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Dynamical symmetries of Markov processes with multiplicative white noise

We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the Landau-Lifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.


Introduction
Stochastic Markov processes in which the noise acts multiplicatively on a function of the variable of interest are manifold. In physics one finds the diffusion of a colloidal particle close to a wall, in chemistry one counts autocatalytic chemical reactions in which the production of a molecule is enhanced by the presence of the same molecules already produced, in economy the Black and Scholes model provides a theory of option pricing. A detailed discussion of the dynamics of single variable and extended systems with multiplicative noise can be found in Ref. [1].
In order to make sense, any multiplicative-noise differential equation, say a Langevin equation, needs to be complemented with a discretisation prescription. This choice becomes crucial for stochastic Markov processes with white noise. For these, the associated Fokker-Planck equation for the probability distribution function of the stochastic variable(s) depends, in general, on the discretisation prescription parameter, say α, and on the function that multiplies the noise, say g. Its stationary solution also depends on α and g [2,3,4,5].
Stratonovich noticed that the qualitative behaviour of the stationary probability distribution can change as a function of the noise strength [2], and thus deviate from the usual Gibbs-Boltzmann distribution which would only involve the potential responsible for the deterministic forces acting, say, on the particle. For instance, for adequately chosen functions g, the stationary probability distribution function can develop new extrema at values of the stochastic variable(s) that are not the ones set by the deterministic forces. Similar effects on extended systems can lead to coarsening phenomena in models without symmetry breaking potentials. The critical values of the noise strength at which the number or the nature of the extrema changes can be interpreted as noise-induced phase transitions [1].
However, one can adopt a different point of view from the one above. The stochastic differential equation can be modified so that the approach to the usual Gibbs-Boltzmann distribution is ensured for any discretisation prescription parameter α and multiplicative function g. This is achieved by adding a drift term to the stochastic equation [given in Eq. (2.8) and necessary even when the Stratonovich usual mid-point prescription is used]. Under this change, the α dependence disappears from the Fokker-Planck equation though the dependence on g remains. Still, the stationary solution with zero-current is the desired g-independent Gibbs-Boltzmann measure and the dynamic approach to it can also be proven.
We present a concise summary of these known, though perhaps not sufficiently assimilated, issues in the two first subsections of Sec. 2, that are supplemented by material in Apps. A, B and C, in the framework of a stochastic differential equation on a single variable.
It is sometimes found in the literature that any multiplicative white noise first-order differential equation can be transformed into a Langevin equation with additive noise, and that in this new framework all subtleties linked to the discretisation can be ignored. This statement is, however, wrong as the new additive-noise equation depends explicitly on the α-prescription used to define the discrete-time dynamics of the departing multiplicative noise equation. The reason is that the chain-rule for the derivative of a function of the stochastic variable has to be used in the transformation between multiplicative and additive noise equations, and this chain-rule involves α and g. Accordingly, the Fokker-Planck equation associated to the resulting additivenoise Langevin equation and its asymptotic solution depend on α and g. This can be cured by adding a drift term to the additive noise Langevin equation that is completely equivalent to the one to be used in the multiplicative noise formalism. We discuss these facts in Apps. B and C.
In Sec. 2.3 we recall the generating functional formalism for stochastic Langevin processes with multiplicative white noise [6,7]. We explain why we do not agree with the path integrals used in Ref. [8] and Refs. [9,10]. Concerning the latter, we show in App. E that non-linear changes of variable within the path integral generate a priori unexpected terms in the action, and invalidate the construction in Ref. [9] later used in Ref. [10]. The nontrivial effects of non-linear transformations within the path integral were already observed in the quantum field theory context [11,12,13,14] and in the stochastic field theory one as well [15,6].
The functional formulation of stochastic processes is well-suited to prove model-independent properties of generic physical observables. In this paper we will use a transformation of the path-integral formulation to show that the equilibrium fluctuation-dissipation theorems and the out of equilibrium fluctuation relations [16,17,18,19,20,21] hold for multiplicative white noise Markov processes with drift terms that ensure their approach to usual Gibbs-Boltzmann equilibrium. This approach is different from the one of other groups, see e.g. [8], [9,10] and [22,23], where no drift terms were added and the asymptotic distribution function involves a non-equilibrium potential. We will also present another transformation that leaves the action (and measure) invariant and serves to derive the Schwinger-Dyson equations for correlation and linear responses.
For simplicity, we will present detailed derivations in the framework of a one-dimensional stochastic equation. The generalisation to higher dimensional problems, and field theories, should then be clear. At the end of the paper, we will apply our results to the dynamics of a magnetic moment governed by the stochastic Landau-Lifshitz-Gilbert equation [24,25] taking advantage of the path-integral formalism developed in Ref. [26].
Recapping, the paper is organised as follows. In Sec. 2, we recall some features of the Langevin, Fokker-Planck and path-integral formulations of stochastic Markov processes with multiplicative white noise in the framework of one-dimensional problems. We re-derive Crooks relation [27] between path probabilities for forward and backward stochastic processes, now extended to take into account non-trivial issues due to the discretisation of Markov stochastic processes with multiplicative white noise. We prove equilibrium and out-of-equilibrium fluctuations theorems. Complements to these sections are given in the Appendices. Section 3 is devoted to the application of these ideas to the Landau-Lifshitz-Gilbert stochastic equation. Finally, in Sec. 4 we open some ways for future research.

One-variable stochastic Markov processes
In this Section, we define the one-dimensional problem that we use as a framework to recall a number of important features that, sometimes, appear in confusing terms in the literature. We also discuss a time-reversal symmetry of the equilibrium generating functional and we use it to derive equilibrium relations such as the fluctuation-dissipation theorem. Out of equilibrium, this symmetry is broken and we use the resulting symmetry-breaking terms to derive fluctuation relations. We then discuss another symmetry of the generating functional, valid in and out of equilibrium, that is useful derive the Schwinger-Dyson equations for correlations and linear responses.

The Langevin equation
Let us consider a real single variable x, the dynamics of which is governed by the following Langevin equation [7] d t where ξ(t) is a Gaussian white noise with zero mean ξ(t) = 0 and variance The noise is said to be multiplicative because it acts multiplicatively on g(x), a function of the stochastic variable. This stochastic differential equation makes sense only when complemented with a discretisation prescription to define at which point g(x) should be evaluated. This is relevant since each pulse ξ(t) yields a discontinuity in x and therefore the value of x at which g(x) is evaluated (and hence the size of the discontinuity) is a priori not well defined. Without restricting the generality of the foregoing, we work with the generic α prescription [3,4,5] which corresponds, in discrete time, to with dW n ≡ ξ n dt, dW n = 0 and dW n dW m = 2Dδ nm dt for the statistics of the noise, and with α a real parameter 0 ≤ α ≤ 1. Note that the discretisation used in the argument of f is irrelevant in the continuous-time limit. Equation (2.1) can be taken into a form in which the noise appears additively but the dependence on the discretisation parameter α is still present in the new equation, see Apps. B and C. As explained in Refs. [3,4,5] and recalled in App. A, the chain-rule for the time-derivative of a function F of the variable x depends on the stochastic equation governing the time evolution of x; in this case it reads where d t x ≡ dx/dt. The usual chain-rule of conventional calculus is recovered only in the case α = 1/2, the so-called Stratonovich mid-point prescription.
(Note that the chain-rule is independent of the 'force' f (x), that is to say, it will take the same form for the Langevin equation with the drift term, Eq. (2.11), to be discussed below.)

The Fokker-Planck equation
The Fokker-Planck equation for the Langevin equation (2.1) in a generic α-prescription reads [3,4,15,23] (Note the difference between this equation and the one used in Ref. [8].) Once supplemented by an initial condition P i (x) = P (x, 0), this equation describes the deterministic evolution of the probability density P (x, t) of finding x at time t. It can be written in the form of a continuity equation ∂ t P = ∂ x J. Its stationary solution with vanishing current, J = 0, is where x represents the indefinite integral over x ′ and Z is a normalisation constant [3,4,23]. The approach to this asymptotic form can be proven with the construction of an H-function as in Ref. [28], or with the mapping of the Fokker-Planck operator into a Schrödinger one and the analysis of its eigenvalue problem [29]. Clearly, the fact that P st depends on α and g shows that these can have highly non-trivial consequences on the transient dynamics and the asymptotic stationary properties of the system [1,2,30].
However, if we allow ourselves to consider the special 'drift' force with the short-hand notation V ′ ≡ ∂ x V and g ′ ≡ ∂ x g, the Fokker-Planck equation loses any dependence on α, Importantly, since physical observables are computed using P (x, t), this implies that the physics of Eq. (2.1) with the drift force in Eq. (2.8) does not depend on the prescription parameter α. This can also be proven using the BRST symmetry of the generating functional [22] or with a perturbative analysis [31]. Moreover, the asymptotic solution of the Fokker-Planck equation in Eq. (2.9) simply reads independently of α and g. P GB stands for the Gibbs-Boltzmann equilibrium distribution function in the canonical ensemble of statistical mechanics. Therefore, in order to describe the Markovian stochastic dynamics of a physical quantity subject to multiplicative white noise and that reaches a usual Gibbs-Boltzmann equilibrium measure one needs to work with the drifted Langevin equation in a generic α-prescription. Several features of this equation should be remarked. First, there is a non-trivial drift force even in the Stratonovich mid-point (α = 1/2) prescription. Second, the extra term is not equal to the one in the chain-rule (2.5). Moreover, this equation is equivalent to the original 'undrifted' Eq. (2.1) with a post-point prescription α = 1.
Although it is conventional to work with Langevin equations of the form of Eq. (2.1) or Eq. (2.11) in which the left-hand side (lhs) is solely given by the time derivative, d t x, it can be illuminating to re-write the latter as where we re-parametrised g(x) ≡ 1/k(x). Equation (2.12) has the form claimed in Ref. [32] for the Markovian overdamped dynamics of particles subject to forces deriving from a potential V and interacting with a bath of oscillators via a non-linear coupling K(x) with K ′ (x) ≡ k(x). The exact integration over the degrees of freedom of the oscillators gives rise to a viscous friction force, here in the left-hand side (lhs), as well as the multiplicative noise in the right-hand side (rhs). The drift force was not discussed in [32] since the focus in this paper was on non-Markovian dynamics (either because of the presence of a coloured noise or inertia) for which case no drift force is needed to ensure the convergence to the usual Gibbs-Boltzmann distribution [3,4,5]. We can therefore re-interpret the time derivative, d t x, in the lhs of Eq. (2.1) or Eq. (2.11) as originating from the dissipative interaction with the same equilibrium bath that is responsible for the random noise ξ(t).

The path-integral formulation
The probability distribution for a given trajectory of x, with initial condition x(−T ) at time −T distributed according to P i (x(−T )), and governed by Eq. (2.1) is where we used the short-hand notation for the time integral in the exponential that runs over the (symmetrised for convenience) time interval [−T , T ], and x t = x(t) for the time-dependent functions. We introduced the time-dependent auxiliary field ix t to exponentiate the δ-function that imposes that x t be a solution to the Langevin equation. We wrote the latter constraint in the compact form Eq t [x, ξ; α] = 0. The Jacobian J is given by (2.14) The calculations detailed in App. D yield [33,34] This action coincides with the form given in Ref. [7] but differs from the one in Ref. [8] that lacks the first α-dependent term (that should be present even for α = 1/2). For reasons explained in App. E, we disagree with the construction of the path integral presented in Refs. [9,10]. For the drift force that ensures the approach to the usual Gibbs-Boltzmann equilibrium, Eq. (2.8), the MSRJD action reads

Fluctuations
The stochastic nature of the dynamics is responsible for fluctuations of the field (here x t ) and more generally of all the possible physical observables that depend on this field [i.e., any A(x)]. Amongst the few universal results that apply to these dynamics, there is a class of exact relations between the path probabilities that are very precious since they lead to strong relations between observables. In a functional formalism, these relations can be proven by 1) making use of physical symmetries or broken symmetries of the system or its dynamics, 2) exploiting the invariance of the generating functional under a dummy linear change of integration variables. In this Subsection, we shall discuss these relations in the context of stochastic Markov processes with multiplicative white noise as defined in Eq. (2.11).

Relation between path probabilities
Let us consider the cases in which the force f depends on a set of externally controlled, possibly time-dependent, parameters λ t . The stochastic process is characterised by the path integral (2.15) that expresses the joint probability distribution, P [x, ix; α, λ], of the time series {x t , ix t } of the physical and the auxiliary fields, in the α-prescription, and under the set of parameters λ t . Following Crooks [27], we ask how does P [x, ix; α, λ] compare to the probability distribution of the transformed time-dependent variables {T x t , T ix t } in another discretisation prescription, α, and, possibly, under a transformed set of parameters, λ t . By choosing adequately the transformation rules T x, T ix, α and λ we will obtain relations of the type We have distinguished the notation for the transformation of the dynamical fields, T x, T ix, from the changes in the discretisation parameter, α, and the external time-dependent parameter, λ t . The relation (2.17) implies, for the average of a generic function A of the physical and auxiliary fields (but, for simplicity, not of their time-derivatives) Moreover, if the measure over the transformed fields can be related to the one over the original ones with a unit Jacobian, and if the domain of integration at each time slice, here the real axis, is unchanged or can be taken back to the real axis, then the relation above becomes With different choices of the function A, and their associated B, one can derive various relation. In particular, choosing A = 1, In Sec. 2.4.2, we will identify the transformation T eq , associated with the time-reversal invariance of equilibrium dynamics, that leaves the probability density invariant (∆S = 0) whenever the system is subject to equilibrium conditions, meaning initial conditions drawn from the Gibbs-Boltzmann distribution P GB ∝ e −V /D and dynamics given by the Langevin equation Eq. (2.11), with a drift force deriving from the same potential V , and in contact with a thermal bath at the same temperature such that k B T = D. We later use this invariance to derive generic properties of equilibrium dynamics, such as the fluctuation-dissipation theorem (Sec. 2.4.3). Out of equilibrium, ∆S = 0, and we will derive in Sec. 2.4.4 various fluctuation relations that have been extensively studied in recent years [16,17,18,19,20,21].

The time-reversal transformation
We look for the invariance of the generating functional that corresponds to the time-reversal invariance of the equilibrium dynamics. Although the action functional in Eq. (2.16) is relatively cumbersome, the identification of the correct field transformation that leaves it invariant can be simplified by the fact that one expects the time-reversal invariance to hold for the system and its environment separately. In other words, we expect the terms in the action that have their origin in the coupling to the bath to transform independently from the rest of the action. We collect them in where, to simplify notations, we defined and, we recall, D = k B T = β −1 . For a field x t corresponding to a physical quantity x that is even under time-reversal transformation (such as the particle's position), the transformation of the physical field must naturally be x t → x −t . The expression of S diss in Eq.(2.22) suggests that we look for a transformation such that d This is only true if we simultaneously transform the discretisation parameter α → 1 − α. Altogether, we are led to propose the following transformation of the dynamical field x t and its associated auxiliary field ix t complemented with the transformation of the discretisation parameter It is easy to check that S diss is indeed invariant under this transformation: We now have to check that the remaining terms in the action functional are also invariant under the proposed transformation. In the potential case with no time-dependent parameter (λ t = ct) and a drift force ensuring the convergence to the usual Gibbs-Boltzmann equilibrium measure, the remaining terms are gathered into and they transform as We recognise that the first boundary term, ln P i (x T ), needs to be taken back to the initial time, −T , if one wants to recover the original S det+jac [ix, x; α]; the second and last terms are already part of S det+jac [ix, x; α]; rewriting 1 − α = α + (1 − 2α), the fourth term produces the last piece needed to fully reconstruct the original S det+jac [ix, x; α]. All in all, we have Using the explicit form of d (α) t x t in Eq. (2.23), we can simplify this expression as follows Replacing the term in d t x t V ′ t above by using the chain-rule of stochastic calculus recalled in Eq. (2.5) [23], Finally, with initial conditions drawn from the Gibbs-Boltzmann distribution we end the proof of the full invariance of the equilibrium action functional in Eq. (2.16) under the transformation T eq given in Eq. (2.25): and ∆S = 0. Note that to achieve this invariance, there was a subtle interplay between the contributions coming from the deterministic part of the action and the ones coming from the α-dependent Jacobian. This invariance of the action functional yields the following relation between path probabilities (2.35) After the transformation T eq , the domain of integration ofx t at each time slice of the generating functional is shifted from the real axis to the complex line with a constant imaginary part iD −1 d t x t . Using the analyticity of exp S[x; ix; α], one can return to an integration over the real axis by closing the contour at both infinities and by dropping the contributions of the vertical ends that vanish owing to the term D(ix t ) 2 . Note also that the Jacobian associated to the change of variables {x, ix} → {T eq x, T eq ix} is unity. Finally, D[x,x] = D[T eq x, T eq ix] and we obtain the following relation between the forward and backward path probabilities where we defined The relation (2.36) is valid whenever the system is in thermal equilibrium.

The fluctuation-dissipation theorem
The fluctuation-dissipation theorem is a model-independent relation between the linear response and the correlation of spontaneous equilibrium fluctuations of a given observable. The linear response of x with respect to a previous perturbation is defined as where the infinitesimal perturbation h couples linearly to the field x in such a way that the potential V → V h = V −hx and, therefore, In the path-integral formulation, the linear response is given by where the action has been modified as (2.40) Therefore, the linear response is expressed as a correlation function reading where the average has to be taken with the measure given by the unperturbed action S[x, ix; α]. The subindex α expresses the fact that the stochastic process is defined with a discretisation parameter α. Exchanging momentarily α by 1 − α one has Take now the expression in Eq. (2.42) and perform the change variables in the path integral from {x t , ix t } to {T eq x t , T eq ix t }: where T eq g = g(T eq x t ) and similarly for T eq g ′ . Using that S[T eq x, T eq ix; 1 − α] = S[x, ix; α], and applying the transformation T eq defined in Eq. (2.25) to the function of x t and ix t ′ to be averaged, one has Identifying R α (−t, −t ′ ) in the first term in the rhs and using now d Using the fact that the physics cannot depend on the discretisation parameter [see the discussion below the drifted Fokker-Planck Eq. (2.9)], we can drop the irrelevant index α (or 1 − α) in the linear response and the correlation function and We apply the transformation T eq once again on the correlation function in the rhs to show C(−t, −t ′ ) = C(t, t ′ ). Owing to the time-translational invariance of equilibrium dynamics, C(t, t ′ ) = C(τ ) and and to the causality of the response R(τ ) = 0 for τ < 0, we obtain the celebrated fluctuation-dissipation theorem (FDT) Here Θ(τ ) is the Heaviside step function.

Broken symmetry and fluctuation theorems
There are various ways to drive a system out of equilibrium, e.g., by changing an external parameter in the potential in time, λ t , or by using initial conditions that are not in equilibrium, P i (x −T ) = P GB (x −T ). In these cases, the associated time-reversal symmetry of the dynamics is broken and the action functional is no longer invariant under the field transformation T eq . In practice, this means that ∆S defined in Eq. (2.17) does not vanish and the very same transformation technique that we used earlier can now be used to derive exact out-of-equilibrium relations between path probabilities, the so-called fluctuation relations.
The time-reversed dynamics corresponds to evaluating the action functional in the transformed fields T eq x t and T eq ix t , the discretisation prescription parameter α = 1 − α, and the time-reversed protocol (2.47) In order to evaluate ∆S, one first notices that the dissipative part of the action S diss does not depend upon the applied force nor the initial condition. Therefore, it remains invariant under T eq . However, as S det+jac depends on both the initial distribution and the force, we expect it to yield ∆S = 0. More precisely, the terms contributing to ∆S are given in Eq. (2.30) where V ′ t and V ′′ t are the first and second derivative with respect to the variable x of the potential V , If the potential depends on the time-dependent parameter λ t , its total time-derivative expressed in Eq. (2.31) acquires an extra term, and we use this new relation to replace d t x ∂ x V t in the first term in Eq. (2.30): Here, we made explicit the dependence on the time-dependent parameter λ t of the potential and the initial probability distribution function. The second term in the rhs is related to the work done by the time-dependent potential force For generic P i we cannot simplify further the last four terms in the rhs of Eq. (2.49) and ∆S is the stochastic entropy, defined as the sum of the Shannon entropy [ln P i (x T , λ T ) − ln P i (x −T )] and the heat transfer (βQ = β∆V − βW and β = D −1 ).
If, instead, we assume that the system is initially prepared in the Gibbs-Boltzmann distribution at temperature k B T = D = β −1 , under a potential V (x −T , λ −T ), we find The last two terms can be regrouped into In conclusion, we obtain the following relation between the forward and backward path probabilities where we defined Multiplying both sides of Eq. (2.56) by A, a generic observable which can depend on x and ix, and summing over all paths, one obtains where the subscripts F and B stand for averaging with the forward and backward path probability distributions defined in Eq. (2.57).

Schwinger-Dyson equations
We end the analysis of the single-variable problem by presenting an easy derivation of the Schwinger-Dyson equations linking correlation and linear responses that is based on the use of another set of transformation rules that leave the action and measure invariant and hold in general.

Out-of-equilibrium symmetry
Let us consider the most generic out-of-equilibrium situation, i.e. work with the original Langevin Eq. (2.1) without making any assumption on the force f , that can possibly be time-dependent. We recall that the MSRJD action functional associated to the dynamics reads

This action is invariant under the transformation
with no need to change the parameter α. We do not reproduce here the proof of invariance as it is rather straightforward.

Ward-Takahashi identities
Let us use this invariance of the action functional in the expression of the linear response to a perturbation h such that We obtain If one uses the drift force given in Eq. (2.8) to ensure convergence to equilibrium with a Gibbs-Boltzmann measure P GB ∝ e −V /D , this yields (2.63)

Composition of T eq and T eom
For equilibrium conditions, when working with the drifted Langevin Eq. (2.11), both transformations T eq and T eom are symmetries of the generating functional and therefore, their composition complemented with the transformation of the discretisation parameter is also a symmetry of the generating functional. Starting from the expression of the linear response, we derive Using the causality of the linear response, applying the transformation one more time, using the time-translational invariance of equilibrium dynamics, and D = k B T = β −1 we finally obtain the relation This is a generalisation of the relations found in Refs. [37,38,39] for additive noise stochastic processes. The special interest of this kind of relation is that it allows to compute the linear response, notably in a numerical evaluation, without any need to apply a perturbation, by taking advantage of the expression of R as the sum of two correlation terms.

Many-variable stochastic Markov processes
In Sec. 2, we focused on a stochastic differential equation on a single variable x. In a more general situation, the stochastic variable can be a multidimensional vector. In this Section, we focus on the stochastic Landau-Lifshitz-Gilbert (sLLG) equation [24,25], a Langevin equation describing the dynamics of a classical magnetic moment, a 3d vector M, in contact with an environment. We analyse the time-reversal transformation of the magnetisation and auxiliary vector that leaves the action invariant under equilibrium conditions. We derive some of its consequences, such as the fluctuation-dissipation theorem. Furthermore, we also analyse the out-ofequilibrium dynamics driven by a spin-polarised current of electrons and we derive the corresponding fluctuation relations.

The sLLG magnetisation dynamics
In Ref. [26] we gave a detailed presentation of the stochastic Landau-Lifshitz-Gilbert (sLLG) equation that describes the dynamics of a magnetic moment M under the influence of a deterministic local magnetic field H eff and a thermal environment responsible for both dissipation and a fluctuating local magnetic field H(t). Studying this equation in various discretisation prescriptions, we showed that unless the Stratonovich mid-point is used, a drift term is needed to conserve the magnetisation modulus in the course of time, |M| = M s , and to ensure the approach to Gibbs-Boltzmann equilibrium in the absence of non-conservative and time-dependent forces. Numerical checks of this fact were discussed in Ref. [40]. In Ref. [26] we also constructed the path-integral formalism for the generating functional in the Cartesian and spherical coordinate systems. Adapting the results of Sec. 2, i.e. identifying the field transformation that generalises the one in Eq. (2.25) to the physical problem at hand, we derive the corresponding fluctuation-dissipation theorem as well as the fluctuation relations.
In the so-called Gilbert formulation, the sLLG equation reads [41] D (α) This equation has to be understood in a generic discretisation prescription, and satisfies D t . The second term in D (α) t is necessary to ensure the conservation of the modulus of the magnetic moment, M s = |M|, and the approach to the Gibbs-Boltzmann equilibrium, see Eq. (3.6), in the absence of non-conservative forces and time-dependent magnetic fields [26]. We note that D (α) t in the last term between the square brackets in Eq. (3.1) can be simply replaced by d t as the second term in Eq. (3.2) does not contribute due to the vector product with M.
The chain-rule for the time-derivative of a function of the magnetisation vector governed by Eq. (3.1) reads [26]  The sLLG equation depends on several parameters. γ 0 = γµ 0 is the product of γ, the gyromagnetic ratio relating the magnetisation to the angular momentum, and µ 0 , the vacuum permeability constant. The gyromagnetic factor is given by γ = µ B g/ (in our convention γ > 0 for the electronic spin) with µ B Bohr's magneton and g Lande's g-factor.
The Gaussian white noise H(t) acts multiplicatively on the magnetisation. It has zero average and correlations characterised by the diffusion constant D, We assume that the environment is in equilibrium at the temperature k B T ≡ β −1 yielding the Einstein relation with V the volume of the system and η the friction coefficient that also appears in the last, dissipative, term in the rhs of Eq. (3.1). Indeed, the term −ηD (α) t M induces dissipation in the form introduced by Gilbert [42]. The deterministic magnetic field H eff collects conservative and non-conservative contributions. The former can be derived from a potential energy density U as H c eff = −µ −1 0 ∇ M U whereas the latter does not admit such a representation. U can possibly have contributions from an externally applied magnetic field H ext (that we assume to be constant for simplicity) and from a local magnetic field typically generated by the anisotropy potential of the local crystal structure (the so-called crystal field) A timely example of a non-conservative H nc eff is the so-called spin-torque exchange. In the context of spintronics, the manipulation of the local magnetisation is performed by circulating a spin-polarised current of electrons through the ferromagnetic material. This can exchange angular momentum with the magnetisation via the spin-torque term where χ t is a time-dependent parameter that is proportional to the externally controlled current J t , and p t is a unit vector indicating the spin polarisation of We will prove the time-dependent fluctuation-dissipation theorem for this problem below (see also [43]). For simplicity, we study the dynamics for a system initially prepared in equilibrium with a Gibbs-Boltzmann distribution P i (M −T ) = P GB (M −T ). In particular, we set χ −T = J −T = 0. In the absence of a drive, H nc eff = 0 ∀t and for a time-independent effective field H eff , the system remains in thermal equilibrium. However, a finite drive H nc eff = 0 or a time-dependent effective field H eff (t) push the magnetic moment out of equilibrium.

The path-integral formulation
As shown in Ref. [26], the generating functional reads The action functional can be expressed as the sum where P ⊥ ij ≡ M 2 s δ ij − M i M j is the projector on the plane perpendicular to M and ∂ j is a short-hand notation for ∂ M j . S det encodes the initial conditions and the deterministic forces. S diss encodes the hybridisation with the thermal bath. The term in iM t imposes the conservation of the modulus of the magnetisation vector. S jac stems from the unicity of the solution to Eq. (3.1) once the initial conditions and the noise history are specified. For magnetic fields that are independent of the magnetisation vector, H eff = H ext , S jac vanishes.
The two-time correlation between any function of M, A(M), and, say, one component of the magnetisation, M j , reads The linear response of the same observable A measured at time t to a previous perturbation H that modifies the potential energy per unit volume U

Fluctuations
Equilibrium dynamics are ensured as long as the system is initially prepared in equilibrium at a given temperature T and under conservative forces, and that it is subsequently evolved under the same time-independent potential forces and in contact with an environment at the same temperature. In our setup, this corresponds to setting the drive to zero, i.e. H nc eff = 0 at all times, and taking H ext constant and V ani not explicitly dependent on time.

The time-reversal transformation
One can prove that the action in Eqs. (3.11)-(3.14), and more generally the full generating functional, are invariant under the following variable and discretisation parameter transformations This discrete symmetry of the dynamical action encodes all the features of equilibrium dynamics and it is broken out of equilibrium.
Case without anisotropy potential, V ani = 0. Let us start the proof by treating this simpler case in which H eff = H ext . It is easy to see, similarly to the one-dimensional example of Sec. 2, that the dissipative terms in S diss are invariant independently of the other terms of the action functional. We do not reproduce here this calculation that is quite straightforward. We simply mention that we do not need to use α → 1−α since in this case S diss does not depend on α explicitly. Let us now discuss the invariance of S det and S jac . Since H eff = H ext , S jac vanishes and is therefore trivially invariant under T eq . For S det , we look separately at the terms proportional to iM ⊥ t and iM t . For the former Given the symmetry property U(−M, −H ext ) = U(M, H ext ), we have Z(−H ext ) = Z(H ext ). We first change t → −t as the integration variable in the temporal integrals and we next rearrange terms to write The first integral has the original form. The second integral can be computed directly, as H ext is independent of M. One recovers the boundary terms at T and −T ; one cancels the first term in the rhs, the other one builds the initial probability weight. The term that imposes the spherical constraint is invariant on its own if we use Note the different transformation rules on iM ⊥ and iM . The transformation α → 1 − α is needed so that the two terms in D Case with an anisotropy potential, V ani = 0. Let us now examine the generic case in which there is an anisotropy potential and S jac no longer vanishes. The analysis of the dissipative part of the action is identical to the one we discussed in the previous subsection. As for the combined contributions S det + S jac , we proceed as follows. We start from Eq. (3.20) conveniently generalised to take into account the fact that H c eff = −µ −1 0 ∇ M U: We use now Eq. (3.3) in the last term and we obtain +βV U(M T , H ext ) − βV U(M −T , H ext ) after integrating the total time derivative. Using the property U(−M T , −H ext ) = U(M T , H ext ), the first term cancels the first term in the rhs in (3.23) and the second term reconstructs the exponential weight in the initial distribution. We therefore have where we replaced D by its definition in Eq. (3.5), while We notice that the first term is what we need to build S jac [M, α] and the last term cancels the remaining one in Eq. (3.25). The invariance of the term imposing the spherical constraint works in the same way as in the V ani = 0 case.
We have therefore completed the proof of invariance of the action under the transformation in (3.17).

The fluctuation-dissipation theorem
Applying this symmetry to the linear response, see Eq. (3.16) we obtain where A r is the time-reversed observable of A. In the last step, we applied the transformation another time to the last term. All averages are taken with the unperturbed action measured in the original variables and with the α parameter, what we would call S[M, iM; α]. Using the causality of the response, R ArM j (−t, −t ′ ) = 0 for t > t ′ and the time-translational invariance of equilibrium dynamics, one may simplify the expression above to where we introduced τ = t − t ′ and Θ(τ ) is the Heaviside step function. Note that this relation applies to any observable A. Higher order fluctuationdissipation relations of this kind can by easily derived.

Broken symmetry and fluctuation theorems
In this three-dimensional vector problem, the spin-torque due to a spinpolarised current χ t = 0 for t > −T is a non-conservative force that drives the magnetic moment out of equilibrium. It gives rise to the following contribution to the deterministic action, and to an extra term in the Jacobian, (3.30) When evaluated with the time-reversed variables (ensuring notably that H torq is odd under time reversal), the deterministic part of the action functional corresponding to the time-reversed dynamics reads We see that the term that is generated cannot be partially integrated away, as in the potential case, since H torq is not the gradient of a potential. The spin-torque contribution to the Jacobian transforms as (3.34) The rest of the action remains invariant under this transformation as it was in the absence of the spin-torque term and with no time-dependent parameter dependencies in the effective field H ext . Ultimately, we obtain the work performed by the spin-torque term. This result implies the relation where we defined the forward and backward path probability distributions

Conclusions and outlook
In this paper, we studied Markov stochastic processes with multiplicative white noise and adequately drifted to ensure their approach to the usual Gibbs-Boltzmann measure under equilibrium conditions. In this respect, our viewpoint is different from the one in Ref. [23] where no drift force was added to the stochastic differential equation and the dynamics approached a non-Gibbs-Boltzmann stationary state. The stochastic processes studied in this paper can also be represented as a path integral in an enlarged functional space including Grassmann variables, by adapting the construction presented in Ref. [23]. In this formulation and under equilibrium conditions, there is also a supersymmetryà la Parisi-Sourlas [44,45] that yields, through the corresponding Ward identities, a few exact relations on the equilibrium dynamics such as the fluctuationdissipation theorems.
In recent years, thermodynamic relations and concepts in out-of-equilibrium stochastic processes have been searched for. In this paper, we showed how fluctuation theorems can be derived for Markov processes with multiplicative white noise that, under no external drive, do reach thermal equilibrium with the environment. One could ask whether the effective temperature idea [46] applies to multiplicative noise processes that are not able to reach equilibrium with their surroundings and whether once set out of equilibrium they would satisfy fluctuation theorems in the way discussed in Ref. [47] for additive noise processes. Exploring the stochastic thermodynamics [20,48] proposals for multiplicative noise Markov processes should also be an interesting research project.

A Stochastic calculus A.1 Chain-rule
We examine the time-derivative of a generic function, [F (x n+1 ) − F (x n )]/dt, with the stochastic variable x governed by the Langevin equation with multiplicative white noise, , and x n in F (x n ), around the generic α pointx n = αx n+1 + (1 − α)x n we obtain that in the limit dt → 0 becomes This expression is written as the generalised chain-rule [3,4]

A.2 From the α to the Stratonovich prescription
One can transform a stochastic equation in the generic α-prescription into one in the Stratonovich mid-point prescription by simply expanding the arguments of f and g around the latter points. More precisely, let us start from an equation in the α-prescription i.e. x n = x n + α(x n+1 − x n ). The Stratonovich mid-points are x S n = x n + 1 2 (x n+1 − x n ), and α and S points are related by Expanding now Eq. (A.4) around the S mid-points one finds where we dropped contributions of O(dt 3/2 ) and we used (dW n ) 2 = 2Ddt. The function g that multiplies dW n is evaluated now at the S point x S n and in this sense this is an equation in the Stratonovich prescription.
This same strategy can be followed to transform an equation from the α to the α ′ -prescription at the price modifying the force with an adequate drift term.

B From multiplicative to additive noise
It is often found in the literature that a multiplicative-noise process can be mapped to an additive-noise process, and that in the latter formulation all subtleties linked to the discretisation prescription can be simply forgotten.
Here we show that, while indeed such a mapping exists [49], the discretisation used to define the original multiplicative noise process enters the additivenoise process in the form of a non-trivial drift force.
Let us re-parametrise the original equation of motion, Eq. (2.1) with g(x) ≡ 1/k(x), such that the origin of the multiplicative noise and of the velocity in the lhs can be tracked back to a non-linear coupling to a thermal bath of oscillators, see the discussion below Eq. (2.12). We write it as the equation (in the α-prescription) where we also re-parametrized f by f (x)k 2 (x) → f (x) such that f can now be thought as a true force, in units of Newtons, possibly deriving from a potential f (x) = −V ′ (x). If we now divide this equation by k(x), we obtain an equation in which the noise appears additively. However, it has to be treated with great care because the term k(x)d t x hides subtleties associated with the discretisation.
Equation (B.1) is defined in the α-prescription, for which unusual rules of calculus apply. Re-writing it momentarily as one can show that the associated chain-rule reads [see proof in Gardiner's book [3] and recalled in Eq. (E. 2)] can be re-written as Let us now perform the change of variable from x to u according to the non-linear transformation to get the additive-noise process where we introducedf (u) ≡ f (x(u))/K ′ (x(u)). We finished mapping the original multiplicative noise process in Eq. (B.1) to an additive-noise Langevin equation with a simple time-dervivative d t u in the lhs, but the first term in the rhs is non-trivially inherited from the discretisation of the original multiplicative-noise process and depends explicitly on α and k. It only vanishes for α = 1/2 (Stratonovich calculus) or k ′ = 0 (additive noise in the original Langevin equation).

C Stationary distribution and drift term
We revisit here the need for a drift term to ensure the approach to the Gibbs-Boltzmann distribution in the asymptotic long-time limit by working with the Langevin equation approach exclusively. We show that, given the generic multiplicative-noise equation (B.1), the stationary distribution is not of Boltzmann form unless a force is added to the conservative force, consistently with what we found with the Fokker-Planck approach in Sec. 2.2.
Case of a Stratonovich prescription. Let us first treat the simpler case of α = 1/2. We learned above that Eq. (B.1) can be re-written as the additive noise equation of motion The stationary probability distribution of the stochastic variable u governed by Eq. (C.1) is and N a normalisation constant. Switching back the stochastic variable x governed by Eq. (B.2), the corresponding probability distribution can be recovered as This corresponds to the usual Gibbs-Boltzmann distribution if we allow ourselves to work with With this choice, the equation of motion becomes If we re-parametrize back to the original notations of Eq. (2.1), i.e. k → 1/g and f → f /g 2 , we obtain which is Eq. (2.11) in the case α = 1/2.
Generic α-prescription. The generalisation to the generic α case is straightforward. Equation (B.1) can be re-written as Eq. (B.6) with a stationary probability distribution P st (u) = Ne −β V k (u) for u and Switching back the the stochastic variable x, via u = K(x), the corresponding stationary probability distribution is This corresponds to the usual Gibbs-Boltzmann distribution if we allow ourselves to work with The Langevin equation becomes If we re-parametrize back to the original notations of Eq. (2.1), i.e. k → 1/g and f → f /g 2 , we obtain

D Path integral construction
We sketch a construction of the path integral for the Langevin equation of motion with multiplicative white noise: We use a continuous time notation with the discretisation subtleties being encoded in the choice of the value of the Heaviside theta-function at zero, Θ(0) = α.
The explicit calculation of the Jacobian yields . Note that if g(x) = ct the noise appears explicitly in the functional under det tt ′ . After some simple algebra, J can be factorised as and the first factor can be discarded in the normalisation. We can now rewrite the second factor with the help of the identity det(1+C ξ ) = exp Tr ln(1+ C ξ ) with the causal function C ξ (x, t, t ′ ) = Θ(t − t ′ )A(x, ξ), where we highlighted the dependence of C ξ on the noise by adding a subscript ξ to C. The ln(1 + C ξ ) can now be expanded in Taylor series. Usually, the causal structure of C (that is also usually noise-independent) truncates the series at first order in C. However, in this explicitly noise dependent case one needs to be careful and also keep the quadratic order [50]: , ξ(t ′′ )). Using now Θ(0) = α, and simplifying notations such asẋ = d t x, g ′ (x(t)) = g ′ , C 2 ξ (x, t, t ′ ) = C 2 ξ and dt = , P [x] reads (D.5) (To alleviate the notation we do not write here the time-dependent of the functions in the action, as we do in the main text.) Before performing the integration over ξ that involves let us translate the noise by a function of the variables x and ix, ξ → ξ + 2D (ixg − αg ′ ), in the functional integral. Notice that ξ ∈ R but ix ∈ iR.
We can restore the original integration domain using the analyticity of the exponential that is zero on the boundary thanks to the term −(4D) −1 ξ 2 . The functional integral in (D.6) transforms into a new path integral Keeping the terms in C 2 ξ+2D(ixg−αg ′ ) that are quadratic in the noise and yield a δ(t − t ′ ) contribution within the Tr tt ′ under the noise average, and using the notation . . . dt dt ′ Θ(t−t ′ )Θ(t ′ −t)g ′ (x(t))g ′ (x(t ′ )) ξ(t)ξ(t ′ ) = e −Dα 2 g ′2 .
Altogether we obtain P [x] ∝ D[x] e S[x,ix] with the action (to which we should add the contribution from the initial measure). This is the same action derived by Lau and Lubensky [7] who used a slightly different approach in which the equation of motion was reformulated as This is convenient because the noise does not appear explicitly in the Jacobian, although its effect subtly re-appears along the calculation

E Non-linear change of variables
In this section we examine the transformation of the stochastic path-integral under generic non-linear changes of variables. Similar questions were posed in the quantum field theory context [11,12,13,14] and in the stochastic field theory one [15]. We start by showing that the transformation is reversible and poses no problem at the level of the Langevin equation, but it generates non-trivial extra terms in the action when applied at the level of the path integral of the generating functional.

E.1 In the Langevin equation
Let us start from the x equation of motion in the form presented in Eq. (2.1): with the associated x-chain-rule [3] also recalled in the main text. We call Eq. (E.1) the x-eom. Let us consider the function u(x) and derive the corresponding equation of motion for u, that we call the u-eom. The chain-rule (E.2) applied to u yields the u-eom, As a trivial check, we re-transform the latter u-eom to the x-eom by applying the u chain-rule to the function x(u): If we replaceu by (E.4) and we use d u = 1/u ′ d x to replace we recover d t x = f + gξ, which is the expected x-eom.
We have therefore proven that the transformation from x to u and back to x is 'reversible' at the level of the Langevin equations. The same calculations could have been done for the drifted stochastic equations or within the Fokker-Planck formalism (the latter can be found in Ref. [49]).

E.2 In the path-integral
Here, the question we want to answer is whether, starting from the generating functional corresponding to the dynamics of u, we can transform variables to recover the generating functional corresponding to the dynamics of x.
We recall that the MSRJD action functional corresponding to the x-eom recalled in Eq. (E.1), reads Starting from the stochastic process for the variable u governed by Eq. (E.6), one constructs its generating functional by following the same steps as done in App. D for the variable x. One finds for all t, t ′ > −T . We now change variables from u to x using the identities We use the chain-rule for the time-derivative of the function u(x) given in Eq. (E.3), and the action becomes (E.14) The Jacobian of this change of variables is and cancels Jûṽ for all t, t ′ > −T , the remaining t = t ′ = −T contribution helps building the initial probability measure ln P i (x −T ) and the integration measure over x −T . Rearranging terms The identity dz e (iσz) 2 −iza−izb = e − a 2 4σ 2 − ab with σ 2 = Dg 2 , a = 2αDg 2 u ′′ /u ′ and b =ẋ − f + 2Dαgg ′ allows us to recast the terms in iṽ and the path integral as We notice that the terms in f cancel. By taking the explicit x-derivative of the last term we obtain −αD(1 − 2α) [2gg ′ u ′′ + g 2 u ′′′ ] and the term in α 2 here cancels with the one in the second line. One is left with if, instead of the expression for S given in Eq. (E.9), the action functional had read S + S diff with