the present paper is to analyze the large deviations properties of these various diffusion processes in a unified framework. We first consider the Level 1 concerning time-averaged observables over a large time-window $T$ : we write the first rescaled cumulants for generic observables and we identify the specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed-generator. The explicit large deviations at Level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.

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arXiv2023

Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels

For boundary-driven non-equilibrium Markov models of non-interacting particles in one dimension, either in continuous space with the Fokker-Planck dynamics involving an arbitrary force $F(x)$ and an arbitrary diffusion coefficient $D(x)$, or in discrete space with the Markov jump dynamics involving arbitrary nearest-neighbor transition rates $w(x \pm 1,x)$, the Markov generator can be transformed via an appropriate similarity transformation into a quantum supersymmetric Hamiltonian with many remarkable properties. In particular, the mapping from the boundary-driven non-equilibrium dynamics towards some dual equilibrium dynamics [J. Tailleur, J. Kurchan and V. Lecomte, J. Phys. A 41, 505001 (2008)] can be reinterpreted via the two corresponding quantum Hamiltonians that are supersymmetric partners of each other, with the same energy spectra. We describe the consequences for the spectral decomposition of the boundary-driven dynamics, and we give explicit expressions for the Kemeny times needed to converge towards the non-equilibrium steady states. Finally, we analyze the large deviations at various levels for empirical time-averaged observables over a large time-window $T$. We start with the always explicit Level 2.5 concerning the joint distribution of the empirical density and of the empirical flows, and we then consider the contractions towards lower levels. In particular, the rate function for the empirical current alone can be explicitly computed via the contraction from the Level 2.5 using the properties of the associated quantum supersymmetric Hamiltonians.

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