About the Optimal FPE for Non-linear 1d-SDE with Gaussian Noise: The Pitfall of the Perturbative Approach

Abstract

This paper deals with the problem of finding the Fokker Planck Equation (FPE) for the single-time probability density function (PDF) that optimally approximates the single-time PDF of a 1-D Stochastic Differential Equation (SDE) with Gaussian correlated noise. In this context, we tackle two main tasks. First, we consider the case of weak noise and in this framework we give a formal ground to the effective correction, introduced elsewhere (Bianucci and Mannella in J Phys Commun 4(10):105019, 2020, https://doi.org/10.1088/2399-6528/abc54e), to the Best Fokker Planck Equation (a standard “Born-Oppenheimer” result), also covering the more general cases of multiplicative SDE. Second, we consider the FPE obtained by using the Local Linearization Approach (LLA), and we show that a generalized cumulant approach allows an understanding of why the LLA FPE performs so well, even for noises with long (but finite) time scales and large intensities.

1 Introduction

In the statistical physics framework, complex physical processes, deterministic by definition, are often modeled by stochastic differential equations (SDE), i.e., ordinary differential equations (ODE), where a stochastic forcing is added to mimic unresolvable fast fluctuating degrees of freedom. Any fundamental approach to physical phenomena should start from a Hamiltonian representation of the whole underlying system or model. However, typically the whole system is too complex (see, for example, ENSO [1,2,3]), or the part of interest of the whole system is characterized by only one long time scale. Then, it makes sense to approximate the whole dynamics by, in principle, a much simpler non-linear stochastic differential equation for the part of interest. The typical SDE will be non-Hamiltonian and will be given by

$$\begin{aligned} {\dot{x}}= - C(x) - \epsilon \xi (t), \end{aligned}$$

(1)

where x is the variable of interest, $-C(x)$ is the unperturbed drift field, $\xi (t)$ is a stochastic driving with zero average, a finite correlation time $\tau$ ,¹ and a normalized autocorrelation function $\varphi (t,u)=\langle \xi (t)\xi (u)\rangle _\xi /\langle \xi ^2\rangle _\xi$; $\langle \ldots \rangle _\xi$ denotes average over the realizations of $\xi (t)$. We set $\varphi (u):=\varphi (t,t-u)$, discarding the possible transient regime that would make the statistical properties of $\xi (t)$ non stationary. We also define $\tau =\int _0^\infty \varphi (u) \text {d}u$: note that the value of this integral can be much smaller than the decorrelation time between events $\xi (t)$ when the function $\varphi (u)$ decays oscillating with time. The parameter $\epsilon$ controls the perturbation strength and the corresponding diffusion coefficient, i.e., $D_0:=\epsilon ^2 \langle \xi ^2\rangle _\xi \, \tau$. Without loss of generality, we may assume that $\langle \xi ^2\rangle _\xi \, \tau =1$, so that $D_0=\epsilon ^2$.

In the nonlinear case the SDE (1) presents some challenges, both from the physical point of view, as an interpretation of a given observed phenomenon, and from the mathematical point of view. We must use caution to avoid falling into meaningless results, as we will see.

From a physical point of view, if ${\dot{x}}= - C(x)$ is assumed to be the “phenomenological” equation for a macroscopic observable x (for example, the charge on a condenser in an electric circuit), it is clear that $\xi (t)$ cannot represents the thermal fluctuations. This is because, if we take the average over the $\xi (t)$ realizations, the resulting equation would not be equivalent to setting $\xi (t)=0$. Thus, the non-linear SDE (1) would not have a physical basis. Starting from ${\dot{x}}= - C(x)$, the correct approach would be to introduce the statistics due to the thermal fluctuations through the proper Master Equation instead of a generalized Langevin equation. This point was investigated in details by Van Kampen [4]. However, it often happens that SDE (1) is the genuine result of a procedure of contraction/elimination of fast or unobservable variables of a complex system (e.g., by using the Mori approach [5,6,7], or by exploiting the temporal multiscale property of the system, see, for example [8,9,10]). In such a case, SDE (1) represents a good qualitative (and often quantitative) description of the complex phenomenon under study (see, for example [11] or [12]).

From a mathematical point of view, it is well known that if $\xi (t)$ of (1) is a Gaussian Markovian stochastic process (i.e., a Gaussian white noise), an equivalent FPE for the Probability Density Function (PDF) of the x variable of interest can be easily obtained, but paying the price of having to deal with non-differentiable functions (the famous Itô versus Stratonovich controversy [13]). Here, on the other hand, we will always be dealing with a Gaussian correlated noise with a finite correlation time $\tau$, a case for which the equivalent FPE structure usually breaks down, but which we would still like to recover within certain conditions. Our focus on correlated noise is due to the well-known fact that, unfortunately, white noise is often an oversimplification of the real driving acting on a system of interest. Correlated noise (often termed “colored” in the literature) is very common in continuous systems, and its importance has been recognized in a large number of very different situations, like for instance statistical properties of dye lasers [14,15,16,17] and laser technology [18], chemical reaction rate [19,20,21,22], optical bistability [23, 24], stochastic resonance [25] large scale Ocean/Atmosphere dynamics [2, 26], non-linear energy harvesting systems [27], sensors design [28], ecosystems [29,30,31], medical sciences [32,33,34], neural systems [35,36,37], material science [38, 39], systems driven by a quasi monochromatic or narrow banded noise [40, 41], and many others. Typically, the noise in the above papers was assumed to be exponentially correlated in time, but there are notable exceptions, like [39,40,41].

One can recover a FPE in the colored noise case, by embedding the original 1-D system in a higher dimensional space, for example by using one of the extensions of the Mori approach [5, 42]. However, in general, such a space is infinite-dimensional, thus an approximated truncation process must be implemented, following a systematic approach as the “reduced” model theory (RMT) [6, 43, 44]. In any case, relevant information regarding the statistics of the variable of interest x, such as the equilibrium PDF or the Mean First Passage Time (MFPT), is unlikely to be obtained from a multidimensional FPE, thus, increasing in number of degrees of freedom often does not allow us to gain information about the statistical quantities of interest. For this reason, we consider here the approximate FPE for the PDF, which we can get directly from SDE (1), in the case of colored noise.

Several techniques have been developed to deal with the correlation time of the noise in nonlinear 1D-SDE, with the aim of eventually obtaining this effective FPE. They can be grouped in three main categories that correspond to three general techniques: the cumulant expansion technique [45,46,47], the functional-calculus approach [32, 48,49,50,51,52,53,54] and the projection-perturbation methods (e.g., [6, 21, 55, 56]). Each of these methods leads to a formally exact evolution equation for the PDF of the driven process, and the different descriptions are therefore equivalent. The exact formal results do not lend themselves to calculations nor give a FPE structure, therefore they require that approximations be made. The approximations made within these various formalisms involve truncations and/or partial resummations of infinite power series with respect to $\epsilon$ and $\tau$, which are typically the small parameters in the problem. Not surprisingly, it has been argued [54] that the effective FPE obtained from the different techniques are identical, if the same approximations are made (time scale separation, weak perturbation, Gaussian noise etc.). The results of the approximations can be grouped in three categories: the “Best Fokker Plank Equation” (BFPE) obtained by Lopez, West and Lindenberg [54] from a standard perturbation method, where $\epsilon$ is the small parameter and $\tau$ is finite but (in general) not limited; the “Local Linearization Assumption” (LLA) FPE, that formally can be considered as a small $\tau$ expansion of the BFPE, and that has been obtained in different ways, e.g. by exploiting an ad-hoc projection procedure [57], or by using functional- calculus [12, 49,50,51,52,53, 58,59,60]; and, finally, the gen-FPE, that makes use of moments of the unknown response PDF [32, 48] and that, improving the old cited functional-calculus approach, leads to a non linear FPE for values of $\epsilon ^2 \tau$ enough large to include most of the cases of interest. It is also worth mentioning the Unified Colored Noise Approximation (UCNA) [61, 62], a filtering approach introduced for a general stochastic dynamic systems driven by an Ornstein- Uhlenbeck process (termed red noise in the literature). The approach is based on two steps: taking advantage of the simple characteristics of red noise, the number of degrees of freedom is increased by one, to obtain a multidimensional white noise SDE. Then, under the condition of small or large correlation times $\tau$, the number of degrees of freedom is reduced back to the original one, thanks to the white nature of the noise. In the limit of small $\tau$ the equivalent FPE coincides, of course, with the LLA one, but for large $\tau$ it is different. In general, filtering approaches have the drawback of increasing the dimensionality of the FPE equation when there is not a large scale separation between the correlation time of the noise and the dynamics of the system of interest.

A separate consideration deserves some interesting recent works on the gen-FPE [63, 64, 64]. The approximation scheme is based on an extension of the Novikov-Furutsu theorem and on a stochastic Volterra-Taylor functional expansion around the instantaneous values of appropriate response moments [64]. The results are not limited to the red noise case and are in excellent agreement with numerical simulations of the SDE in both the transient and long-time regimes, for any correlation function of the stochastic perturbation (assuming the system is stable). But, they are intrinsically limited to the case of 1-D correlated Gaussian noise and the gen-FPE has a nonlinear/non-local structure. Although this latter fact does not pose many difficulties in the numerical simulations (the nonlinearity and the non-locality appear in the diffusion coefficient(s)), it does not allow to directly use standard simple analytical tools developed for linear FPE as the eigenvalues approach or the Mean First Passage techniques (see, e.g., [65, Chap.5]).

In this work we focus our attention on the “standard” (i.e., linear and local) FPE to be associated, with good approximation, to 1D-SDE with a weak general correlated Gaussian noise, “weak” meaning really that we deal with expansions where the noise intensity is treated as a small parameter, to be compared to the other relevant appropriate quantities, depending on the physical problem under study. In practice, given that we will use a perturbative-projective approach, a weak noise is a noise such that the second order term ($O(\epsilon ^2\tau )$) is much larger than the next non identically zero term (either the third order term ($O(\epsilon ^3\tau ^2)$) or the fourth order term ($O(\epsilon ^4\tau ^3))$). Moreover we want to use a method that leaves open the possibility of including non Gaussian perturbations. For this reason we will deal, mainly, with the BFPE, a “Born-Oppenheimer”-like result (i.e., holding for weak noises), facing some known inconsistency problems, already highlighted in [66], where a “naive” recipe to solve them has been suggested.

Here will clarify the origin of these issues, which are related to the behavior of the unperturbed system of interest (i.e., Eq. (1) with $\epsilon =0$), so supporting the arguments of [66] and we will also extend the treatment to the case of multiplicative noise. As we will see, for Eq. (1), not even the continuity equation, or Reynold’s transport theorem, is guaranteed.

We can provide a physical intuition of the origin of these issues: let us consider the following system:

$$\begin{aligned} {\dot{x}}=v,\,\,\,\,{\dot{v}} =-U'(x)- \gamma v+f(t),\,\,\,\,U'(x)\equiv \frac{\partial U(x)}{\partial x}, \end{aligned}$$

(2)

where f(t) is a white noise with $\langle f(t) \rangle _{f} = 0$. For example, Eq. (2) describes a chemical reaction process in a solvent, with U(x) a reaction potential characterized by the barrier $E_b$ (Fig. 1)). The reaction rate k has the famous general Arrhenius-like structure $k=A \exp \left( -E_{b}/k_{B} T\right)$.

Fig. 1

Schematic drawing of a typical reaction potential in arbitrary units. The reactants are the particles with “coordinate” x smaller than $x^*$. The reactants move under the influence of the potential U(x) and of the “thermal bath”. When a reactant reaches the point $x^*$ the chemical reaction takes place, and it disappears. Thus $x^*$ is an absorbing point, and $E_b$ is the corresponding energy barrier

It is well known that if noise and friction are related to each other via the fluctuation-dissipation theorem, then a system, such as (2), can be considered Hamiltonian type [56, 67,68,69,70,71,72,73]. However, despite its simplicity, statistical quantities, like the A coefficient of the Arrhenius law, may be hard to derive. In this case, it is convenient to somehow reduce the number of degrees of freedom to one so that standard approaches, such as the MFPT technique, can be applied. One way to do that is to consider the under-damped or the over-damped limits. In the latter case, the contraction of the velocity variable results equivalent to [74, 75] considering that the acceleration is locally vanishing (i.e., $\dot{v}=0$ in (2)), leading to SDE (1), where $C(x)= U'(x)/\gamma$ and $\epsilon \, \xi (t):=-f(t)/\gamma$.

We will show that this apparently safe reduction to one degree of freedom leads to issues, from both a physical and a formal point of view, and we will assess when, moving from the original Hamiltonian system (2) to the 1-D-model (1), these issues arise. As an example in the over-damped case, the issue is related the assumption of a friction so large that the velocity immediately reaches a steady state (i.e., $\dot{v}=0$): this assumption breaks down if $U'(x)$ grows more than linearly with x and if the noise is large enough to push the trajectories towards large values of x. We will obtain the correction to the BFPE proposed in [66] when these issues are accounted for.

The paper is organized as follows: in Sect. 2 we provide a formal justification of the corrections to the BFPE suggested in [66]. In Sects. 3–4 we extend the results of [66] to SDE with multiplicative noises. Sect. 5 discusses the LLA FPE. In Sect. 6, we study the Stratonovich model, and in Sect. 7 we draw some conclusions. Appendices include additional theoretical considerations and comparisons with numerical simulations.

2 The Origin of the Correction to the BFPE Lies in the Anomalous Unperturbed Dynamics

Any perturbation approach applied to a weakly forced dynamical system, leads to results (the transport coefficients) expressed in terms of convolutions between functions of the perturbing variable and some functions of the unperturbed trajectories (a brief and simple introduction to this topic is given in Appendix A). If there is a large time scale separation between the perturbing force and the unperturbed system dynamics, then the possibly non linear nature of the latter does not plays any role and the issues we are considering in this paper do not appears. This is the case where it is possible to use, for example, the standard procedure of elimination of fast variables (e.g., [6]). On the other hand, if the time scale of the noise is not so much shorter than that of the unperturbed system, the possible nonlinearity of the unperturbed dynamics comes into play.

We are going to show that it is just the unperturbed non linear dynamics which can generate the issues we mentioned in the Introduction.

We begin considering the 1-D system (1) with $\epsilon =0$, from which, using the continuity equation, the unperturbed Liouvillian is obtained ($\partial _x:=\partial /\partial x$):

$$\begin{aligned} \mathcal{L}_a=\partial _x C(x). \end{aligned}$$

(3)

If Eq. (3) is the unperturbed Liouvillian of the PDF (in this deterministic case defined by some initial “ensemble”), it follows that the unperturbed evolution of the same PDF is

$$\begin{aligned} P_0(x,t)=e^{\mathcal{L}_at}P(x,0). \end{aligned}$$

(4)

From (4), with $x_0(x,t)$ the unperturbed evolution starting from the initial position x, the following formal equalities are obtained:

$$\begin{aligned} P_0(x,t)&=e^{\mathcal{L}_at}P(x,0) =e^{\mathcal{L}_at}P(x,0)e^{-\mathcal{L}_at}e^{\mathcal{L}_at} =e^{\mathcal{L}_a^\times t}\left[ P(x,0)\right] e^{\mathcal{L}_at} \nonumber \\&=P(x_0(x,-t),0)e^{\mathcal{L}_at} =P(x_0(x,-t),0)\left( e^{\mathcal{L}_at}C(x)\frac{1}{C(x)}\right) \nonumber \\&=P(x_0(x,-t),0)e^{\mathcal{L}_a^\times t}\left[ C(x)\right] \left( e^{\mathcal{L}_at} \frac{1}{C(x)}\right) \nonumber \\&=P(x_0(x,-t),0)C(x_0(x,-t)) \left[ \frac{1}{C(x)}+\left( \partial _xC(x)\frac{1}{C(x)}\right) +...\right] \nonumber \\&=P(x_0(x,-t),0)\frac{C(x_0(x,-t))}{C(x)}, \end{aligned}$$

(5)

where for any operator/function $\mathcal{A}$ and $\mathcal{B}$, we defined $\mathcal{A}^\times [\mathcal{B}]:=[\mathcal{A},\mathcal{B}]=\mathcal{A} \mathcal{B}- \mathcal{B}\mathcal{A}$; moreover we used the Hadamard’s lemma for exponentials of operators: $e^{\mathcal{A}t}\mathcal{B}e^{-\mathcal{A}t}=e^{\mathcal{A}^\times t}\left[ \mathcal{B}\right]$ (see Appendix B and [47] for details). If it does exist, the “inverse” of (5) reads:

$$\begin{aligned}&P(x,0)=e^{\mathcal{-L}_at}P_0(x,t)=P_0(x_0(x,t),t)\frac{C(x_0(x,t))}{C(x)}. \end{aligned}$$

(6)

Finally, we note that the quantity $C(x_0(x,t))/C(x)$ of (5) and (6) is the Jacobian of the variables transformation $x\Rightarrow x_0(x,t)$. Indeed

$$\begin{aligned} J(x_0(x,t),x)&:=\frac{\partial }{\partial x}x_0(x,t)= \frac{1}{C(x)}\left( C(x)\frac{\partial }{\partial x}x_0(x,t)\right) \nonumber \\&=- \frac{1}{C(x)}\dot{x}(x_0(x,t))=\frac{1}{C(x)}C(x_0(x,t)). \end{aligned}$$

(7)

Also, according to Euler theorem, $\partial _t J(x_0(x,t),x) = -J(x_0(x,t),x) C'(x_0(x,t))$.

The pitfall that leads to the inconsistency of the standard BFPE (a negative, or, in some cases, imaginary diffusion coefficient) rests on the fact that for strongly dissipative fluxes (more than linear) we have singularities in the Jacobian of the forward evolution (5). For example, for $C(x)=x^3$ we have $x_0(x,t)=x/\sqrt{1+2tx^2}$, from which $C(x_0(x,-t))/C(x)=1/(1-2tx^2)^{3/2}$ that diverges for $t\rightarrow 1/(2 x^2)$. Thus, if we have an initial Gaussian PDF of the form $P(x,0)=\exp [-x^2/2]/\sqrt{2\pi }$, which is well defined for $-\infty<x<\infty$, from (5) we obtain that it evolves as:

$$\begin{aligned} P_0(x,t)&=e^{\mathcal{L}_at}P(x,0) = \frac{\exp \left[ -\frac{x^2}{2(1-2tx^2)}\right] }{\sqrt{2\pi }(1-2tx^2)^{3/2}}, \end{aligned}$$

(8)

where, for any time $t>0$, we have that $|x |$ cannot exceed $1/\sqrt{2t}$: this means that under this flow any initial (even infinite) domain is shrunk, in a finite time, to the same time dependent limited range $|x |< 1/\sqrt{2t}$.

From a formal point of view this implies that the Liovillian operator alone is not enough to completely define the time evolution of density functions: we need to somehow include the information related to the domain of existence of the time dependent PDF.

Defining $\Theta [(x_1,x_2)]\equiv \Theta (x-x_1) \Theta (x_2-x)$, where $\Theta (\cdot )$ is the Heaviside function, we obtain the right unperturbed temporal evolution of the PDF by imposing the conservation of the normalization:

$$\begin{aligned} \int \Theta [(a,b)]\,P(x,0) \text {d}x&=\int \Theta [(x_0(a,t),x_0(b,t)]\,P_0(x,t) \text {d}x \nonumber \\&=\int \Theta [(x_0(a,t),x_0(b,t)]\,e^{\mathcal{L}_at}P(x,0) \text {d}x \end{aligned}$$

(9)

with (a, b) the initial domain of the PDF. If $\Theta [(x_0(a,t),x_0(b,t)]\ne \Theta [(a,b)]$, we must make sure that the operator $e^{\mathcal{L}_at}$ takes care of the evolution of the support of the initial PDF. For example, following this prescription, Eq. (8) becomes:

$$\begin{aligned} \text {PDF}(t)&=\Theta [(x_0(a,t),x_0(b,t)]\,P_0(x,t) =\Theta [(x_0(a,t),x_0(b,t)]e^{\mathcal{L}_at}P(x,0)\nonumber \\&=\Theta (1-2tx^2) \frac{\exp \left[ -\frac{x^2}{2(1-2tx^2)}\right] }{\sqrt{2\pi }(1-2tx^2)^{3/2}}, \end{aligned}$$

(10)

that is well defined and properly normalized for any $x \in {\mathbb {R}}$.² Notice that in this case, the fact that the Jacobian diverges for $\left| x\right| \rightarrow 1/\sqrt{2 t}$ leads to the bi-modal character of the PDF (10), peaked at $x=\pm 1/\sqrt{2 t}$ (see Fig. 2), a strange behavior for a system derived from the model (2) with purely quartic potential.³

Fig. 2

The PDF (10), i.e., the evolution of the initial PDF $P(x,0)=\exp [-x^2/2]/\sqrt{2\pi }$, under the unperturbed flux given by $\dot{x}(t)=-x(t)^3$. Solid line: $t=0$, dashed line $t=1$, dotted line $t=2$, dot-dashed line $t=4$ (see text for details)

To stress once more the attention we must pay when considering 1-D systems with strongly dissipative drift, we note that the backwards evolution operator $e^{\mathcal{-L}_at}$, that is commonly used to pass to the interaction representation, in general does not conserve the norm (see Appendix C for details).

Now, let us return to the SDE (1). Using any perturbation approach where $\epsilon$ is the small parameter of expansion and assuming no restriction on the noise correlation time $\tau$, we get the so called BFPE (see Appendix A and [66], Sect. 2 for a brief and simple derivation of the BFPE). However, allowing for Eq. (9), we attach the function $\Theta [(x_0(a,t),x_0(b,t)]$ to the unperturbed evolution operator $e^{\mathcal{-L}_at}$: hence, the diffusion coefficient of the BFPE becomes:

$$\begin{aligned} D(x,t)= \epsilon ^2 \frac{C(x)}{\tau } \int _0^{t}du\, \frac{ \Theta [(x_0(a,u),x_0(b,u)]}{C(x_0(x;-u))}\varphi (u). \end{aligned}$$

(11)

We note that the constraint given by $\Theta [(x_0(a,u),x_0(b,u)]$ is equivalent to the condition $u<{{\bar{u}}}(x)$, where ${{\bar{u}}}(x):= \int _{x}^{\pm \infty } \frac{1}{ C(y) } \; dy$ [the upper extreme of integration is $+\infty$ ($-\infty$) for x larger (smaller) than the largest (smallest) zero of C(x)] is the time at which the backwards unperturbed evolution $x_0(x,-u)$ diverges [66]. In other words, in (11) we can safely set $\Theta [(x_0(a,u),x_0(b,u)]=\Theta ({\bar{u}}(x)-u)$, leading to an expression which coincides with the diffusion coefficient of the corrected BFPE (cBFPE) introduced in [66]:

$$\begin{aligned} D(x,t)_{cBFPE}= \epsilon ^2 \frac{C(x)}{\tau } \int _0^{t}du\, \frac{ \Theta ({\bar{u}}(x)-u)}{C(x_0(x;-u))}\varphi (u). \end{aligned}$$

(12)

This yields a formal justification of the “heuristic” recipe proposed in [66] to cure the anomalies of the BFPE associated to the 1D-SDE (1).

We conclude this section considering again the SDE (1) and focusing on the case $x>x_l$, where $x_l$ is the largest zero of C(x) (if any exists, otherwise we do not constraint x, see Fig. 9 for example). If $\lim _{x \rightarrow \infty } C(x) \sim x^\alpha$ with $\alpha >1$, it follows that, starting from $x>x_l$, the backwards unperturbed evolution $x_0(x;-u)$ has a vertical asymptote at $u={\bar{u}}(x):= \int _{x}^{\infty } \frac{1}{ C(y) }\text {d}y$, thus the function $\Theta ({\bar{u}}(x)-u)$ plays an effective role in the integral that defines the stationary diffusion coefficient $D(x;\infty )$, i.e, Eq. (12) with $t\rightarrow \infty$:

$$\begin{aligned} D(X,\infty )_{cBFPE}= \epsilon ^2 \frac{ 1}{\tau } C(X)\left( \int _0^{{\bar{u}}(X)}du\, \frac{ 1}{C(X_0(X;-u))}\varphi (u) \right) =D(X,{\bar{u}}(X))_{BFPE}. \end{aligned}$$

In this situation the convergence of the integral does not depend on the decay (i.e., large time) characteristics of the correlation function $\varphi (u)$: the rapid decay of the unperturbed system dynamics of interest guarantees the existence of a diffusion coefficient over long times.

3 The Multiplicative SDE

Let us consider the one-dimensional SDE with multiplicative noise⁴:

$$\begin{aligned} {\dot{x}}= - C(x)- \epsilon \, I(x) \xi (t) \end{aligned}$$

(13)

where I(x) is a given function of the system variable x and $\xi (t)$ is a Gaussian colored noise. SDE (13) have been extensively studied in literature with the aim of obtaining an effective FPE for the PDF(x). In particular, in the last twenty years, it has been shown that simple 1-D SDE, included in the class of systems represented by (13), are important models for generating power laws in a variety of fields such as in statistical physics, economics, biology etc. [2, 10, 77,78,79,80,81,82,83]. However, this important literature has not captured the issues stressed in the Introduction and in Sect. 2, which are present also in the case of multiplicative noise. Our aim is to remedy this situation.

From (13), it follows that for any realization of the process $\xi (u)$, with $0\le u\le t$, the time-evolution of the PDF of the whole system, i.e., $P_{\xi }(x,t)$, satisfies the following PDE:

$$\begin{aligned} \partial _t P_{\xi }(x,t)=\mathcal{L}_a\,P_{\xi }(x,t)+ \epsilon \, \xi (t) \mathcal{L}_I\, P_{\xi }(x,t) \end{aligned}$$

(14)

in which the unperturbed Liouville operator $\mathcal{L}_a$ is the same of (3) and the Liouville perturbation operator is

$$\begin{aligned} \mathcal{L}_I:=\partial _x I(x). \end{aligned}$$

(15)

As mentioned in Sect. 2, we must consider the PDF support evolution. When the Gaussian stochastic forcing is a white noise, $\langle \xi (t) \xi (t')\rangle _\xi =2\, \delta (t-t')$, SDE (13) is completely equivalent to the following FPE for the reduced P(x, t):

$$\begin{aligned} \partial _t P(x,t)=\partial _x C(x) P(x,t) + \epsilon ^2 \partial _x I(x)\partial _x I(x) P(x,t), \end{aligned}$$

(16)

where we have used the Stratonovich interpretation for the differentials of Wiener processes. If it exists, the equilibrium PDF is given by

$$\begin{aligned} P_{eq}(x)=\frac{1}{Z} \frac{1}{I(x)} e^{-\int ^x \frac{C(y)}{\epsilon ^2I^2(y)} \, \text {d}y} \end{aligned}$$

(17)

where Z is a normalization constant. On the other hand, if the stochastic processes $\xi (t)$ has a finite correlation time $\tau$ the FPE structure usually breaks, similarly to what happens in the additive noise case. There are however two important exceptions:

• the function C(x) is linear and I(x) is a constant (the Gaussian nature of the noise is linearly transferred to the variable of interest [22, 84]);

• $C(x)=k I(x),$ with k constant, which maps into the trivial case $\dot{y}(t)=-k+\epsilon \xi (t)$ after the change of variable $\text {d}y=\text {d}x/I(x.)$

Apart from these particular cases, when the correlation time of the noise is not zero and the noise intensity is weak, it is possible to obtain an effective (i.e., not exact, but “optimal”) FPE for the reduced PDF, written as

$$\begin{aligned} \partial _t {P}(x,t) = \partial _x C(x) P(x,t)+ \epsilon ^2 \, \partial _x I(x) \partial _x H(x,t)\,P(x,t), \end{aligned}$$

(18)

with a proper function H(x, t) (any perturbation procedure will work, see Appendix A). Once the FPE (18) is obtained, the stationary PDF, if it exists, is

$$\begin{aligned} P_s(x)=\frac{1}{Z}\frac{1}{H(x)} e^{-\int ^x \frac{C(y)}{\epsilon ^2 H(y)I (y)} \, dy}. \end{aligned}$$

(19)

We stress that when the non-linear perturbation term of (13) is too strong, the motion becomes so unstable that no equilibrium PDF can be reached.

4 The Corrected BFPE for the Multiplicative Noise Case

By comparing (18) with the BFPE result of (A12) we obtain the BFPE coefficient $H(x,t)_{BFPE}$ as

$$\begin{aligned} H(x,t)_{BFPE}= \frac{1}{\tau } C(x) \left( \int _0^{t}\text{ d }u\, \frac{I(x_0(x;-u)) }{C(x_0(x;-u))}\, \varphi (u)\right) . \end{aligned}$$

(20)

Following Sect. 2 and the result of [66], this expression must be corrected inserting $\Theta ({{\bar{u}}}(x)-u)$ in the kernel, with ${\bar{u}}(x) := \int _{x}^{\pm \infty } \frac{1}{ C(y) }\text {d}y,$ to limit the time range of the backwards trajectory to values smaller than ${{\bar{u}}}(x)$:

$$\begin{aligned} H(x,t)_{cBFPE}= \frac{1}{\tau } C(x) \left( \int _0^{t}\text{ d }u\, \frac{I(x_0(x;-u)) }{C(x_0(x;-u))}\Theta ({\bar{u}}(x)-u)\, \varphi (u)\right) . \end{aligned}$$

(21)

For times much larger than the decay times appearing in the integrand (which depends both on the relaxation property of the unperturbed system of interest and of $\varphi (u)$) the H function (21) becomes

$$\begin{aligned} H(x;\infty )_{cBFPE}= \frac{1}{\tau } C(x) \left( \int _0^{{\bar{u}}(x)}\text{ d }u\, \frac{I(x_0(x;-u))\, }{C(x_0(x;-u))}\varphi (u)\right) . \end{aligned}$$

(22)

Eq. (18), jointly with (22), gives (a generalized version of) the BFPE of Lopez, West, and Lindenberg [54], cured here by introducing $\Theta ({\bar{u}}(x)-u)$. We recall that ${\bar{u}}(x)$ depends only on the unperturbed velocity field C(x) of the backward flow, and is finite when the unperturbed backward trajectory $x_0(x,u)$ has a vertical asymptote at $u={{\bar{u}}}(x)$ (solid line in Fig. 3).

Fig. 3

Two unperturbed backwards trajectories $x_0(x;-u)$ (dashed lines), guided by the backward drift field $C(x)=-x+x^3$ (inset). The function $x_0(x;-u)$ can be obtained inverting the relation $u= \int _{x}^{x_{0}(u)} \frac{1}{C(y)}\text {d}y$ (see Eq. (13)). The field C(x) has a stable fixed point in $x=0$, with the basin of attraction [0, 1), and an unstable fixed point in $x=1$, with no other poles for the function 1/C(x). Hence considering ${\bar{u}}(x):= \int _{x}^\infty \frac{1}{C(y)}\text {d}y$, the backwards evolution for $0\le x<1$ (dashed green line) extends to infinity, whereas for $x>1$, the backward trajectory (dashed red line) has a vertical asymptote at the finite time ${{\bar{u}}}(x)$. Physically, this means that under the flow induced by C(x), any point x in the interval (0, 1) evolved within the interval (0, 1) (i.e., crossing finite and well-defined x values) for all previous times, whereas a point $x>1$ came from $x \rightarrow \infty$ over a finite time: this implies that in the case $x>1$ any integral involving times should be limited to the time range needed to reach the given finite $x>1$ value from infinity. Colors and line types are the same of Figure 5 of [66], which refers to the same case

One subtle point that marks a difference between the present multiplicative case and the additive one of Sect. 2 and [66], is that in the latter case, for which $I(x)=1$, if ${\bar{u}}(x)$ is finite, the coefficient $H(x,t)_{BFPE}$ of Eq. (20) exhibits un-physical behaviors: $H(x,t)_{BFPE}$ turns out to be no longer positively defined or even not a real number. On the contrary, in the present multiplicative case, it can happen that even if ${{{\bar{u}}}}(x)$ is finite, $H(x,t)_{BFPE}$ remains real and positive for any t. This is because, for $u \ge {{{\bar{u}}}} (x)$, the function $I(x_0(x;-u))$ in the integrand of (20) can in principle suppress any “singular” behaviour of the Jacobian function $C(x)/C(x_0(x;-u))$. The archetypal case where this happens is the Stratonovich model, as we will see in Sect. 6. However, even if this happens, we must use the Heaviside function within the integral of the H function, since it correctly tracks the unperturbed evolution of the boundaries of the PDF domain. In fact, without this correction, the BFPE for the Stratonovich model would lead to completely incorrect results (see Sect. 6).

Finally we stress that while the term $I(x_0(x;-u))$ can apparently cure the problem for the BFPE, it can lead to additional issues. In fact, due to the $I(x_0(x;-u)$ term, the kernel of Eqs. (20)–(22) might diverge. This is not an “artifact” of the BFPE approach but rather a physical constraint: when the non-linear perturbation term of Eq. (13) is too strong, the motion becomes so unstable that no equilibrium PDF can be reached. In general, the multiplicative term I(x) introduces additional x dependencies and constraints that must be taken into account (see Appendix E for details). Here we highlight only the result that generalizes the additive noise case: if for large $|x|$, $C(x)\sim |x|^\alpha$ with $\alpha >1$ and $I(x)\sim |x|^\beta$, then for the integral of $H(x;\infty )$ in (22) to converge it is sufficient that the condition $\beta <2 \alpha -1$ holds. Thus, it does not depend on the decay characteristics of the correlation function $\varphi (u)$. The physical reason for this can be again traced back to the fact that the rapid relaxation of unperturbed trajectories makes the time integral defining the function H finite, regardless of the decay property of $\varphi (t)$.

5 The LLA FPE

The corrected BFPE obtained in Sect. 4 is what we can get from a perturbative approach (i.e., for “enough” weak noise). This somehow might suggest that in general perturbative aapproaches and reductions to 1-D systems are limited to small noise intensities. However, as pointed out in [66], numerical simulations of SDE (1) show that the LLA FPE, which is also based on a perturbative approach, gives very good results even for large intensity of the noise, at least when $C(x)=\sinh (x)$ or $C(x)=x^3$. Very good agreement between numerical simulations and LLT theory has been confirmed also in other cases.

We briefly recall here how the LLA FPE has been obtained in the literature through a perturbation approach and leave instead to the appendix F how to derive, by contrast, the LLA FPE without the assumption of a weak noise with short correlation time.

The LLA FPE can be formally obtained from the BFPE (thus, under the assumption of weak noise) in the case of a large time-scale separation between the dynamics of the slow system of interest and that of the fast perturbation $\xi (t)$. To show that, we introduce the following function $\Pi (x)$ [57]:

$$\begin{aligned} \Pi (x) :=C(x) \left( \frac{C'(x)}{C(x)}-\frac{I'(x)}{I(x)}\right) \end{aligned}$$

(23)

from which we have

$$\begin{aligned} e^{\int _0^u \Pi (x_0(x;u'))\text {d}u'}= \frac{ C(x)}{C(x_0(x,u))}\frac{I(x_0(x,u))}{I(x)}, \end{aligned}$$

(24)

being $\frac{\text {d}}{\text {d}u}x_0(x,u)=-C(x_0(x,u))$). Expanding the logarithm of the l.h.s. of (24) in a power series of u, and using Eq. (21) we obtain:

$$\begin{aligned} H(x,t)_{cBFPE} = \frac{1}{\tau } I(x)\left( \int _0^{t}\text{ d }u \,e^{-\Pi (x) u-\Pi '(x) C(x) \frac{u^2}{2}+O(u^3)}\,\Theta ({\bar{u}}(x)-u) \,\varphi (u) \right) . \end{aligned}$$

(25)

When the unperturbed motion is slow compared to the decay process of the correlation function $\varphi (t)$, or if the derivatives of $\Pi (x)$ can be locally neglected (the local linearization assumption-LLA) we can truncate the power series in the r.h.s. of (25) to first order, and we no longer need to stop the backward evolution to times smaller than ${{\bar{u}}}(x)$. The H coefficient for the LLA FPE is then obtained:

$$\begin{aligned} H(x,t)_{LLA}:= \frac{1}{\tau } I(x) \left( \int _0^{t}\text{ d }u\, e^{-\Pi (x)u} \, \varphi (u)\right) \xrightarrow [t \rightarrow \infty ]{} \frac{1}{\tau } I(x) {\hat{\varphi }}(\Pi (x)), \end{aligned}$$

(26)

where ${\hat{\varphi }}(\cdot )$ is the Laplace transform. Notice that Eq. (26) generalizes Eq. (32) of [66] for the multiplicative noise case. If $\varphi (t)=e^{-t/\tau }$, Eq. (26) together with the FPE (18), exactly gives the LLA FPE of Grigolini [57] and Fox [50].

On the other hand, as already mentioned, we prove in Appendix F that the LLA FPE can also be obtained without the assumption of weak noise with a short correlation time, as long as the noise is Gaussian: in this case, the LLA for the function $\Pi (x)$ is enough to guarantee that the reduced PDF associated with SDE (13) exactly satisfies the LLA FPE. Note that this general result was not grasped by Fox [49, 50], Hänggi [51,52,53] and the Barcelona group [12, 58,59,60]: all these groups obtained the (same) LLA FPE through a perturbative functional-calculus under the assumption of weak noise, an assumption which turns out to be unnecessary. This implies that for Gaussian noise the LLA FPE might be applicable even for not small noise intensities, going well beyond the range one would expect under the expectation that it was originally derived using a perturbation approach. We warn, however, that there might be cases when the $H(x,t)_{LLA}$ coefficient of (26) does not exist or is negative (see Appendix F for details): in these cases, as shown in Section 4, we must fall back to using the cBFPE which implies that in these situations we are restricted to weak noises.

6 The Stratonovich Model

We now consider the specific case of the Stratonovich model with different Gaussian noises and we compare the numerical simulations of this model with the BFPE, the cBFPE and the LLA FPE results, respectively.

The Stratonovich model, initially developed for the study of fluctuations in electrical circuits [85] and has been widely studied in many different situations (e.g., [86] and references therein) is defined by the following SDE ($\alpha >0$):

$$\begin{aligned} {\dot{x}}(t)=x(t)-\alpha x(t)^3+\epsilon x(t) \xi (t). \end{aligned}$$

(27)

We shall restrict our study to $x \ge 0$. Comparing (27) with (13), we get $C(x)=-x+\alpha x^3$ (see Fig. 4), and $I(x)=x$.

Fig. 4

The function $-C(x)$ for the Stratonovich model of Eq. (27), with $\alpha =1$. $x_0=0$ is an unstable equilibrium point, where $x_l=1/\sqrt{\alpha }$ is the stable one

For a white noise, $\langle \xi (t)\xi (0) \rangle =2\delta (t)$, the FPE (16) is exact, and from Eq. (17) we obtain the stationary PDF ($D_0:=\epsilon ^2$):

$$\begin{aligned} P _ { st } (x) = N\, x ^ {\frac{1-D_0}{D_0}} e^{ - \frac{ \alpha }{ 2 D_0 } x ^ { 2 } }, \end{aligned}$$

(28)

where N is a normalization constant, and where we used the Statonovich interpretation for the stochastic calculus, given that we are interested in the correlated noise case in what follows. What makes Stratonovich’s model particularly interesting is that, as we can see from Eq. (28), it exhibits a noise-induced phase transition at $D_0=1$. For $D_0\ge 1$ $P _ { st } (x)$ diverges at $x = 0$ and monotonically decreases for $x>0$, while for $D_0< 1\, P _ { st } (x)$ vanishes at $x=0$ and $x \rightarrow \infty$, with a maximum at $x_m=\sqrt{(1-D_0)/\alpha }$. The second derivative of $P _ { st } (x)$ undergoes an abrupt (infinite) change around $D_0=1$.

It is interesting to see how and if this transition is modified when correlated noise is considered. To achieve this, we employ the projection/cumulant perturbation approach, both in the standard version leading to the BFPE result (20) and in the corrected version proposed in Eq. (21). Additionally, we will consider the LLA-FPE result of (26). It will become evident that, except in cases where the time scale of the noise is exceedingly small, the standard results, while analytically straightforward, lead to a fundamentally incorrect equilibrium PDF.

The evaluation of the theoretical results is performed by comparing the different equilibrium PDFs that we obtain from these different approximation schemes with the numerical simulations of the Stratonovich model (27), using Gaussian noises with three different time correlations:

• 1. a “standard” Ornstein-Uhlenbeck (OU) process with correlation time $\tau$, i.e., $\varphi (t)=\exp (-t/\tau )$;

• 2. a noise with stretched exponential correlation function, i.e. $\varphi (t)=e^{(-t/\vartheta )^s}$, from which $\tau =\vartheta \, \Gamma \left( 1+\frac{1}{s}\right)$ ; we shall set $s=1/2$, thus $\tau =2\vartheta$;

• 3. a noise with a power law-like correlation function, i.e., $\varphi (t)=\frac{1}{1+(t/p)^a}$, from which $\tau =p/(a-1)$; we shall set $a=5/2$, thus $\tau =\frac{2 p}{3}$.

Note that case 1 is a special case of 2, with $s=1$. In Fig. 5 we summarize the comparison between numerical simulations (symbols) and theoretical prescriptions, focusing on the shape of the numerically observed PDF in the various cases, and the expected theoretical ones. The figure shows, in the variables $\tau$ and D, the phase space of the different PDFs shapes. A more detailed comparison between the PDFs obtained from the simulations and the theoretical ones expected using the different prescription for the effective diffusion coefficient can be found in Appendix H. We observed three types of PDFs, indicated with three different symbols in Fig. 5: a square indicates a PDF like the one shown in insets marked “(a)” (the PDF goes to zero for $D \rightarrow 0, \infty$, with a single maximum for finite D’s), a filled circles a PDF like the one shown in insets marked “(b)” (the PDF diverges of $D \rightarrow 0$, and monotonically goes to zero for $D \rightarrow \infty$) and an empty circle a PDF like the one shown in insets marked “(c)” (the PDF diverges of $D \rightarrow 0$, and goes to zero for $D \rightarrow \infty$, but non monotonically).

Fig. 5

Phase diagram corresponding to the different types of stationary PDF of the FPE (18) for different values of $\tau$ and D for the Stratonovich model (27). Upper panel: OU noise ($s=1$). Middle panel: stretched exponential ($s=0.5$). Bottom panel: power law. See text for details. Left: the BFPE result. Center: the cBFPE result. Right: the LLA result. Symbols correspond to numerical simulations. Squares stands for PDF like that of the inset a, full circles like that of inset b and circles like that of inset c. For $D>1$ it is sometime quite difficult, from numerical simulations, to distinguish the case of monotone decreasing (full circles) from that where there is a maximum (circles). In such cases we have filled the circles with a gray color

We begin comparing the result of the numerical simulations with the prediction of the BFPE. First, we note that for $x\gg x_l$ , we have $C(x)\sim x^3$, therefore we expect that the unperturbed evolution must be considered with care, given the warnings detailed in the Introduction. For this model the unperturbed trajectories are given by $x_0(x,u)= x\,e^u/\sqrt{\left( e^{2 u}-1\right) \alpha x^2+1}$, which means that the support of any initial set converges to the limit range $x\le \sqrt{1/\alpha }$ exponentially fast in time and for $x>\sqrt{1/\alpha }$ the backward evolution $x_0(x;-u)$ and the corresponding Jacobian diverge at ${{\bar{u}}}(x)=\frac{1}{2}\log \left( \frac{ \alpha x^2}{ \alpha x^2-1}\right)$ (see Fig. 6).

Despite these “anomalies” in the unperturbed dynamics, the H coefficient calculated according to the BFPE approach, Eq. (20) looks quite regular:

$$\begin{aligned} H(x,t)_{BFPE}&= \frac{1}{\tau } C(x) \left( \int _0^{t}\text{ d }u\, \frac{I(x_0(x;-u))}{C(x_0(x;-u))}\, \varphi (u)\right) \nonumber \\&=x\int _0^t \text{ d }u\, \left( 1-\alpha \left( 1-e^{-2 u}\right) x^2\right) \varphi (u) \end{aligned}$$

(29)

which, for $t\rightarrow \infty$, leads to

$$\begin{aligned} H(x,\infty )_{BFPE}= x\int _0^\infty \text{ d }u\, \left( 1-\alpha \left( 1-e^{-2 u}\right) x^2\right) \varphi (u). \end{aligned}$$

(30)

As we have already observed, this fact is due to the interaction function $I(x_0(x;-u))$, that “regularizes” the behaviour of the kernel of $H(x,t)_{BFPE}$. From Eq. (30) we obtain

$$\begin{aligned} H(x,\infty )_{BFPE}=x(1-l(\tau ) x^2) \end{aligned}$$

(31)

where

$$\begin{aligned} l(\tau ):= \left\{ \begin{array}{cll} &{}\frac{2 \tau }{1+2\tau }&{} \text {for } s=1 \\ &{}1-\frac{2-\sqrt{\pi } e^{\frac{1}{4 \tau }} \sqrt{\frac{1}{\tau }} \text {erfc}\left( \frac{1}{2 \sqrt{\tau }}\right) }{4 \tau }&{} \text {for } s=1/2\\ &{}6 \tau -6 \sqrt{3 \pi } e^{3 \tau } \tau ^{3/2} \text {erfc}\left( \sqrt{3\tau }\right) &{} \text {for the power law }, \end{array} \right. \end{aligned}$$

(32)

where $\text {erfc}(u):=1-\text {erf}(u)$ is the standard complementary error function.

Thus, a $\tau$ dependent constraint in the support of the PDF appears, which keeps the diffusion coefficient non negative: $x\le 1/\sqrt{1 - l(\tau )}$ (see Fig. 7). This constraint looks “reasonable” because it could be interpreted as a bound for the values of $\tau$, related to the values of x, in order to keep the time scales of the system of interest “enough” larger than that of the noise. But, as we shall see in a moment, it disappears in the corrected coefficient (21), so it is artificially introduced by not considering the correction to the BFPE.

Fig. 6

Unperturbed trajectories $x_0(x,u)=\frac{ x\,e^u}{\sqrt{\left( e^{2 u}-1\right) x^2+1}}$ for the Stratonovich model of Eq. (27) with $\alpha =1$, for different initial position x. For $x<1$ the backward evolution is not divergent

Fig. 7

Both the $H_{BFPE}$ of (31) (orange) and the $H_{cBFPE}$ of (34) (blue) for the case of a standard OU noise ($s=1$, left) and stretched exponential correlation function ($s=0.5$, right). For $x\le \sqrt{(1+l(\tau ))}$, $H_{BFPE}$ is negative, therefore is not physically acceptable. On the other hand, the $H_{cBFPE}$ is always positive. The case of power law correlation function does not differ qualitatively

Using Eqs. (31)–(32) in the stationary PDF (19) we obtain

$$\begin{aligned} P_{st,BFPE}=N\, x^{\frac{1-D_0}{D_0}} \left( \frac{1-l(\tau )}{1-x^2 l(\tau )}\right) ^{1-\frac{1-l(\tau )}{2 D_0 l(\tau )}} \end{aligned}$$

(33)

where $D_0=\epsilon ^2 \langle \xi ^2 \rangle \tau =\epsilon ^2$, since we assumed $\langle \xi ^2\rangle =1/\tau$). From Eq. (33), the BFPE approximation predicts that the phase transition at $D_0=1$, present in the white noise case, should be present also for $\tau > 0$, and it should still be located at $D_0=1$. This phase transition concerns the behavior of the PDF for $x\rightarrow 0^+$. Therefore the correction to the BFPE, that in the present Stratonovich case are relevant only for $x>\sqrt{\alpha }=1$, does not change the overall picture. On the other hand, this conclusion could have been obtained directly looking at Eqs. (19) and (31). In fact, assuming we consider the same drift C(x) and the same interaction function I(x), the behaviour of the stationary PDF close to $x=0$ depends, in turn, on the behaviour of the $H(x;\infty )$ coefficient for x close to zero, that is $H(x;\infty )=x+O(x^3)$ in all the three cases of Eq. (31).

From (33) we obtain the phase diagrams shown in the leftmost column of Fig. 5, where the insets show the theoretical PDF obtained using the different theoretical prescriptions.

Looking at the insets of the leftmost plate, we see that, apart from the transition at $D_0=1$ of the PDF for $x\rightarrow 0^+$, the predicted theoretical phase diagram is quite more complex than in the case of white noise ($\tau \rightarrow 0$): the white noise phase diagram corresponds to the transition from a PDF line in inset (a) to a PDF line in inset (b); this is correctly reproduced by the simulations. However, as $\tau >0$, we see that the theory predicts additional phase transition lines between other PDF (insets (d) and (e)). The simulations, however, do not support these theoretical predictions: there are no other phase transitions apart from the one observed for $D_0=1$. The simulations show that the numerical PDFs are, depending on the parameters, the ones shown in insets (a), (b) and (c), whereas PDFs like in insets (d) and (e) are never observed. On the other hand, the BFPE fails to predict the existence of PDFs in the form of inset (c), which are found in the simulations.

We now turn to the cBFPE. Given that in this model ${{\bar{u}}}(x)=\frac{1}{2}\log \left( \frac{ x^2}{ x^2-1}\right)$ is finite only for $x>1$, the correction to the H coefficient is relevant only for the same range. Therefore we have

$$\begin{aligned} H_{cBFPE}(x;\infty )= \left\{ \begin{array}{cll} &{} H_{BFPE}(x;\infty ) &{} \text {for } x\le 1\\ &{}H_{BFPE}(x;\infty )+ l_1(x;\tau ) &{} \text {for } x>1 \end{array} \right. \end{aligned}$$

(34)

where

$$\begin{aligned} l_1(x;\tau ):= \left\{ \begin{array}{ll} \frac{2 \tau x^2+1}{1+2\tau } \left( \frac{ x^2-1 }{ x^2 }\right) ^{\frac{1}{\tau }+1}\qquad \qquad \qquad \qquad \qquad \text {for }s=1 \\ \\ x^2 \left[ -\frac{1}{4 \tau } \sqrt{\frac{\pi }{\tau } } e^{\frac{1}{4 \tau }} \text {erf}\left( \sqrt{\log \left( \frac{x^2}{x^2-1}\right) } +\frac{1}{2 \sqrt{\tau }}\right) -\frac{1}{2 \tau } e^{-\sqrt{\frac{1}{\tau }\log \left( \frac{x^2}{x^2-1}\right) }}\right. \\ \left. +\frac{1}{4 \tau }\sqrt{\frac{\pi }{\tau } } e^{\frac{1}{4 \tau }} +e^{-\sqrt{\frac{1}{\tau }\log \left( \frac{x^2}{x^2-1}\right) }} \left( \sqrt{\frac{1}{\tau }\log \left( \frac{x^2}{x^2-1}\right) }+1\right) \right] &{}\\ -\frac{e^{-\sqrt{\frac{1}{\tau }\log \left( \frac{x^2}{x^2-1}\right) }} \left( 2 \tau \sqrt{\frac{1}{\tau }\log \left( \frac{x^2}{x^2-1}\right) }+2 \tau -1\right) }{2 \tau } \qquad \text {for }s=1/2\\ -6 \sqrt{3 \pi } e^{3 \tau } \tau ^{3/2} x^3 \text {erfc}\left( \sqrt{3 \tau +\log \left( \frac{x^2}{x^2-1}\right) }\right) \\ +\frac{6 \sqrt{3} \tau ^{3/2} x \left( x^2-1\right) }{\sqrt{3 \tau -2 \tanh ^{-1}(2 x+1)-2 \tanh ^{-1}(1-2 x)}} \qquad \qquad \text {for power law} \end{array} \right. \end{aligned}$$

(35)

As we can see in Fig. 7, the proposed correction leads to a H coefficient which is positive for any values of x and $\tau$.

The corresponding phase diagrams for the related stationary PDF are shown in the middle plates of Fig. 5. The phase transition at $D_0=1$ is still present, but now the stationary PDF goes always to zero smoothly for large x. Moreover, for $0\le D_0< 1$ the stationary PDF has always one maximum, irrespectively of the value of $\tau$, while for $D_0>1$ the PDF is bimodal for $\tau > rsim 0.25$, $\tau > rsim 0.12$ and $\tau > rsim 0.16$, in the OU, stretched exponential and power law cases, respectively. This is a much more “realistic” behaviour that agrees well with the numerical simulations of the SDE (27). For large $D_0$ values the phase line separating PDFs like (b) from those like (c) departs from the simulation results, but this is not surprising given that the cBFPE is based on a perturbation approach on the noise intensity.

Note that the line separating PDFs like (b) from those like (c) does not define a phase transition but a transition from a bimodal PDF with a maximum to a monotone decreasing one. This transition is very smooth thus with the numerical simulation it is hard to detect it.

Finally, the $H_{LLA}(x;\infty )$ of Eq. (26) for the Stratonovich model is:

$$\begin{aligned} H_{LLA}(x;\infty )= \left\{ \begin{array}{cll} &{} \frac{x}{2 \alpha \tau x^2+1} &{}\text {for }s=1 \\ &{}\frac{1}{4 \tau x^2}\left( 2 x-\sqrt{\frac{\pi }{\tau }} e^{\frac{1}{4 \tau x^2}} \text {erfc}\left( \frac{1}{2 \sqrt{\tau } x}\right) \right) &{} \text {for }s=1/2,\\ &{}x \left( 6 \sqrt{3 \pi } \tau ^{3/2} x^3 e^{3 \tau x^2} \left( 1-\text {erf}\left( \sqrt{3} \sqrt{\tau } x\right) \right) \right. \\ &{} \; \; \left. -6 \tau x^2+1\right) &{} \text {for power law} \end{array} \right. \end{aligned}$$

(36)

and, as opposed to the additive case with the same unperturbed velocity field reported at the end of Appendix G, it is now defined for any values of x and $\tau$. A Taylor expansion of the three expression in the r.h.s. of (36) gives $H_{LLA}(x;\infty )=x+O(x^3)$, thus the phase transition at $D_0 = 1$ is still preserved for any $\tau$ in all the three cases.

The corresponding diagram for the stationary PDF is plotted in the rightmost plates of Fig. 5. We see that it does not differ qualitatively from the cBFPE result, but for $D_0>1$ the LLA results match better what observed in digital simulations.

Actually, the LLA FPE remain always very close to what is obtained from the numerical simulation of the SDE of Eq. (13). This is clearly shown in Appendix H where we compare the PDFs obtained from the numerical simulations and the analytical results. We conclude that for the Stratonovich model of (27), the agreement between the numerical simulations and the LLA FPE is really impressive, for any $D_0,\tau$ values. This is a general result holding when the noise is Gaussian, and when the LLA does not leads to the issues outlined in Appendix G.

7 Conclusions

In this paper we have shown that great care must be taken when using a perturbation projection method to find an approximate FPE for the PDF of non-linear 1-D SDEs. In fact, for such systems the non-linearity of the unperturbed velocity field leads to rather singular dynamics (which has little to do with the real phenomenon one would like to mimic) that, in turn, yields issues. Thus it happens that any initial ($t=0$) PDF with a support $x\in [a,b]$, under a nonlinear drift field “collapses” to a PDF with a t-dependent existence domain, independent of the initial a and b extremes. This fact requires that we include this time dependent domain into the Liouville equation, leading to the correction to the BFPE (the cBFPE) proposed here and heuristically introduced in [66]. If this correction is not taken into account, in the case of additive noise we obtain a diffusion coefficient that may be negative or even a non-real number. In the multiplicative case (13), it may happen that the function I(x), which represents the system state dependence of the noise intensity masks the problems present in the additive case, i.e., it allows the BFPE diffusion coefficient to be positive and real, even without introducing the proposed correction. However, even in this case our correction must be considered, otherwise we obtain results that are both “physically” inconsistent, and not in agreement with numerical simulations, as the examples considered in Sect. 6 show. It is noticeable that in the Stratonovich model, the finite correlation time of the noise introduces, with respect to the white noise case, a third type of PDF: a sort of “merged” one between the two types we have in the white noise case for $D_0<1$ and $D_0>1$, respectively, and that emerges for $D_0>1$ and sufficiently large values of $\tau$. This fact, confirmed by the numerical simulations, is not captured by the standard BFPE.

In this work, we have also shown, once again, that the LLA FPE gives a stationary PDF that agrees very well with the numerical simulation of the underlying SDE, far beyond the weak noise limit and for any values of the noise correlation time $\tau$. This approximation corresponds to assuming a locally (i.e., x dependent) exponential relaxation of the kernel of H(x, t) and it is closely related to the over-damped dynamics hypothesis, which is often at the origin of the derivation of the 1D-SDE. By exploiting a recent generalization of the cumulant theory [87,88,89], here we also justify this fact. Indeed, as it is shown in Appendix: F, the “local linearization approximation” causes all generalized cumulants to vanish exactly, except the second one, which corresponds, precisely, to the LLA FPE for the PDF of x. It should be also appreciated that the LLA FPE leads to a very simple expression of the diffusion coefficient.

Despite these benefits, even the LLA FPE has weaknesses, as it can happens, for some drift C(x) and interaction function I(x) of the SDE (13), that the $H(x,\infty )_{LLA}$ coefficient (26) diverges or becomes negative. For example, it is apparent that this is typical for the additive case, when C(x) is non-monotonic and the time scale of the noise is not short compared with the drift dynamics (see Appendix G for a couple of examples). Moreover, since the generalized cumulant argument above, invoked to justify the effectiveness of the LLA FPE, is based on the assumption that the noise is Gaussian, it is reasonable to assume that for non-Gaussian perturbations (e.g., deterministic and chaotic drivings) the LLA FPE cannot perform better than the BFPE, corrected as proposed here (and limited to weak noises).

In our opinion, the best approach to mimic the phenomenon we are interested in should be that of resting, as much as possible, on SDEs that have an Hamiltonian “origin”, as that of (2), or without the issues here described. For systems with a non linear drift, this means to rely on at least a two-dimensional SDE. When a 1D SDE is highly desirable (for example, to use the MFPT technique), the reduction to one degree of freedom must be done very carefully, evaluating the right procedure on a case-by-case basis: in general the LLA FPE is the first choice, for any intensity and correlation time of the noise. If the LLA FPE cannot be used, for small noise intensities the cBFPE must be considered.

Appendices

Appendix A The Standard BFPE

From (13) or (1) (in the additive noise case), it follows that, for any realization of the process $\xi (u)$, with $0\le u\le t$, the time-evolution of the PDF of the whole system, which we indicate with $P_{\xi }(x,t)$, satisfies the following PDE:

$$\begin{aligned} \partial _t P_{\xi }(x,t)=\mathcal{L}_a\,P_{\xi }(x,t)+ \epsilon \, \xi (t) \mathcal{L}_I\, P_{\xi }(x,t) \end{aligned}$$

(A1)

in which the unperturbed Liouville operator $\mathcal{L}_a$ is

$$\begin{aligned} \mathcal{L}_a:=\partial _x C(x) \end{aligned}$$

(A2)

and the Liouville perturbation operator is

$$\begin{aligned} \mathcal{L}_I:=\partial _x I(x). \end{aligned}$$

(A3)

A standard step of the perturbation method is to introduce the interaction representation, by which (A1) becomes

$$\begin{aligned} \partial _t {\tilde{P}}_\xi (x,t) = \epsilon \xi (t)\,\tilde{\mathcal{L}}_I(t) {\tilde{P}}_\xi (x,t), \end{aligned}$$

(A4)

where

$$\begin{aligned} {\tilde{P}}_\xi (x,t):= e^{-\mathcal{L}_a t}P_\xi ( x,t), \end{aligned}$$

(A5)

and

$$\begin{aligned} \tilde{\mathcal{L}}_I(t):= e^{-\mathcal{L}_a t}\mathcal{L}_I e^{\mathcal{L}_a t} =e^{-\mathcal{L}_a^\times t} [\mathcal{L}_I]. \end{aligned}$$

(A6)

In [47] $\tilde{\mathcal{L}}_I(t)$ of (A6) is also called the Lie evolution of the operator $\mathcal{L}_I$ along the Liouvillian $\mathcal{L}_a$, for a time $-t$ (see also Appendix B). A trivial but useful fact is that the Lie evolution of a product of operators/functions is the product of the Lie evolution of the individual operators/function:

$$\begin{aligned} e^{\mathcal{A}^\times t} [\mathcal{B}\mathcal{C}]=e^{\mathcal{A}^\times t} [\mathcal{B}]\,e^{\mathcal{A}^\times t} [\mathcal{C}]. \end{aligned}$$

(A7)

Thus from (A2)–(A3) and (A7) we obtain

$$\begin{aligned} e^{-\mathcal{L}_a^\times t} [\mathcal{L}_I]&=e^{-\mathcal{L}_a^\times t} [\partial _x ]e^{-\mathcal{L}_a^\times t} [I(x)] =e^{-\mathcal{L}_a^\times t} \left[ \partial _x \frac{C(x)}{C(x)}\right] I(x_0(x,u))\nonumber \\&= \partial _x C(x)\, e^{-\mathcal{L}_a^\times t}\left[ \frac{1}{C(x)}\right] I(x_0(x,u)) =\partial _x \,\frac{ C(x)}{C(x_0(x,u))}I(x_0(x,u)). \end{aligned}$$

(A8)

where we have used (A2) and (A7). Integrating (A4) and averaging over the realization of $\xi (t)$, we get

$$\begin{aligned} {\tilde{P}}(x,t) = \langle \overleftarrow{\exp } \left[ \epsilon \int _0^t du \; \xi (u) \tilde{\mathcal{L}}_I (u) \right] \rangle P(x;0) \end{aligned}$$

(A9)

where $\overleftarrow{\exp } [\ldots ]$ is the standard chronological ordered exponential (from right to left), $P(x,t) := \langle P_{\xi } (x,t)\rangle$ and we assumed that $P_{\xi } (x;0) = P(x;0)$, i.e., at the initial time $t=0$ $P_{\xi } (x;0)$ does not depend on the possible values of the process $\xi$, or that we wait long enough to make the initial conditions irrelevant. The result of (A9) is exact, no approximations have been introduced at this level.

The r.h.s. of (A4) can be considered as a sort of generalized moment generating function for the fluctuating operator $\xi (u) \tilde{\mathcal{L}}_I$ to which it is possible to associate a generalized cumulant generating function [87]:

$$\begin{aligned} \langle \overleftarrow{\exp } \left[ \epsilon \int _0^t du \; \xi (u) \tilde{\mathcal{L}}_I (u) \right] \rangle P(x;0):= \overleftarrow{\exp } \left[ \mathcal{K}(\epsilon ,t) \right] P(x;0) \end{aligned}$$

(A10)

with $\mathcal{K}(\epsilon ,t)=\sum _{i=1}^\infty \epsilon ^i\mathcal{K}_i(t)$. Keeping up to the second generalized cumulant $\mathcal{K}_2(t) := \int _0^t \text {d}u_1\int _0^{u_1} \text {d}u_2\, \langle \xi (u_1) \tilde{\mathcal{L}}_I (u_1)\xi (u_2) \tilde{\mathcal{L}}_I (u_2)\rangle$, assuming without loss of generality that $\langle \xi \rangle =0$, and exploiting (A9) we arrive to the following result [87, see the example in Section 4.4.2]:

$$\begin{aligned} \partial _t {\tilde{P}}(x,t)= & {} \left( \partial _t \mathcal{K}_2(t)\right) {{\tilde{P}}}(x,t)\nonumber \\= & {} \epsilon ^2\langle \xi ^2\rangle \, \tilde{\mathcal{L}}_I(t)\int _0^t du \; \tilde{\mathcal{L}}_I(u) \varphi (u){{\tilde{P}}}(x,t) \end{aligned}$$

(A11)

which coincides with the usual one obtained using a second order in $\epsilon ,$ Zwanzig-like projection approach [21, 55, 56, 72, 90].

Getting rid of the interaction picture and exploiting (A3) and (A6), from (A11) we have

$$\begin{aligned} \partial _t {P}(x,t)&= \mathcal{L}_a P(x,t)+ \epsilon ^2 \langle \xi ^2\rangle \, \partial _x I(x) \int _0^{t}du\, e^{\mathcal{L}_a^\times u}[\partial _x I(x)] \varphi (u){P} (x,t) \nonumber \\&=\mathcal{L}_a P(x,t)+ \epsilon ^2 \langle \xi ^2\rangle \, \partial _x I(x) \int _0^{t}du\, \partial _x \nonumber \\&\quad \frac{ C(x)}{C(x_0(x;-u))}I(x_0(x;-u)) \varphi (u){P} (x,t) \end{aligned}$$

(A12)

where in the last line we have also used (A8).

We stress again that the result (A12) is standard in the sense that it can be obtained starting from (A1) and using any perturbation approach, where $\epsilon$ is the small parameter (assuming a finite, but not necessarily small, correlation time $\tau$), as the Zwanzig projection method hereafter cited. We have used the generalized cumulant approach, that is based on the identification of the r.h.s. of (A9) with a generalized (operator value) characteristic function, because, according to the theory developed in [87], it gives a sound justification of the second order truncation of the full series of generalized cumulants. In other words, expanding the argument of the exponential of the r.h.s. of (A10), the generalized cumulant approach avoids the problem of facing a power series that can be non absolutely convergent, that is typical for projection perturbation procedures (see Appendix D).

Thus, for weak enough noise intensity $\epsilon$, the BFPE looks like the best possible approximation we can get from a perturbation approach to the general SDE of (13). However, this is not the case: the diffusion coefficient of (A12) turns out to be wrong, for the reasons we have illustrated in the Introduction Section.

Appendix B

In literature (e.g., [47]), $e^{\mathcal{A}^\times t}\left[ \mathcal{B}\right]$ is called the Lie evolution of the operator $\mathcal{B}$ along $\mathcal{A}$, for a time t. In (5), we also used the fact that the Lie evolution of a function f(x) along the Liouvillian $\mathcal{L}_a$ gives the backward evolution of f(x), along the flux generated by the adjoint of the same Liouvillian (see proposition 2 of [47]):

$$\begin{aligned} e^{\mathcal{L}_a^\times t}[f(x) ]=f(x_0(x,-t)). \end{aligned}$$

From the Hadamard’s lemma, it is obvious that the Lie evolution of a product of operators is the product of the Lie evolution of the individual operators:

$$\begin{aligned} e^{\mathcal{A}^\times t} [\mathcal{B}\mathcal{C}]=e^{\mathcal{A}^\times t} [\mathcal{B}]\,e^{\mathcal{A}^\times t} [\mathcal{C}]. \end{aligned}$$

(B13)

Appendix C

The fact that an initial infinite set shrunk, in a finite time, to a time dependent finite x-range, is a real problem when we consider the backwards evolution operator $e^{\mathcal{-L}_at}$, i.e., when we use the interaction representation (as it is usual for a time dependent perturbation approach). More precisely, similarly to equation Eq. (6), the following result holds:

$$\begin{aligned} {{\tilde{P}}}(x,t):=e^{\mathcal{-L}_at}P(x,t) =P(x_0(x,t),t)\frac{C(x_0(x,t))}{C(x)}. \end{aligned}$$

(C1)

Contrarily to the backward Jacobian $C(x_0(x,-t))/C(x)$ of (5), the forward Jacobian $C(x_0(x,t))/C(x)$ of (C1) is usually smooth and well behaved also for a strongly dissipative drift. However, in this latter case the operator $e^{\mathcal{-L}_at}$ does not conserve the norm. A practical example may serve for illustration. Let us consider again the case of cubic drift $C(x)=x^3$, from which $x_0(x,t)=x/\sqrt{1+2tx^2}$, thus $C(x_0(x,t))/C(x)=1/(1+2tx^2)^{3/2}$. At time t, let us apply the backward evolution to a Gaussian function with a given standard deviation $\sigma (t)$. From (C1) we get

$$\begin{aligned} P(x,t)=\frac{\exp {\left[ -\frac{x^2}{2\sigma ^2(t)}\right] }}{\sqrt{2\sigma ^2(t)\pi }}\Rightarrow {{\tilde{P}}}(x,t)= \frac{\exp \left[ -\frac{x^2}{2\sigma ^2(t)(1+2tx^2)}\right] }{\sqrt{2\sigma ^2(t)\pi }(1+2tx^2)^{3/2}}. \end{aligned}$$

(C2)

The integration of ${{\tilde{P}}}(x,t)$ in $[-\infty ,\infty ]$ yields a time-dependent value as shown in Fig. 8.

Fig. 8

The value of $Z=\int _{-\infty }^\infty \frac{\exp \left[ -\frac{x^2}{2\sigma ^2(t)(1+2tx^2)}\right] }{\sqrt{2\sigma ^2(t)}\pi (1+2tx^2)^{3/2}}\text {d}x$ versus time t, see text for details

Appendix D

The projection-perturbation approach splits the full Liouvillian operator $\mathcal L$, associated to Eq. (13), into a perturbation $\epsilon \, \xi (t)\mathcal{L}_I$ and an unperturbed part $\mathcal{L}_a$. The Fokker-Planck structure is recovered at the second order in the interaction strength $\epsilon$, whereas perturbation terms of higher orders ($\sim \epsilon ^n \partial ^n \partial x^n$ with $n>2$) break the FPE structure. As for any time-dependent perturbation procedure, in general we expect that the FPE describes well the dynamics of the PDF only for times much smaller than $(\epsilon ^2 \tau )^{-1}$ while the same is not guarantee for larger times. However we are typically interested in “macroscopic” dynamics, like the relaxation and diffusion processes emerging after the coupling of the unperturbed system of interest with the booster, which takes times order of (usually larger than) $(\epsilon ^2 \tau )^{-1}$. Thus, in principle, the stationary PDF ($=\lim _{t\rightarrow \infty } P(x,t)$) of the FPE obtained from the projection-perturbation approach, for finite $\tau$ values, could be wrong. In effect, it has been shown that in some cases, as for the dichotomic noise, that the second order perturbation approach leads to an exact result in the white noise limit but is completely invalid for colored noise” [91].

It is easy to show using a simple non rigorous argument why a perturbation approach on the parameter $\epsilon$ should fail in estimating the stationary PDF in cases when the unperturbed system of interest is dissipative and deterministic. Consider Eq. (13) with $I(x)=1$

$$\begin{aligned} {\dot{x}}= - C(x) + \epsilon \, \xi (t). \end{aligned}$$

(D1)

In the projection-perturbation approach an exact master equation for P(x, t) is obtained writing the partial differential equation (PDE) in power series of differential operators of order n, with coefficients $\tau ^{n-2}\epsilon ^n$:

$$\begin{aligned} \partial _t P(x,t)&=\left\{ \partial _x C(x) + a_2(t)\, \epsilon ^2 \partial _x^2 + a_3(t)\tau \epsilon ^3 \partial _x^3+ a_4(t)\tau ^2 \epsilon ^4 \partial _x^4+\ldots \right. \nonumber \\&\quad \left. \ldots + a_n(t)\, \tau ^{n-2}\epsilon ^n \partial _x^n+\ldots \right\} P(x,t)+R[P(x,0)] \end{aligned}$$

(D2)

where the “residual” R[P(x, 0)] is a term that depends on the PDF at the initial time $t=0$. It is assumed that this term can be disregarded either because the initial time is so far in the past that the statistics features of the process at times t are uncorrelated with those at $t=0$, or because the PDF of the whole system is factorized at the initial time $t=0$ , so that the perturbing system is at equilibrium. For the same reasons and also because it is not relevant in this context, we assume that the coefficients $a_i$ do not depend on time and on x (which would be unreasonable, of course, for non linear systems of interest, but it is done here for the sake of the argument). Then, the FPE is obtained disregarding all the terms with $n>2$, justified by the fact that $\epsilon$ is “small”:

$$\begin{aligned} \partial _t P_2(x,t)=\left\{ \partial _x C(x)+ a_2\, \epsilon ^2 \partial _x^2 \right\} P_2(x,t). \end{aligned}$$

(D3)

For $\epsilon \rightarrow 0$, the stationary PDF of the dissipative system of Eq. (D1) is, of course, a delta function: $P_{s,0}(x)=\delta (x)$. Thus, the “width” of the unperturbed PDF must vanish for $\epsilon \rightarrow 0$. The FPE of Eq. (D3) shares this property. If for $\epsilon \rightarrow 0$ the PDF converges to the delta function, it follows that, for $\epsilon \rightarrow 0$, in any open interval around $x=0$ the n-th partial derivative of the stationary PDF shall diverge as $\sim \epsilon ^{-n}$ and this fact, in turn, means that we cannot justify any truncation of the series of Eq. (D2). We arrive to the same conclusion also in another way: we assume that the FPE of Eq. (D3) holds true. From this simple FPE we get directly the following stationary PDF: $P_{s,2}(x)\sim \exp \left[ -\frac{\int C(x) \text{ d }x}{a_2 \epsilon ^2}\right]$. Inserting this stationary PDF in Eq. (D2), we directly see that the first two terms of the series on the r.h.s. of course vanishes, but, for $\epsilon \rightarrow 0$ the n-th term diverges as $\epsilon ^{-n}$. Therefore we can conclude that in general for a not linear velocity field C(x) and a not vanishing correlation time $\tau$, the Zwanzig projection perturbation approach alone cannot guarantee that the stationary PDF ($=\lim _{t\rightarrow \infty } P(x,t)$) of the FPE is a good approximation of the “true” one. However, without going through mathematical details, is it clear that the divergence of the n-th differential term of Eq. (D2) when applied to the stationary PDF of the FPE (D3), is due to the singular behaviour of the same PDF for $\epsilon \rightarrow 0$. The situation would be completely different when the system of interest is not dissipative (e.g., Hamiltonian) and the interaction with the “rest of the system” is of Hamiltonian type, or it satisfies a detailed balance principle, or, in other words, it gives rise to both fluctuations and dissipation processes that are related to each other by the standard Linear Response Theory (LRT) [42]. In such cases, the “width” of the stationary PDF of the FPE does not depends on $\epsilon$ but rather on the ratio between the induced diffusion coefficient and the induced dissipation (friction) coefficient, and both are proportional to $\epsilon ^2$. For example, for a Hamiltonian system of interest (e.g. a reactive coordinate) interacting with a standard thermal bath (a solvent), the equilibrium is the Canonical PDF at the thermodynamic temperature of the bath.

However, in many situations, like for dye lasers or in geophysics (just to quote a couple of examples), the reduced models we are interested are intrinsically dissipative, and the fluctuations are due to external perturbations. Therefore, the fluctuation dissipation relationship does not hold, and then it becomes relevant what is the appropriate FPE which should be used.

Appendix E

Let us consider here the behavior of C(x) for x “large”). There are three main different cases (see Fig. 9):

Fig. 9

The different possible behaviors of the asymptotic velocity field (large values of x) C(x), for backward evolution, including the possible last two equilibrium points. See the text for details

• 1. $x > x_l$, where $x_l$ is the largest zero of C(x), and ${{\bar{u}}}(x)=\infty$ (i.e., $C(x)\approx kx^\alpha$ with $0 < \alpha \le 1$, for $x\rightarrow \infty$),

• 2. $x > x_l$ and ${{\bar{u}}}(x)$ is finite (i.e., $C(x)\approx kx^\alpha$ with $\alpha >1$, for $x\rightarrow \infty$),

• 3. x is between two last zeros of C(x), with $x_0$ denoting the stable one.

Fist, let us discuss the cases (1) and (2). For both of them, the backward evolution leads to a $x_0(x,-u)$ going to infinity. To include the contribution of the interaction function, we set $I(x)\approx hx^\beta$ for $x\rightarrow \infty$.

For an enough large $x'$ we can write

$$\begin{aligned} u(x,x_0)&:=\int _{x}^{x_0(x,-u)}\frac{dx}{ C(x)}\approx u(x,x') +\int _{x'}^{x_0(x,-u)}\frac{dx}{ k\,x^{\alpha }}\nonumber \\&=u(x,x') +\left\{ \begin{array}{ll} \frac{1}{k}\log (\frac{x_0(x,-u)}{x'}) &{}\text {for } \alpha =1\\ \frac{ (x_0(x,-u))^{1-\alpha }}{k(1-\alpha )}-\frac{(x')^{1-\alpha }}{k(1-\alpha )} &{}\text {for } \alpha \ne 1 \end{array}\right. \end{aligned}$$

(E1)

and inverting the above results for the two cases we get

where $u':=u(x,x')\le u$. From (E2b) it is clear that for $\alpha >1$ $x_0(x;-u)\rightarrow \infty$ for $u-u'\rightarrow {{\bar{u}}}=[1/k(\alpha -1)x']^{\alpha -1}$, while for $0<\alpha \le 1$, $x_0(x;-u)\rightarrow \infty$ for $u\rightarrow \infty$. Inserting (E2a) and (E2b) in (22) we get

$$\begin{aligned}&H(x;\infty )_{cBFPE}\approx H(x;u')_{BFPE}\nonumber \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{\infty }\text{ d }u\,\,e^{-k(1-\beta ) (u-u')}\, \varphi (u)&\text {for}\; \alpha =1 \end{aligned}$$

(E3)

$$\begin{aligned}&H(x; {\bar{u}}(x))_{cBFPE}\approx H(x;u')_{BFPE}\nonumber \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{{\bar{u}}(x)}\text{ d }u\,\left[ \frac{1}{x^{\alpha -1}}-k \,(\alpha -1)(u- u')\right] ^{\frac{\alpha -\beta }{\alpha -1}} \varphi (u)&\text {for}\; \alpha \ne 1, \end{aligned}$$

(E4)

respectively.

From (E3) and (E4) it emerges that the interaction function I(x) might make the system unstable. However, in the (1) case, i.e., ${{\bar{u}}}(x)=\infty$ (that implies ${\varvec{\alpha \le 1}}$), if $\beta \le \alpha$, the integrals of (E3) and (E4), in general, exists (with the assumption that $\int _0^\infty \varphi (u)\text {d}u<\infty$). If $\beta > \alpha ,$ $\varphi (u)$ must decays quickly enough to compensate the blowing up of the rest of the integrand. In other words, in this case the time scale separation between the dynamics of the system of interest and that of the (fast) perturbation must be large enough. More precisely, for $\alpha =1, \varphi (u)$ must decay exponentially faster than $e^{-k(\beta -1)u}$, while for $\alpha <1$ any exponential decay of $\varphi (u)$ is acceptable.

In the (2) case, i.e., ${\varvec{\alpha >1}}$, we have ${{\bar{u}}}(x)=u'+\frac{1}{x^{\alpha -1}\,k(\alpha -1)}<\infty$, thus $u'<u<\bar{u}<\infty$ and (E4) can be written as:

$$\begin{aligned} H(x; {\bar{u}}(x))_{cBFPE}&\approx H(x;u')_{BFPE}\nonumber \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{{\bar{u}}(x)}\text{ d }u\,\left[ k \, (\alpha -1)({{\bar{u}}}(x)-u)\right] ^\frac{\alpha -\beta }{\alpha -1}\,\varphi (u), \end{aligned}$$

(E5)

from which we see that the integral converges if $\beta <2 \alpha -1$. Note that if $\beta =0$ (the additive noise case), we get $\alpha >1/2$ that is always satisfied given that we assumed $\alpha >1$. In other words, according to (E5) the following result seems to hold true: For $\alpha >1$ and x larger than the largest zero of C(x), the characteristics of the decay of the correlation function do not matter to ensure convergence of the integral that defines the diffusion coefficient of (22).

This is a really strong statement, implying that the time scale of the dynamics of the system of interest could even be shorter than the time scale of the dynamics of the stochastic perturbation: physically, the integral of (22) converges thanks to the fast decaying of the unperturbed dynamics. The reader should also appreciate that the condition $\beta <2 \alpha -1$, valid for the case (2), is weaker than that of the case, (1), i.e., $\beta <\alpha$.

In the  (3) case, if C(x) is a continuous function, the backward evolution remains in the basin of attraction of some stable point (between $x_0$ and $x_l$ in Fig. 9), thus there are not asymptotes for $x_0(x,-u)$. In other words, ${{\bar{u}}}(x)$ is infinite, that means that we do not have to correct the BFPE result. However, it is interesting to see, for the different cases hereafter considered, which is the constraint for the time scale separation between booster and system of interest dynamics. To fix the idea, we assume that $x_0$ and $x_b$ are two consecutive zeros of C(x) ($x_b=x_l$ in Fig. 9). If C(x) is a regular function, in the backward evolution one of these zeros is stable, while the other is unstable. Without loss of generality, we assume that $C(x)<0$ for $x\in (x_0,x_b)$, thus $x_0$ is the stable one: $x_0(x,-u)\rightarrow x_0$, as in Fig. 9. We also assume that for $x\rightarrow x_0$ we have $C(x)\approx -k(x-x_0)^\alpha$. Therefore, for any $x'>x_0$ sufficiently close to $x_0$ so that $C(x')\approx -k(x'-x_0)^\alpha$ and for $x_0<x_0(x,-u)<x'<x<x_b$, we have

$$\begin{aligned} u(x,x_0)&=\int _{x}^{x_0(x,-u)}\frac{dx}{ C(x)}\approx u(x,x') -\int _{x'}^{x_0(x,-u)}\frac{dx}{ k(x-x_0)^{\alpha }}\nonumber \\&=u(x,x') +\left\{ \begin{array}{ll} \frac{1}{k}\log (\frac{x'-x_i}{x_0(x,-u)-x_0}) &{}\text {for } \alpha =1\\ -\frac{ (x_0(x,-u)-x_0)^{1-\alpha }}{k(1-\alpha )}+\frac{(x'- x_0)^{1-\alpha }}{k(1-\alpha )} &{}\text {for } \alpha \ne 1 \end{array}\right. \end{aligned}$$

(E6)

where $u':=u(x,x')$ and, by construction, $u>u'$. From (E6), we directly get

$$\begin{aligned} x_0(x,-u)&\approx x_0+(x'-x_0)e^{-k(u-u')}&\text {for } \alpha =1 \end{aligned}$$

(E7)

$$\begin{aligned} x_0(x,-u)&\approx x_0+\frac{1}{\left[ k (\alpha -1) ( u-{{\tilde{u}}})\right] ^\frac{1}{\alpha -1}}&\text {for } \alpha >1 \ \end{aligned}$$

(E8)

$$\begin{aligned} x_0(x,-u)&\approx x_0+\left[ k (1-\alpha ) ( {{\tilde{u}}} - u)\right] ^\frac{1}{1-\alpha }&\text {for } \alpha < 1 \end{aligned}$$

(E9)

with ${{\tilde{u}}}:=u'-\frac{1}{k(\alpha -1)(x'- x_0)^{\alpha -1}}$, thus, $u\ge u'>{{\tilde{u}}}$ for $\alpha >1$ and ${{\tilde{u}}}> u'$ for $\alpha <1$. On the other hand, for $\alpha >1$ Eq. (E8) says that the unperturbed backwards evolution takes an infinite time to reach the stable point $x_0$. Inserting Eqs. (E7)–(E8) in (22) we obtain

$$\begin{aligned}&H(x;\infty )_{BFPE}\approx H(x;u')_{BFPE}\nonumber \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{\infty }\text{ d }u\,I(x_0(x;-u)) \frac{e^{k(u-u')}}{ k(x'-x_0)} \varphi (u)&\text {for}\; \alpha =1\nonumber \\&H(x;\infty )_{BFPE}\approx H(x;u')_{BFPE}\nonumber \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{\infty }\text{ d }u\,I(x_0(x;-u))\left[ k (\alpha -1) ( u-{{\tilde{u}}})\right] ^\frac{\alpha }{\alpha -1} \varphi (u)&\text {for}\; \alpha >1 \end{aligned}$$

(E10)

from which it is clear that for $\alpha \ge 1$ the existence of the integral depends of the decay characteristics of the correlation function $\varphi (u)$ with respect to the behaviour of the rest of the integrand, which diverges. Physically, this requires a time scale separation between the dynamics of the system of interest and that of the perturbation to ensure convergence of the integral.

Concerning the case $\alpha <1$, from (E9), we see that the backward unperturbed evolution reaches the stable point $x_0$ in a finite time $u={{\tilde{u}}}$, i.e., $C(x_0(x,-u))\rightarrow 0$ for $u\rightarrow {{\tilde{u}}}$ and $C(x_0(x,-u))= 0$ for $u>{{\tilde{u}}}$. This fact implies that if $I(x_0)\ne 0$, then $H(x,t)=H(x,{{\tilde{u}}})$ for $t>\tilde{u}$.

If the function I(x) also vanishes at $x_0$ as $I(x)\approx h(x-x_0)^\beta$ [this is the case of the Stratonovich model in Sect. 6, where $x_0=0$, $\beta =1$], then we have

$$\begin{aligned} H(x;\infty )_{BFPE}&\approx H(x;u')_{BFPE} \\&\quad +\frac{1}{\tau } C(x)\frac{1}{ (x'-x_0)} \int _{u'}^{\infty }\text{ d }ue^{-k(\beta -1)(u-u')}\, \varphi (u)&\text {for}\; \alpha =1\\ H(x;\infty )_{BFPE}&\approx H(x;u')_{BFPE} \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{\infty }\text{ d }u \left[ k (\alpha -1) ( u-{{\tilde{u}}})\right] ^\frac{\alpha -\beta }{\alpha -1}\varphi (u)&\text {for}\; \alpha >1\\ H(x;\infty )_{BFPE}&\approx H(x;u')_{BFPE} \\&\quad +\frac{1}{\tau } C(x) \int _{u'}^{{{\tilde{u}}}}\text{ d }u \left[ k (1-\alpha ) ( {{\tilde{u}}} - u)\right] ^\frac{\beta -\alpha }{1-\alpha }\varphi (u)&\text {for}\; \alpha <1 \end{aligned}$$

from which we see that

• for $\alpha = 1$ the convergence is reached when $\beta \ge 1$ irrespectively of the decays characteristics of the correlation function $\varphi (u)$, while when $\beta < 1$ the correlation function $\varphi (u)$ must decay at least exponentially with a decay time shorter than $k(1-\beta )$.

• For $\alpha >1$ and $\beta >2 \alpha -1$, the integral exists independently of $\varphi (u)$, otherwise, for $\beta <2 \alpha -1$, any exponential decaying of $\varphi (u)$ is acceptable.

• For $\alpha <1$ we must have $\beta > \alpha$ and the decays features of $\varphi (u)$ does not matter because the integrand is exactly zero for $u>\tilde{u}$.

Appendix F The LLA From the Cumulant Theory

We show in this appendix that, if $\Pi '(x)$ can be neglected, we obtain the LLA FPE (i.e., the FPE (18) with LLA diffusion coefficient of Eq. (26)), without using the weak noise assumption. The proof is based on the results of some recent works which extend the concept of cumulants [87,88,89]. We recall that from these results (see [87] Section 4.4.2, and [88], Section 6.2) if the operator $\Omega (t):=\epsilon \, \xi (t)\tilde{\mathcal{L}}_I(t)$ is O-Gaussian, i.e., it is M-Gaussian where the M projection is the standard time ordering that defines the time ordered exponential, then the FPE (A11) is an exact result. However, in [89] Section 4 it was also pointed out that, in general, for $\Omega (t)$ to be O-Gaussian is not sufficient that $\xi (t)$ is Gaussian in the ordinary sense (this is the reason for which the PDF of a stochastic system perturbed by a Gaussian noise usually does not satisfies a FPE), due to the fact that the O-cumulants of such operator $\Omega (t)$ are not proportional to the cumulants of $\xi (t)$. For the same reason an O-Central Limit Theorem cannot be invoked to obtain, as a limiting result, a FPE (see [89] for details about this point).

However, in the same Section 4 of [89], it is shown that, on the contrary, in the special cases where $\left[ \tilde{\mathcal{L}}_I(u_i),\tilde{\mathcal{L}}_I(u_j)\right] = 0$, the Gaussian nature of $\xi (t)$ implies the O-Gaussian nature for $\Omega (t)$. Comparing (A6) with (24) we easily obtain ($u_i\ge u_j$):

$$\begin{aligned} \left[ \tilde{\mathcal{L}}_I(u_i),\tilde{\mathcal{L}}_I(u_j)\right] =-\partial _x \,I(x)^2 e^{\int _{u_i}^{u_j}\Pi (x_0(x;v))\text {d} v} \int _{u_i}^{u_j} \Pi '(x_0(x;v))\text {d} v, \end{aligned}$$

(F1)

therefore, we see that the commutator in Eq. (F1) vanishes in two cases:

• 1. imposing $u_i-u_j$ to be very small (actually, $u_i-u_j<<\tau$ because terms like (F1) appear in the cumulant expansions inside the convolution integral with the correlation function of $\xi (t)$);

• 2. considering $\Pi '(x)\sim 0$.

Thus, if either of these conditions is valid, irrespectively of the value of $\epsilon$, $\Omega (t):=\epsilon \, \xi (t)\tilde{\mathcal{L}}_I(t)$ is O-Gaussian, that implies that the PDF of x satisfies the FPE (18), where H(x, t) is obtained by the second O-cumulants of $\Omega (t)$, i.e., is the BFPE result of (20). Incidentally, these very two conditions were invoked in Sect. 5, to obtain the the LLA result (26) for the BFPE one.

We conclude this appendix observing that if $I(x)\propto C(x)$ (in this case the SDE (13) would represent a Kubo stochastic oscillator [92]), then $\Pi (x)=\Pi '(x)=0$ (see Eq. (23)), thus also in this case we get a LLA FPE which is exact and which coincides with the FPE of Eq. (16) obtained for white noise, with $\tau =\int _0^\infty \varphi (u)\text {d}u$.

Appendix G

Let us consider, for simplicity, the additive SDE of (1). In this case, the $H(x,t)_{LLA}$ coefficient of (26) becomes $H(x,t)_{LLA}={\hat{\varphi }}(C'(x))$. If the slope of C(x) change sign with x and if $\tau$ is not small enough, then ${\hat{\varphi }}(C'(x))$ could diverge or be negative for some x value. Let’s take the crucial example of Brownian motion in a periodic potential, for instance: $C(x)=k \sin (x)$. Moreover, let us assume that the perturbation to the brownian particle is give by a red noise, i.e., a standard Ornstein-Uhlenbeck process, with correlation time $\tau$. This is a heuristic model with applications in various branches of science and technology, like the diffusive dynamics of atoms and molecules on crystal surfaces [93]. In this case ${{\bar{u}}}(x)=\infty$ $\forall x$, therefore, we do not have the problems illustrated above concerning the projection perturbation approach. However, if $k\tau >1$ the LLA cannot be used, because it would lead to the diffusion coefficient

$$\begin{aligned} D(x,\infty )_{LLA}=\epsilon ^2\, \frac{1 }{k\tau \,\cos ( x)+1} \end{aligned}$$

(G1)

that diverges for $x=2 \pi c_1\pm \arccos \left( -\frac{1}{k\tau }\right) ,c_1\in {\mathbb {Z}}$.

Another important example is that of the same SDE (1), but now with $C(x)=-x(t)+\alpha x(t)^3$ (the same drift of the Stratonovich model (27)), and $\xi (t)$ again a OU noise with correlation time $\tau$. In this case the corrected BFPE diffusion coefficient, given in (12) results quite complicate, involving Hypergeometric functions, but it is positive and well defined for any x and $\tau$ values. On the contrary, the LLA result is simple:

$$\begin{aligned} D(x,\infty )_{LLA}=\epsilon ^2\, \frac{1 }{\tau \left( 3 x^2-1\right) +1} \end{aligned}$$

(G2)

but clearly cannot be accepted for $\tau \ge 1/(1-3x^2)$.

Appendix H: Some Numerical Results

In this Appendix we compare the plots of the stationary PDFs obtained from the numerical simulation (blue dishes) of the Stratonovic SDE (27) with the BFPE result of (31) (blue line), the cBFPE result of (34) (orange line) and the LLA result of (36) (green line), for three different types of Gaussian noises characterized by different autocorrelation functions:

• 1. pure exponential, i.e., $\varphi (t)=e^{-t/\tau }$; this noise is a Ornstein Uhlenbeck (OU) process,

• 2. stretched exponential, i.e., $\varphi (t)=e^{(-t/\vartheta )^s}$, from which $\tau =\vartheta \Gamma \left( 1+\frac{1}{s}\right)$ ; we shall set $s=1/2$

• 3. power law ($\varphi (t)=\frac{1}{1+(t/p)^a}$), from which $\tau =p/(a-1)$; we shall set $a=5/2$.

Each one of these three cases has been simulated and compared with the analytical results for different values of $D_0=\epsilon ^2$ and $\tau =\int _0^\infty \varphi (t) \text {d}t$.

The random processes with the required correlation functions were generated with the algorithms of [94]. The SDE is turned into an ODE, where the random stochastic process is numerically integrated assuming it is a given time function (generated via Fourier transform, see [94]), with a scheme which would be the Heun algorithm in case of white noise (see [95] for details). We checked that for the OU case the results obtained with these approach agreed with what is obtained using more standard approach for the integration of SDE driven by an OU process (see [95] for standard algorithms).

We found that for $D,\tau$ values not very small, the BFPE gives wrong results. For small and intermediate values of $D,\tau$ the cBFPE performs well, and for any values of $D,\tau$ the LLA FPE agrees very well with the numerical simulations.

First, let us consider the OU noise case.

$D=0.2$, $\tau =0.98$:

$D=0.75$, $\tau =0.2$:

$D=1.37$, $\tau =0.6$:

$D=2.60$, $\tau =0.2:$

Then, let us consider the stretched exponential case, with $s=0.5$.

$D=0.21$, $\tau =0.12$:

$D=0.2$, $\tau =0.5$:

$D=0.2$, $\tau =0.98$:

$D=0.59$, $\tau =0.98$:

$D=0.72$, $\tau =0.12$:

$D=1.21$, $\tau =0.12$:

$D=1.38$, $\tau =0.98$:

$D=2.65$, $\tau =0.12$:

$D=2.56$, $\tau =0.99$:

Finally, let us not consider the power law case, with $a=2.5$.

$D=0.11$, $\tau =0.11$:

$D=0.10$, $\tau =0.49$:

$D=0.10$, $\tau =0.62$:

$D=0.5$, $\tau =0.23$:

$D=0.77$, $\tau =0.11$:

$D=0.7$, $\tau =0.62$:

$D=1.22$, $\tau =0.11$:

$D=1.28$, $\tau =0.36$:

$D=1.47$, $\tau =0.62$:

Then, let us consider the stretched exponential case, with $s=0.5$.

$D=0.21$, $\tau =0.12$:

$D=0.2$, $\tau =0.5$:

$D=0.2$, $\tau =0.98$:

$D=0.59$, $\tau =0.98$:

$D=0.72$, $\tau =0.12$:

$D=1.38$, $\tau =0.98$:

$D=2.65$, $\tau =0.12$:

$D=2.56$, $\tau =0.99$: