The two-particle irreducible effective action for classical stochastic processes

Let us generalize equation (3) to an action functional comprising an arbitrary number of time-dependent fields ϕ_a (t) with indices a = 1, ..., n, denoted collectively by ϕ (t). Then we can define a moment-generating or partition functional as

$$\begin{align}\hfill Z[\boldsymbol{j}]& =\left\langle \mathrm{exp}\left({\int }_{0}^{{t}_{f}}\mathrm{d}t\,\,{\boldsymbol{j}}^{T}(t)\boldsymbol{\phi }(t)/c\right)\right\rangle \hfill \\ \hfill & =\int \mathcal{D}\boldsymbol{\phi }\,\mathrm{exp}\left({\int }_{0}^{{t}_{f}}\mathrm{d}t\,\,{\boldsymbol{j}}^{T}(t)\boldsymbol{\phi }(t)/c-S[\boldsymbol{\phi }]/c\right),\hfill \end{align} \tag{ 4 }$$

where we have introduced a constant c that formally plays the same role as ℏ in QFT, and j (t) is the source field. The moment-generating functional Z, in turn, gives rise to the scaled cumulant-generating function (CGF)

$$\begin{equation}\mathcal{W}[\boldsymbol{j}]=c\,\mathrm{ln}\,Z[\boldsymbol{j}],\end{equation} \tag{ 5 }$$

where formally one should take c → 0 [29]. However, since we are usually interested in the case of finite c⁻¹, we do not include this limit into the definition of $\mathcal{W}(j)$. In QFT, the analogous procedure is to let ℏ → 0, which leads to the standard loop expansion in the limit of small quantum fluctuations [7].

For asymptotically small c, the path integral in equation (4) becomes very strongly peaked, such that its main contribution comes from the saddle point, and one may approximate

$$\begin{equation}\mathcal{W}[\boldsymbol{j}]\approx \underset{\boldsymbol{\phi }}{\mathrm{sup}}\left({\int }_{0}^{{t}_{f}}\mathrm{d}t\,\,{\boldsymbol{j}}^{T}(t)\boldsymbol{\phi }(t)-S[\boldsymbol{\phi }]\right).\end{equation} \tag{ 6 }$$

Now equation (6) suggests defining the so-called one-particle irreducible (1PI) effective action Γ as the Legendre transform [29] of the scaled CGF with respect to the first cumulant:

$$\begin{equation}{\Gamma}[\bar{\boldsymbol{\phi }}]\ :\!=\underset{\boldsymbol{j}}{\mathrm{sup}}\left({\int }_{0}^{{t}_{f}}\mathrm{d}t\,\,{\boldsymbol{j}}^{T}(t)\bar{\boldsymbol{\phi }}(t)-\mathcal{W}[\boldsymbol{j}]\right).\end{equation} \tag{ 7 }$$

When the scaled CGF is expanded into tree graphs where the functional derivatives of ${\Gamma}[\bar{\boldsymbol{\phi }}]$ act as vertices connected by edges ${\delta }^{2}\mathcal{W}[\boldsymbol{j}]/\delta {j}_{a}(t)\delta {j}_{b}({t}^{\prime })$, then ${\Gamma}[\bar{\boldsymbol{\phi }}]$ is found to contain no graphs that can be disconnected by cutting a single edge [6], for which reason it is called '1PI'.

The 1PI effective action ${\Gamma}[\bar{\boldsymbol{\phi }}]$ is generally not identical to $S[\bar{\boldsymbol{\phi }}]$, as one can see also from the fact that the classical saddle-point ${\boldsymbol{\phi }}_{c}^{\ast }$ defined by $0=\delta S[{\boldsymbol{\phi }}_{c}^{\ast }]/\delta {\boldsymbol{\phi }}_{c}^{\ast }(t)$ (i.e. neglecting fluctuations) may deviate from the true expectation value ⟨ ϕ (t)⟩, e.g. when S is not Gaussian and c is finite. We shall assume that the supremum in equation (7) always exists (and is unique), such that we can calculate the Legendre transform as usual by finding the source j (t) as a function of $\bar{\boldsymbol{\phi }}(t)$ from

$$\begin{equation}\bar{\boldsymbol{\phi }}(t)=\delta \mathcal{W}[\boldsymbol{j}]/\delta \boldsymbol{j}(t),\end{equation} \tag{ 8 }$$

the inverse relation to which is

$$\begin{equation}\boldsymbol{j}(t)=\delta {\Gamma}[\bar{\boldsymbol{\phi }}]/\delta \bar{\boldsymbol{\phi }}(t).\end{equation} \tag{ 9 }$$

After setting the source field on the left-hand side of equation (9) to zero, its solution provides an approximation to ⟨ ϕ (t)⟩ that becomes exact as c vanishes (in which case $\bar{\boldsymbol{\phi }}(t)$ becomes identical to ${\boldsymbol{\phi }}_{c}^{\ast }$) or if ${\Gamma}[\bar{\boldsymbol{\phi }}]$ happens to be known exactly. To first order in c (one-loop order in QFT terms [9, 12]), the 1PI effective action reads

$$\begin{equation}{\Gamma}[\bar{\boldsymbol{\phi }}]=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.+S[\bar{\boldsymbol{\phi }}]+\frac{c}{2}\,\mathrm{Tr}\,\mathrm{ln}\,{\boldsymbol{G}}_{0}^{-1}[\bar{\boldsymbol{\phi }}]+\mathcal{O}\left({c}^{2}\right),\end{equation} \tag{ 10 }$$

where ${\boldsymbol{G}}_{0}^{-1}[\bar{\boldsymbol{\phi }}]$ is the Hessian of the action S. For Gaussian action functionals S, the inverse Green function ${\boldsymbol{G}}_{0}^{-1}$ is independent of $\bar{\boldsymbol{\phi }}$ and equation (10) is exact.

An instructive example is provided by zero-dimensional ϕ⁴ theory, for which an investigation of the quantum (2PI) effective action can be found in reference [30]. In the present classical case, we consider the action

$$\begin{equation}S(\phi )=\frac{{\left(\phi -\mu \right)}^{2}}{2{\sigma }^{2}}+\frac{g}{4!}{\phi }^{4},\end{equation} \tag{ 11 }$$

which describes a normal distribution supplemented by a quartic non-linearity. Equation (10) then evaluates to

$$\begin{align}\hfill {\Gamma}(\bar{\phi })& =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.+S(\bar{\phi })+\frac{c}{2}\,\mathrm{ln}\left({\sigma }^{-2}+g{\bar{\phi }}^{2}/2\right)+\mathcal{O}\left({c}^{2}\right),\hfill \end{align} \tag{ 12 }$$

where ${G}_{0}^{-1}(\phi )={\sigma }^{-2}+g{\bar{\phi }}^{2}/2$. Setting j(t) = 0 in the scalar version of equation (9), we obtain

$$\begin{align}\hfill 0& ={\sigma }^{-2}\left(\bar{\phi }-\mu \right)+\frac{g{\bar{\phi }}^{3}}{6}+\frac{cg\bar{\phi }}{2\left({\sigma }^{-2}+g{\bar{\phi }}^{2}/2\right)}.\hfill \end{align} \tag{ 13 }$$

The fact that the probability density $\mathrm{exp}\left(-S(\phi )/c\right)$ concentrates around the 'mean-field' value, which is the solution to equation (13) for c → 0, is the basic principle behind the tree-level approximation well-known from the field theory of statistical physics [6]. The last term on the right-hand side of equation (13) is the first correction beyond tree level and arises from quadratic fluctuations around the saddle point. Seen as a coupling expansion in g, moreover, solving this equation with a non-linear solver produces a uniform approximation [9] to the exact result that is superior to a naïve expansion of the partition function.

While the 1PI effective action in equation (10) can in principle be used to extract information about cumulants of all orders by means of vertex functions [3], self-consistency will by construction be given only for the first cumulant, i.e. the mean. It is, however, often desirable to also compute the second cumulant (i.e. the variance) self-consistently. This is possible by extending the above construction via an additional Legendre transform with respect to the second moment [11, 12]. For this purpose, we introduce a (symmetric) two-point source K (t, t') into the moment-generating functional. In terms of $\mathcal{W}$, we have

$$\begin{align*}\hfill {\mathrm{e}}^{{c}^{-1}\mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}& =\int \mathcal{D}\boldsymbol{\phi }\,\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{1}{c}{\int }_{0}^{{t}_{f}}\hspace{-3.55658pt}\mathrm{d}t\,\left[{\boldsymbol{j}}^{T}(t)\boldsymbol{\phi }(t)\right.\right.\hfill \\ \hfill & \quad \left.\left.+\frac{1}{2}{\int }_{0}^{{t}_{f}}\hspace{-3.55658pt}\mathrm{d}{t}^{\prime }\,{\boldsymbol{\phi }}^{T}(t)\boldsymbol{K}(t,{t}^{\prime })\boldsymbol{\phi }({t}^{\prime })\right]-S[\boldsymbol{\phi }]/c\right\}.\hfill \end{align*}$$

The doubly Legendre-transformed effective action is then defined as

$$\begin{align*}\hfill {\Gamma}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]& ={\int }_{0}^{{t}_{f}}\hspace{-4.26773pt}\mathrm{d}t\left[{\boldsymbol{j}}^{T}(t)\bar{\boldsymbol{\phi }}(t)+\frac{1}{2}{\int }_{0}^{{t}_{f}}\hspace{-4.26773pt}\mathrm{d}{t}^{\prime }\,\mathrm{Tr}\,\boldsymbol{K}(t,{t}^{\prime })\right.\hfill \\ \hfill & \quad \times \left.\left(c\boldsymbol{G}(t,{t}^{\prime })+\bar{\boldsymbol{\phi }}(t){\bar{\boldsymbol{\phi }}}^{T}({t}^{\prime })\right)\right]-\mathcal{W}[\boldsymbol{j},\boldsymbol{K}],\hfill \end{align*}$$

where $\bar{\boldsymbol{\phi }}(t)$ and c G (t, t') are the first and second cumulant in the presence of the external source fields j (t) and K (t, t'), i.e.

$$\begin{align}\hfill {\bar{\phi }}_{a}(t)& =\frac{\delta \mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}{\delta {j}_{a}(t)},\hfill \\ \hfill {G}_{ab}(t,{t}^{\prime })& =\frac{{\delta }^{2}\mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}{\delta {j}_{a}(t)\delta {j}_{b}({t}^{\prime })}.\hfill \end{align} \tag{ 14 }$$

Setting the source fields to zero, one recovers the expectation values

$$\begin{align}\hfill {\left.\frac{\delta \mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}{\delta {j}_{a}(t)}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}& =\langle {\phi }_{a}(t)\rangle ,\hfill \\ \hfill {\left.\frac{{\delta }^{2}\mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}{\delta {j}_{a}(t)\delta {j}_{b}({t}^{\prime })}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}& ={c}^{-1}\left[\langle {\phi }_{a}(t){\phi }_{b}({t}^{\prime })\rangle -\langle {\phi }_{a}(t)\rangle \langle {\phi }_{b}({t}^{\prime })\rangle \right].\hfill \end{align} \tag{ 15 }$$

The second moment can also be obtained via

$$\begin{align}\hfill {\left.\frac{\delta \mathcal{W}[\boldsymbol{j},\boldsymbol{K}]}{\delta {K}_{ab}(t,{t}^{\prime })}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}& =\frac{1}{2}\langle {\phi }_{a}(t){\phi }_{b}({t}^{\prime })\rangle .\hfill \end{align} \tag{ 16 }$$

Note that it is also possible to interpret the functional derivative with respect to K symmetrically, i.e. δ/δK_ab + δ/δK_ba , which removes the factor of 1/2 in equation (16) and the ensuing equations below. In close connection to the 1PI effective action in equation (10), the 2PI effective action [9, 12] can now be shown to read

$$\begin{equation}{\Gamma}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.+S[\bar{\boldsymbol{\phi }}]+\frac{c}{2}\,\mathrm{Tr}\,\mathrm{ln}\,{\boldsymbol{G}}^{-1}+\frac{c}{2}\,\mathrm{Tr}\,{\boldsymbol{G}}_{0}^{-1}[\bar{\boldsymbol{\phi }}]\boldsymbol{G}+{{\Gamma}}_{2}[\bar{\boldsymbol{\phi }},\boldsymbol{G}],\end{equation} \tag{ 17 }$$

where ${{\Gamma}}_{2}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]=\mathcal{O}({c}^{2})$ contains the non-linear contributions taken into account beyond the quadratic fluctuations around the saddle point. When ${{\Gamma}}_{2}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]$ is represented in terms of Feynman diagrams [8], by construction there appear no diagrammatic graphs that can be cut in two by cutting two edges [9]. Since the graph edges G_ab (t, t') represent particle propagators in QFT, this property is called 'two-particle irreducibility'. Graphical examples of a 2PI diagram and a two-particle reducible diagram are given in figures 1(b) and (c), respectively.

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Note how Γ also depends on G : this is the hallmark of self-consistency for the second cumulant. While to first order in c, equation (17) is equivalent to equation (10), any non-trivial Γ₂ now results in self-consistent corrections to both $\bar{\boldsymbol{\phi }}$ and G . After setting the external sources to zero, these self-consistent solutions are determined via the pair of equations

$$\begin{equation}0=\frac{\delta {\Gamma}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]}{\delta {\bar{\phi }}_{a}(t)},\hspace{25.0pt}0=\frac{\delta {\Gamma}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]}{\delta {G}_{ba}({t}^{\prime },t)},\end{equation} \tag{ 18 }$$

the first of which gives the equation of motion of the fields ${\bar{\phi }}_{a}(t)$, while the second may be written as

$$\begin{equation}\frac{c}{2}{G}_{ab}^{-1}(t,{t}^{\prime })=\frac{c}{2}{G}_{0,\,ab}^{-1}(t,{t}^{\prime })+\frac{\delta {{\Gamma}}_{2}}{\delta {G}_{ba}({t}^{\prime },t)},\end{equation} \tag{ 19 }$$

which we recognize as Dyson's equation upon defining the so-called self-energy

$$\begin{equation}{{\Sigma}}_{ab}(t,{t}^{\prime })=-2{c}^{-1}\frac{\delta {{\Gamma}}_{2}}{\delta {G}_{ba}({t}^{\prime },t)},\end{equation} \tag{ 20 }$$

which can be a functional of both $\bar{\boldsymbol{\phi }}$ and G . Now it is well-known that in terms of Feynman diagrams, the self-energy does not contain any graphs that can be cut in two by cutting a single propagator line G_ab (t, t') (one-particle irreducibility). As Σ_ab (t, t') is defined as the functional derivative of ${{\Gamma}}_{2}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]$ in the present formalism, one can easily prove by contradiction [9] that ${{\Gamma}}_{2}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]$ must indeed be 2PI.

With ${G}_{0,\,ab}^{-1}(t,{t}^{\prime })=\delta (t-{t}^{\prime }){G}_{0,\,ab}^{-1}(t,t)$, equation (19) is then also equivalent to an integro-differential equation of Kadanoff–Baym type [31], i.e.

$$\begin{equation}\delta (t-{t}^{\prime })\mathbb{1}={\boldsymbol{G}}_{0}^{-1}(t,t)\boldsymbol{G}(t,{t}^{\prime })-{\int }_{0}^{{t}_{f}}\mathrm{d}s\,\mathbf{\Sigma }(t,s)\boldsymbol{G}(s,{t}^{\prime }).\end{equation} \tag{ 21 }$$

For illustration, we go back to the zero-dimensional ϕ⁴ action of equation (11). We can again perform a self-consistent expansion in the small parameter c (the equivalent of the well-known loop expansion in ℏ). The new variables of the Legendre transform are now defined as

$$\begin{equation}\bar{\phi }=\frac{\delta \mathcal{W}}{\delta j},\hspace{25.0pt}{\bar{\phi }}^{2}+cG=\frac{\delta \mathcal{W}}{\delta K/2}.\end{equation} \tag{ 22 }$$

By developing $\bar{\phi }$ and G in a formal power series, e.g. $\bar{\phi }={\bar{\phi }}_{0}+c{\bar{\phi }}_{1}+{c}^{2}{\bar{\phi }}_{2}+\mathcal{O}({c}^{3})$, one can then show that up to second order in both c and g [32], the 2PI effective action is

$$\begin{align}\hfill {\Gamma}(\bar{\phi },G)& =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.+S(\bar{\phi })+\frac{c}{2}\left({\sigma }^{-2}G+\mathrm{ln}\,{G}^{-1}+g{\bar{\phi }}^{2}G/2\right)\hfill \\ \hfill & \quad +{c}^{2}\left(\frac{g}{8}{G}^{2}-\frac{{g}^{2}}{12}{\bar{\phi }}^{2}{G}^{3}\right)+\mathcal{O}\left({c}^{3}\right).\hfill \end{align} \tag{ 23 }$$

Clearly, the solutions for the two variables will no longer be independent. This becomes particularly manifest from equation (18), which in this case read

$$\begin{equation}0={\sigma }^{-2}\left(\bar{\phi }-\mu \right)+g{\bar{\phi }}^{3}/6+cg\left(\bar{\phi }G-cg\bar{\phi }{G}^{3}/3\right)/2,\end{equation} \tag{ 24a }$$

$$\begin{equation}0={G}^{-1}-{\sigma }^{-2}-g{\bar{\phi }}^{2}/2-cg\left(G-g{\bar{\phi }}^{2}{G}^{2}\right)/2.\end{equation} \tag{ 24b }$$

Comparing this to the one-loop result in equation (13), the main difference is the self-consistency in the second cumulant, which leads to a coupled set of non-linear equations. To first order in c, equation (24) naturally reduce to equation (13).

A standard construction to find the path integral of a stochastic process consists of the following steps: the sought-for transition probability p(x|x₀) from an earlier point x₀ to a later point x is expressed as the marginal of the joint probability to find the system at a (fixed) number of intermediate points. When the distribution of the noise is assumed to be given, it is then possible to express p(x|x₀) in terms of the characteristic functional of the noise [22], which requires knowledge of the corresponding Jacobian. In this work, Itô regularization is assumed throughout for clarity, such that the Jacobian determinant becomes trivial (for details on this topic, see [1, 8]). The construction of the path integral via the noise characteristic functional introduces a special structure of the corresponding field theory that can be expressed by letting

$$\begin{equation}\boldsymbol{\phi }(t)=\left(\begin{matrix}\hfill x(t)\hfill \\ \hfill z(t)\hfill \end{matrix}\right),\hspace{25.0pt}\boldsymbol{j}(t)=\left(\begin{matrix}\hfill h(t)\hfill \\ \hfill j(t)\hfill \end{matrix}\right),\end{equation} \tag{ 25 }$$

where z(t) is the so-called response field (often written $\hat{x}$), and we have also introduced the respective source fields h(t) and j(t). Note that the source field K (t, t') is now a 2es2 matrix acting on the MSR fields in equation (25). The first cumulants are

$$\begin{equation}{\left.\frac{\delta \mathcal{W}}{\delta h(t)}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}=\langle x(t)\rangle ,\hspace{25.0pt}{\left.\frac{\delta \mathcal{W}}{\delta j(t)}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}=\langle z(t)\rangle =0,\end{equation} \tag{ 26 }$$

where ⟨z(t)⟩ = 0 is a well-known property of the MSR formalism following from the normalization of the partition function [8] (provided only a single one of the two boundary values is held fixed [3]). The second cumulants follow accordingly from

$$\begin{align}\hfill {\left.\frac{{\delta }^{2}\mathcal{W}}{\delta h(t)\delta j({t}^{\prime })}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}& =\frac{1}{c}\langle x(t)z({t}^{\prime })\rangle ,\hfill \\ \hfill {\left.\frac{{\delta }^{2}\mathcal{W}}{\delta h(t)\delta h({t}^{\prime })}\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}& =\frac{1}{c}\left[\langle x(t)x({t}^{\prime })\rangle -\langle x(t)\rangle \langle x({t}^{\prime })\rangle \right].\hfill \end{align} \tag{ 27 }$$

The Green function, consisting of the so-called statistical propagator F(t, t'), the response functions G^R/A (t, t') and the response–response function Q(t, t'), thus becomes

$$\begin{equation*}\boldsymbol{G}(t,{t}^{\prime })=\left(\begin{matrix}\hfill F(t,{t}^{\prime })\hfill & \hfill {G}^{R}(t,{t}^{\prime })\hfill \\ \hfill {G}^{A}(t,{t}^{\prime })\hfill & \hfill Q(t,{t}^{\prime })\hfill \end{matrix}\right)=\left(\begin{matrix}\hfill \frac{{\delta }^{2}\mathcal{W}}{\delta h(t)\delta h({t}^{\prime })}\hfill & \hfill \frac{{\delta }^{2}\mathcal{W}}{\delta h(t)\delta j({t}^{\prime })}\hfill \\ \hfill \frac{{\delta }^{2}\mathcal{W}}{\delta j(t)\delta h({t}^{\prime })}\hfill & \hfill \frac{{\delta }^{2}\mathcal{W}}{\delta j(t)\delta j({t}^{\prime })}\hfill \end{matrix}\right),\end{equation*}$$

where the retarded response function G^R (t, t') = ⟨x(t)z(t')⟩/c encodes the correlation of the system with previous noise and thus must vanish for t' > t. For zero sources, the response–response propagator Q(t, t') vanishes for the same reason as ⟨z(t)⟩. Thus, we find that the Green function provides us with the second cumulants, i.e.

$$\begin{equation}{\left.\left(\begin{matrix}\hfill F(t,{t}^{\prime })\hfill & \hfill {G}^{R}(t,{t}^{\prime })\hfill \\ \hfill {G}^{A}(t,{t}^{\prime })\hfill & \hfill Q(t,{t}^{\prime })\hfill \end{matrix}\right)\right\vert }_{\boldsymbol{j},\boldsymbol{K}=0}=\frac{1}{c}\left(\begin{matrix}\hfill \langle x(t)x({t}^{\prime })\rangle -\langle x(t)\rangle \langle x({t}^{\prime })\rangle \hfill & \hfill \langle x(t)z({t}^{\prime })\rangle \hfill \\ \hfill \langle z(t)x({t}^{\prime })\rangle \hfill & \hfill 0\hfill \end{matrix}\right).\end{equation} \tag{ 28 }$$

The special choice of fields from equation (25) also gives rise to a particular structure of the self-energy matrix. From equation (20), we find that

$$\begin{align}\hfill \mathbf{\Sigma }(t,{t}^{\prime })& =\left(\begin{matrix}\hfill 0\hfill & \hfill {{\Sigma}}^{A}(t,{t}^{\prime })\hfill \\ \hfill {{\Sigma}}^{R}(t,{t}^{\prime })\hfill & \hfill {{\Sigma}}^{F}(t,{t}^{\prime })\hfill \end{matrix}\right)=2{c}^{-1}\left(\begin{matrix}\hfill \frac{\delta {{\Gamma}}_{2}}{\delta F(t,{t}^{\prime })}\hfill & \hfill \frac{\delta {{\Gamma}}_{2}}{\delta {G}^{R}(t,{t}^{\prime })}\hfill \\ \hfill \frac{\delta {{\Gamma}}_{2}}{\delta {G}^{A}(t,{t}^{\prime })}\hfill & \hfill \frac{\delta {{\Gamma}}_{2}}{\delta Q(t,{t}^{\prime })}\hfill \end{matrix}\right),\hfill \end{align} \tag{ 29 }$$

where the vanishing of the upper left component is again a property of the MSR formalism (also true for the Keldysh representation of the analogous quantum effective action [33]). After expressing the diagrams contributing to Γ₂ via the propagator lines belonging to the four (scaled) cumulants F(t, t'), G^R/A (t, t') and Q(t, t'), the equations of motion follow straightforwardly from equation (18). Examples of this are given in the following.

The Ornstein–Uhlenbeck process is the standard example of a stationary, Gaussian Markov process [1, 2]. It follows from the Wiener process by addition of a linear drag and is described by the SDE

$$\begin{equation}\mathrm{d}x(t)=-\lambda x(t)\mathrm{d}t+\sqrt{D}\,\mathrm{d}W(t),\end{equation} \tag{ 30 }$$

where W(t > 0) is again a one-dimensional Brownian motion and λ, D > 0. Heuristically, equation (30) already goes back to Langevin [35], where it described the velocity of a Brownian particle in a viscous fluid under the influence of a random force. The corresponding Onsager–Machlup path integral [19] is given by

$$\begin{equation}\int \mathcal{D}x\,\mathrm{exp}\left\{-\frac{1}{2D}{\int }_{0}^{{t}_{f}}\mathrm{d}t\,{\left({\partial }_{t}x(t)+\lambda x(t)\right)}^{2}\right\},\end{equation} \tag{ 31 }$$

for which equation (3) reduces to the action functional

$$\begin{equation}S[x,z]={\int }_{0}^{{t}_{f}}\mathrm{d}t\,\left[D{z}^{2}(t)/2+\mathrm{i}z(t)({\partial }_{t}x(t)+\lambda x(t))\right].\end{equation} \tag{ 32 }$$

Note again that we employ the Itô convention. We now obtain the inverse Green function as the Hessian of equation (32) and find

$$\begin{equation}{\boldsymbol{G}}_{0}^{-1}(t,{t}^{\prime })=\delta (t-{t}^{\prime })\left(\begin{matrix}\hfill 0\hfill & \hfill -\mathrm{i}{\partial }_{t}+\mathrm{i}\lambda \hfill \\ \hfill \mathrm{i}{\partial }_{t}+\mathrm{i}\lambda \hfill & \hfill D\hfill \end{matrix}\right).\end{equation} \tag{ 33 }$$

As we are discussing a Gaussian process, the classical saddle point of S[x, z] provides the exact solution for the first cumulant (i.e. the mean of x(t)). From the saddle-point conditions

$$\begin{equation}0=\delta S[\bar{x},\bar{z}]/\delta \bar{x}(t)=-\mathrm{i}{\partial }_{t}\bar{z}(t)+\mathrm{i}\lambda \bar{z}(t),\end{equation} \tag{ 34a }$$

$$\begin{equation}0=\delta S[\bar{x},\bar{z}]/\delta \bar{z}(t)=D\bar{z}(t)+\mathrm{i}{\partial }_{t}\bar{x}(t)+\mathrm{i}\lambda \bar{x}(t)\end{equation} \tag{ 34b }$$

we see that a vanishing response-field average $\bar{z}(t)=0$ is a valid solution. As mentioned below equation (26), this is a consequence of the MSR formalism. Plugging this into equation (34b), we come out with ${\partial }_{t}\bar{x}(t)=-\lambda \bar{x}(t)$, which is also called the 'phenomenological' law of motion [19].

Turning to the second cumulant or statistical propagator via equation (21), which in the present case reduces to differential form since the self-energy vanishes, we obtain

$$\begin{align}\hfill 0& =\mathrm{i}{\partial }_{t}F(t,{t}^{\prime })+\mathrm{i}\lambda F(t,{t}^{\prime })+D{G}^{A}(t,{t}^{\prime }).\hfill \end{align} \tag{ 35 }$$

To find a complete solution for F(t, t'), one also needs the adjoint equation operating in the direction of the second time argument t'. Accordingly, we have the two equations

$$\begin{align}\hfill {\partial }_{t}F(t,{t}^{\prime })& =-\lambda F(t,{t}^{\prime })+\mathrm{i}D{G}^{A}(t,{t}^{\prime }),\hfill \\ \hfill {\partial }_{{t}^{\prime }}F(t,{t}^{\prime })& =-\lambda F(t,{t}^{\prime })+\mathrm{i}D{G}^{R}(t,{t}^{\prime }),\hfill \end{align} \tag{ 36 }$$

the second of which can be derived from the first via the symmetry properties F(t, t') = F(t', t) and G^R (t, t') = G^A (t', t). The adjoint equations arise because the second cumulants are two-time functions defined on the domain {(t, t')|t ∈ [0, t_f ], t' ∈ [0, t_f ]}. Causality then results in different equations depending on whether t ≶ t'. The 'equal-time' equation, for which t = t', can be derived by adding equation (36) and then taking the limit t' → t. By transforming to so-called Wigner coordinates $T=\left(t+{t}^{\prime }\right)/2$, τ = t − t', equation (36) can also be rewritten as

$$\begin{equation}{\partial }_{T}F(T,\tau )=-2\lambda F(T,\tau )+\mathrm{i}D\left({G}^{A}(T,\tau )+{G}^{R}(T,\tau )\right),\end{equation} \tag{ 37a }$$

$$\begin{equation}{\partial }_{\tau }F(T,\tau )=\frac{\mathrm{i}D}{2}\left({G}^{A}(T,\tau )-{G}^{R}(T,\tau )\right).\end{equation} \tag{ 37b }$$

The representation in Wigner coordinates is physically relevant because the stationary solutions F* will have the property of being independent of the 'center-of-mass' time T, i.e. F*(t, t') = F*(t − t') = F*(τ). The retarded and advanced response functions G^R/A (t, t') are determined by

$$\begin{align}\hfill \delta (t-{t}^{\prime })& =\left(+/-\right)\mathrm{i}{\partial }_{t}{G}^{R/A}(t,{t}^{\prime })+\mathrm{i}\lambda {G}^{R/A}(t,{t}^{\prime }),\hfill \end{align} \tag{ 38 }$$

respectively. The analytical solution for the advanced function is hence ${G}^{A}(t,{t}^{\prime })={G}^{A}(t-{t}^{\prime })=-\mathrm{i}\theta ({t}^{\prime }-t){\mathrm{e}}^{-\lambda \left({t}^{\prime }-t\right)}$, which is manifestly stationary. The retarded solution follows trivially by employing the above symmetry relation. Finding the solution for F(t, t') for all t, t' > 0 is easiest via equation (37). Observing that the dynamics in T decouples from that in τ, we rewrite equation (37a) for τ = 0, i.e.

$$\begin{equation}{\partial }_{T}F(T,0)=-2\lambda F(T,0)+D,\end{equation} \tag{ 39 }$$

where we have used the identity

$$\begin{equation}{G}^{A}(T,0)+{G}^{R}(T,0)={G}^{A}(t,t)+{G}^{R}(t,t)=-\mathrm{i},\end{equation} \tag{ 40 }$$

which is a property of the MSR formalism [33] and thus model-independent. Now equation (39) is solved by

$$\begin{align}\hfill F(T,\tau )& =\left(F(0,0)-\frac{D}{2\lambda }\right){\mathrm{e}}^{-2\lambda T}+\frac{\mathrm{i}D}{2\lambda }\left({G}^{A}(T,\tau )+{G}^{R}(T,\tau )\right),\hfill \end{align} \tag{ 41 }$$

where τ = 0. However, we may employ equation (41) also as an ansatz for the general solution at arbitrary τ. To see this, observe that because ${\partial }_{\tau }\left({G}^{A}(T,\tau )+{G}^{R}(T,\tau )\right)=\lambda \left({G}^{A}(T,\tau )-{G}^{R}(T,\tau )\right)$ by equation (38), the ansatz is also the solution to equation (37b). But since any point (t, t') can be reached through combined time evolution via equations (39) and (37b), we know that equation (41) must in fact also be the solution to equation (37a) for all τ. This is confirmed upon recasting equation (41) into the form

$$\begin{align}\hfill F(t,{t}^{\prime })& =F(0,0){\mathrm{e}}^{-\lambda \left(t+{t}^{\prime }\right)}-\frac{D}{2\lambda }\left({\mathrm{e}}^{-\lambda \left(t+{t}^{\prime }\right)}-{\mathrm{e}}^{-\lambda \left\vert t-{t}^{\prime }\right\vert }\right),\hfill \end{align} \tag{ 42 }$$

which is the known solution for the second cumulant of the Ornstein–Uhlenbeck process, obtained after solving equation (30) directly by $x(t)=x(0){\mathrm{e}}^{-\lambda t}+\sqrt{D}{\int }_{0}^{t}{\mathrm{e}}^{\lambda \left(s-t\right)}\,\mathrm{d}W(s)$. To find the stationary solution, we let T → ∞ and find ${F}^{\ast }(\tau )=D{\lambda }^{-1}\,{\mathrm{e}}^{-\lambda \left\vert \tau \right\vert }/2$. This completes our discussion of Gaussian processes.

A process that is not Gaussian yet still exactly solvable is defined by the SDE

$$\begin{equation}\mathrm{d}x(t)=\mu x(t)\mathrm{d}t+\sigma x(t)\mathrm{d}W(t),\end{equation} \tag{ 43 }$$

where $\mu ,\sigma \in \mathbb{R}$. The process x(t) is called GBM and foundational for the description of financial markets, where it serves as the stock-price model underlying the Black–Scholes equation [4]. Beyond this, stochastic exponential growth is important in a variety of physical and biological systems, e.g. nuclear fission and population dynamics [36]. We proceed to show that the 2PI effective action can reproduce the exact solution for the first two cumulants of this process. This can serve as a starting point for applying self-consistent perturbation theory to other non-trivial stochastic processes such as the Heston model [5].

The MSR action of the process x(t) in Itô convention is given by

$$\begin{equation}S[x,z]={\int }_{0}^{{t}_{f}}\mathrm{d}t\;[{\sigma }^{2}{x}^{2}(t){z}^{2}({t}^{+})/2+\mathrm{i}z(t)({\mathrm{\partial }}_{t}x(t)-\mu x(t))],\end{equation} \tag{ 44 }$$

where we have introduced time-ordering in the non-quadratic part by defining t^± = t ± ɛ for ɛ → 0⁺. This is not strictly required because the correct regularization of the continuum limit is ensured by the causality properties of the response functions G^R/A . In the evaluation of closed time loops produced by the non-linearity, however, some extra care is helpful to avoid ambiguities. Thus the time-ordering in equation (44) is a reminder that the noise should always run 'ahead' of the system [33]. Note that a comment about the Stratonovich convention is given below equation (51).

Expanding the action around the classical saddle point, we have

$$\begin{align}\hfill S[{x}_{0}+\sqrt{c}\delta x,{z}_{0}+\sqrt{c}\delta z]& =S[{x}_{0},{z}_{0}]+{S}^{(2)}+\mathcal{V},\hfill \end{align} \tag{ 45 }$$

with the classical action S[x₀, z₀] to lowest order. The factors of $\sqrt{c}$ will make the Gaussian part S⁽²⁾/c of the exponent independent of c and at the same time multiply each vertex by c^m/2, where m is the number of legs. Since the expectation values in the propagator lines of the Feynman diagrams are formed with respect to S⁽²⁾/c, this removes the need of introducing the correct prefactors of c by hand. It furthermore ensures that c tracks the number of loops in the loop expansion of the effective action [6]. The effective quadratic part is then given by

$$\begin{align}\hfill {S}^{(2)}/c& ={\int }_{0}^{{t}_{f}}\mathrm{d}t\,\left[{\sigma }^{2}{x}_{0}^{2}\delta {z}^{2}(t)/2+\mathrm{i}\delta z(t)\left({\partial }_{t}-\mu \right)\delta x(t)\right.\hfill \\ \hfill & \quad +\left.2{\sigma }^{2}{z}_{0}{x}_{0}\delta z({t}^{+})\delta x(t)+{\sigma }^{2}{z}_{0}^{2}\delta {x}^{2}(t)/2\right],\hfill \end{align} \tag{ 46 }$$

and the non-quadratic part

$$\begin{align}\hfill \mathcal{V}/c& ={\sigma }^{2}{\int }_{0}^{{t}_{f}}\mathrm{d}t\,\left[\sqrt{c}\left({z}_{0}\delta z({t}^{+})\delta {x}^{2}(t)+{x}_{0}\delta x(t)\delta {z}^{2}({t}^{+})\right)\right.\hfill \\ \hfill & \quad \left.+c\delta {x}^{2}(t)\delta {z}^{2}({t}^{+})/2\right],\hfill \end{align} \tag{ 47 }$$

where the vertex given by the first term does not contribute to the perturbation series because it is proportional to z₀. Observe that terms proportional to powers of z₀ may always be discarded immediately, provided one does not have to differentiate with respect to them at a later point. A graphical representation of the two remaining cubic and quartic vertices is provided in figure 2.

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The inverse Green function is given by the Hessian of S⁽²⁾/c in equation (46), evaluated at the Legendre-transformed fields, i.e.

$$\begin{align}\hfill {\boldsymbol{G}}_{0}^{-1}[\bar{x},\bar{z}](t,{t}^{\prime })& =\delta (t-{t}^{\prime })\left(\begin{matrix}\hfill {\sigma }^{2}{\bar{z}}^{2}\hfill & \hfill -\mathrm{i}{\partial }_{t}-\mathrm{i}\mu \hfill \\ \hfill \mathrm{i}{\partial }_{t}-\mathrm{i}\mu \hfill & \hfill {\sigma }^{2}{\bar{x}}^{2}\hfill \end{matrix}\right)\hfill \\ \hfill & \quad +2{\sigma }^{2}\bar{z}\bar{x}\left(\begin{matrix}\hfill 0\hfill & \hfill \delta ({t}^{+}-{t}^{\prime })\hfill \\ \hfill \delta ({t}^{-}-{t}^{\prime })\hfill & \hfill 0\hfill \end{matrix}\right),\hfill \end{align} \tag{ 48 }$$

where in the second term we have taken care of the time-ordered fields δz(t⁺)δx(t) = δz(t)δx(t⁻) in equation (46).

To find the 2PI diagrams contributing to Γ₂ in equation (17), we first consider the cubic vertex depicted in figure 2(a). For a given number of vertices 2k, $k\in \mathbb{N}$, this vertex has 4k response fields and 2k fields δx(t). Thus for k ⩾ 2, there are at least two response–response propagators Q(s, s') contributing to any connected diagram, where s, s' are internal times. According to equations (28) and (29), after differentiation any of the self-energy components Σ^R/A/F will hence contain at least one vanishing response–response propagator. For k = 1 we find the two diagrams shown in figure 3. The self-energies derivable from figure 3(a) vanish by the same argument we have just given for the case k ⩾ 2, while figure 3(b) induces either a vanishing response–response propagator or a closed retarded–advance loop of the form G^R (s, s')G^A (s, s'), which must also vanish due to causality. To summarize, we have shown that the cubic vertex of figure 2(a) does not yield any non-trivial self-energies.

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Turning to the quartic vertex in figure 2(b), to lowest order we find the two diagrams shown in figure 4, which correspond to the self-consistent Hartree–Fock approximation in the framework of the 2PI effective action. Written explicitly in terms of propagators, these diagrams are given in equation (49). To second-order, the quartic vertex gives rise to the diagrams illustrated in figure 5. Again, because of response–response propagators and closed retarded–advanced loops, these diagrams do not result in non-trivial self-energies.

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This leaves us to investigate any connected, 2PI diagram with l > 2 quartic vertices, where $l\in \mathbb{N}$, as well as all possible combinations of cubic and quartic vertices. The latter diagrams are formed out of 2k cubic and l quartic vertices, which as before rules out all diagrams with k ⩾ 2. We do not attempt a formal proof that all of these diagrams vanish since this is achieved more directly via the recovery of the exact solution for GBM from the diagrams in figure 4.

To still gain some insight into why these diagrams vanish, consider figure 5(a). If we break up one solid and one dashed propagator line, we can insert another quartic vertex from figure 2(b) and connect solid to solid and dashed to dashed legs. Topologically, this produces the only connected, 2PI diagram at l = 3. All other valid diagrams with three quartic vertices follow from the latter by opening and mixing any pair of propagators (in the same way figure 5(b) can be obtained from figure 5(a) by opening two propagator lines and 'twisting' them). Clearly, any of the self-energies derivable from the diagrams with l = 3 must vanish, analogously to those of figure 5.

The simplest diagrams containing both cubic and quartic vertices are produced by cutting the two propagators in the first-order diagrams of figure 4 and connecting the resulting amputated legs in all ways to those of two cubic vertices joined by a single propagator. These diagrams are connected, 2PI, and third order in the coupling σ². As one may verify, all of them contain at least one response–response propagator and a retarded–advanced loop, such that the resulting self-energies again vanish.

Summing up, we assume that the only diagrams contributing non-trivially to the self-energy matrix are those of figure 4, and consequently that the exact 2PI effective action of GBM in Itô convention is found by letting

$$\begin{equation}{{\Gamma}}_{2}=\frac{{c}^{2}{\sigma }^{2}}{2}\int \mathrm{d}t\,Q(t,t)F(t,t)+\frac{{c}^{2}{\sigma }^{2}}{4}\int \mathrm{d}t{\left({G}^{A}({t}^{+},t)+{G}^{R}({t}^{-},t)\right)}^{2},\end{equation} \tag{ 49 }$$

where we keep the second term for completeness (because of causality it does not contribute to the self-energies in the Itô regularization where G^A (t⁺, t) = G^R (t⁻, t) = 0). To go from the vertices on the right-hand side of equation (47) to the functional in equation (49), the diagrammatic rules were as follows: (1) take the power of each vertex at the given order in the expansion and contract the fields in all possible combinations according to Wick's theorem [6], i.e. $\delta {x}^{2}(t)\delta {z}^{2}({t}^{+})\to \left\{\langle \delta {x}^{2}(t)\rangle \langle \delta {z}^{2}({t}^{+})\rangle ,2\langle \delta x(t)\delta z{({t}^{+})\rangle }^{2}\right\}$. (2) Replace the averages by Green functions: $\left\{Q(t,t)F(t,t),{[{G}^{A}({t}^{+},t)+{G}^{R}({t}^{-},t)]}^{2}/2\right\}$, where we also have symmetrized the response functions. (3) Multiply by the prefactors determined by the order of the interaction, in this case c ⋅ cσ²/2.

To derive the phenomenological law of motion [19] for the process x(t), we now take the derivative of ${\Gamma}[\bar{\boldsymbol{\phi }},\boldsymbol{G}]$ according to equation (18) with $\bar{\boldsymbol{\phi }}={(\bar{x},\bar{z})}^{\text{T}}$, i.e.

$$\begin{align}\hfill 0& =\frac{\delta S[\bar{x},\bar{z}]}{\delta \bar{z}(t)}+\frac{c}{2}\int \mathrm{d}{t}^{\prime }\,\mathrm{d}{t}^{\prime \prime }\,\mathrm{Tr}\,\frac{\delta {\boldsymbol{G}}_{0}^{-1}({t}^{\prime },{t}^{\prime \prime })}{\delta \bar{z}(t)}\boldsymbol{G}({t}^{\prime \prime },{t}^{\prime })\hfill \\ \hfill & ={\sigma }^{2}\bar{z}{\bar{x}}^{2}+\left(\mathrm{i}{\partial }_{t}-\mathrm{i}\mu \right)\bar{x}+c{\sigma }^{2}\bar{x}\left[{G}^{A}({t}^{+},t)+{G}^{R}({t}^{-},t)\right].\hfill \end{align} \tag{ 50 }$$

The time-ordering in the response functions ensures that they evaluate to zero, such that with $\bar{z}(t)=0$ we find ${\partial }_{t}\bar{x}(t)=\mu \bar{x}(t)$, as expected for GBM in Itô convention. That $\bar{z}(t)=0$ is indeed a solution can be verified by calculating

$$\begin{align}\hfill 0& =\frac{\delta S[\bar{x},\bar{z}]}{\delta \bar{x}(t)}+\frac{c}{2}\int \mathrm{d}{t}^{\prime }\,\mathrm{d}{t}^{\prime \prime }\,\mathrm{Tr}\,\frac{\delta {\boldsymbol{G}}_{0}^{-1}({t}^{\prime },{t}^{\prime \prime })}{\delta \bar{x}(t)}\boldsymbol{G}({t}^{\prime \prime },{t}^{\prime })\hfill \\ \hfill & ={\sigma }^{2}\bar{x}{\bar{z}}^{2}-\left(\mathrm{i}{\partial }_{t}+\mathrm{i}\mu \right)\bar{z}+c{\sigma }^{2}\bar{z}\left[{G}^{A}({t}^{+},t)+{G}^{R}({t}^{-},t)\right],\hfill \end{align} \tag{ 51 }$$

where the contribution from the first term in equation (48) vanishes because it is traceless. Note that using the Stratonovich convention would lead to a different first cumulant since we are studying an SDE with multiplicative noise. Specifically, one expects a solution $\bar{x}(t)=\bar{x}(0)\mathrm{exp}\left((\mu +{\sigma }^{2}/2)t\right)$, i.e. ${\partial }_{t}\bar{x}(t)=(\mu +{\sigma }^{2}/2)\bar{x}(t)$. This would result from an additional noise-dependent drift term σ² x(t)/2 in the Stratonovich analogue of equation (44), and a modified regularization of the response functions, i.e. G^R/A (t^−/+, t) = G^R/A (t, t) = −i/2 in our notation (s. also reference [8] and section 4.4.2 of reference [1]).

To find the self-energy, we plug Γ₂ from equation (49) into equation (29) to obtain

$$\begin{align}\hfill \mathbf{\Sigma }(t,{t}^{\prime })& =-c{\sigma }^{2}\delta (t-{t}^{\prime })\left(\begin{matrix}\hfill 0\hfill & \hfill {G}^{R}({t}^{-},t)\hfill \\ \hfill {G}^{A}({t}^{+},t)\hfill & \hfill F(t,t)\hfill \end{matrix}\right).\hfill \end{align} \tag{ 52 }$$

Taking into account that G^R (t⁻, t) = G^A (t⁺, t) = 0, the response self-energies Σ^R/A (t, t') vanish, while the lower-left component of equation (21) leads to

$$\begin{align}\hfill 0& =\left(\mathrm{i}{\partial }_{t}-\mathrm{i}\mu \right)F(t,{t}^{\prime })+{\sigma }^{2}\bar{x}{(t)}^{2}{G}^{A}(t,{t}^{\prime })-{\int }_{0}^{{t}_{f}}\mathrm{d}s\,{{\Sigma}}^{F}(t,s){G}^{A}(s,{t}^{\prime }).\hfill \end{align} \tag{ 53 }$$

Writing down also the adjoint (cf equation (36)), we thus have the time-local equations

$$\begin{equation}{\partial }_{t}F(t,{t}^{\prime })=\mu F(t,{t}^{\prime })+\mathrm{i}{\sigma }^{2}\left[{\bar{x}}^{2}(t)+cF(t,t)\right]{G}^{A}(t,{t}^{\prime }),\end{equation} \tag{ 54a }$$

$$\begin{equation}{\partial }_{{t}^{\prime }}F(t,{t}^{\prime })=\mu F(t,{t}^{\prime })+\mathrm{i}{\sigma }^{2}\left[{\bar{x}}^{2}({t}^{\prime })+cF({t}^{\prime },{t}^{\prime })\right]{G}^{R}(t,{t}^{\prime }).\end{equation} \tag{ 54b }$$

Since the scaling constant c was introduced for formal reasons, we may now set c = 1 and transform equation (54) to Wigner coordinates. In the direction of the center-of-mass time T, this leaves us with

$$\begin{align}\hfill {\partial }_{T}F(T,\tau )& =2\mu F(T,\tau )+\mathrm{i}{\sigma }^{2}F(T,0)\left[{G}^{A}(T,\tau )+{G}^{R}(T,\tau )\right]\hfill \\ \hfill & \quad +\mathrm{i}{\sigma }^{2}\left[{\bar{x}}^{2}(T+\tau /2){G}^{A}(T,\tau )+{\bar{x}}^{2}(T-\tau /2){G}^{R}(T,\tau )\right].\hfill \end{align} \tag{ 55 }$$

In the equal-time limit τ → 0, and with the identity in equation (40), this transforms into

$$\begin{align}\hfill {\partial }_{t}F(t,t)& =2\mu F(t,t)+{\sigma }^{2}\left[{\bar{x}}^{2}(t)+F(t,t)\right],\hfill \end{align} \tag{ 56 }$$

which is solved by $F(t,t)={\bar{x}}^{2}(t)\left[\mathrm{exp}\left({\sigma }^{2}t\right)-1\right]$ where $\bar{x}(t)=\bar{x}(0)\mathrm{exp}\left(\mu t\right)$. For t > t', equation (54a) has the solution ${F}^{ > }(t-{t}^{\prime },{t}^{\prime })={F}_{0}\,\mathrm{exp}\left(\mu \left(t-{t}^{\prime }\right)\right)$. With an initial condition F₀ = F(t', t') placed on the equal-time diagonal, we can thus find the general solution

$$\begin{align}\hfill F(t,{t}^{\prime })& =\bar{x}(t)\bar{x}({t}^{\prime })\left[\mathrm{exp}\left({\sigma }^{2}{t}^{\prime }\right)-1\right],\hfill \end{align} \tag{ 57 }$$

which holds for t ⩾ t'. The solution on the upper triangle of the two-time square (t, t') can again be found from symmetry. The general solution of the Itô SDE (43) is (e.g. section 4.4.7 of reference [1])

$$\begin{align}\hfill x(t)& =x(0)\mathrm{exp}\left((\mu -{\sigma }^{2}/2)t\right)\mathrm{exp}\left(\sigma ({W}_{t}-{W}_{0})\right).\hfill \end{align} \tag{ 58 }$$

Since the variance of $({W}_{t}-{W}_{0})+({W}_{{t}^{\prime }}-{W}_{0})=\left({W}_{t}-{W}_{{t}^{\prime }}\right)+2\left({W}_{{t}^{\prime }}-{W}_{0}\right)$ is given by $\langle {\left({W}_{t}-{W}_{{t}^{\prime }}\right)}^{2}\rangle +4\langle {\left({W}_{{t}^{\prime }}-{W}_{0}\right)}^{2}\rangle =t-{t}^{\prime }+4{t}^{\prime }$ for t ⩾ t', one has

$$\begin{align}\hfill \langle x(t)x({t}^{\prime })\rangle & ={\bar{x}}^{2}(0)\mathrm{exp}\left((\mu -{\sigma }^{2}/2)(t+{t}^{\prime })\right)\left\langle \mathrm{exp}\left(\sigma ({W}_{t}-{W}_{0}+{W}_{{t}^{\prime }}-{W}_{0})\right)\right\rangle \hfill \\ \hfill & =\bar{x}(t)\bar{x}({t}^{\prime })\mathrm{exp}\left({\sigma }^{2}\left[t+3{t}^{\prime }-(t+{t}^{\prime })\right]/2\right)=\bar{x}(t)\bar{x}({t}^{\prime })\mathrm{exp}\left({\sigma }^{2}{t}^{\prime }\right),\hfill \end{align} \tag{ 59 }$$

and consequently equation (57) as the solution for the two-time variance of the process x(t). As mentioned above, this also serves as proof that in the MSR formalism, the entire diagrammatic series of GBM vanishes (apart from the single diagram in figure 4(a)).

A natural starting point for the investigation of the performance of the 2PI effective action on a non-linear model is provided by the SDE

$$\begin{align}\hfill \mathrm{d}x(t)& =\left(\alpha x(t)+\beta {x}^{2}(t)\right)\mathrm{d}t+\sqrt{D}\,\mathrm{d}W(t).\hfill \end{align} \tag{ 60 }$$

Note that this choice furthermore allows for a direct comparison to previous work involving the 1PI effective action [3]. The corresponding classical potential V(x) = −(αx²/2 + βx³/3) is plotted in figure 6 for both positive and negative values of α. The fixed points of this map are located at ${x}_{1}^{\ast }=0$ and ${x}_{2}^{\ast }=-\alpha /\beta$. Stability is given whenever α + 2βx < 0, which is true for x < 0 at α > 0 and $x< \left\vert \alpha \right\vert /\beta$ at α < 0. It is thus simplest to set α > 0 and confine the stable region to negative positions. Physically, the escape of a trajectory to infinity can serve, among other things, as a model of a firing neuron [3]. The MSR action of equation (60) in Itô form is then given by

$$\begin{align}\hfill S[x,z]& ={\int }_{0}^{{t}_{f}}\mathrm{d}t\,\left[Dz{(t)}^{2}/2+\mathrm{i}z(t)\left({\partial }_{t}x(t)-\alpha x(t)-\beta x{({t}^{-})}^{2}\right)\right],\hfill \end{align} \tag{ 61 }$$

where we have again applied the appropriate time-ordering in the non-linear part. The graphical representation of the interaction vertex is shown in figure 7(a).

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One may now ask in which small parameter the expansion is to be performed, i.e. which parameter is to play the role of the so-far purely formal constant c? The answer is found by rewriting equation (61) in the following way:

$$\begin{align*}S[x,z]& =\frac{\alpha }{\beta }{\int }_{0}^{(\beta /\alpha )\;\alpha {t}_{f}}\mathrm{d}s\left[\frac{D}{2\alpha }z{(s)}^{2}+\mathrm{i}z(s)\left(\frac{\beta }{\alpha }{\mathrm{\partial }}_{s}x(s)-x(s)-\frac{\beta }{\alpha }x{({s}^{-})}^{2}\right)\right],\end{align*}$$

where s is a now unitless parameter. Thus we see that β/α can play the role of a small number dividing the exponent, under the condition that all frequencies are measured in units of the drift α, and that both the integration interval and the time derivative are rescaled by β/α. Practically, of course, we may simply work with equation (61) and set c = 1, which we do in the following (this does not imply that β/α = 1). Proceeding as before, we find an inverse Green function

$$\begin{align}\hfill {\boldsymbol{G}}_{0}^{-1}(t,{t}^{\prime })& =\delta (t-{t}^{\prime })\left(\begin{matrix}\hfill -2\mathrm{i}\beta \bar{z}\hfill & \hfill -\mathrm{i}{\partial }_{t}-\mathrm{i}\alpha \hfill \\ \hfill \mathrm{i}{\partial }_{t}-\mathrm{i}\alpha \hfill & \hfill D\hfill \end{matrix}\right)\hfill \\ \hfill & \quad -2\mathrm{i}\beta \bar{x}\left(\begin{matrix}\hfill 0\hfill & \hfill \delta ({t}^{+}-{t}^{\prime })\hfill \\ \hfill \delta ({t}^{-}-{t}^{\prime })\hfill & \hfill 0\hfill \end{matrix}\right).\hfill \end{align} \tag{ 62 }$$

Analogously to equation (51), this does not yield a contribution to the equation of motion for $\bar{z}(t)$. Instead of equation (50), we now find

$$\begin{align}\hfill 0& ={\partial }_{t}\bar{x}-\alpha \bar{x}-\beta {\bar{x}}^{2}-\beta F(t,t).\hfill \end{align} \tag{ 63 }$$

Note that this equation contains the resummed statistical propagator F instead of the bare one. It thus results in a self-consistently corrected steady-state ${\bar{x}}^{\ast }$, specified by the quadratic equation $0=\alpha {\bar{x}}^{\ast }+\beta {\left({\bar{x}}^{\ast }\right)}^{2}+\beta {F}^{\ast }$, the solution to which is

$$\begin{equation}{\bar{x}}^{\ast }=-\frac{\alpha }{2\beta }\pm \frac{1}{2\beta }\sqrt{{\alpha }^{2}-4{\beta }^{2}{F}^{\ast }}.\end{equation} \tag{ 64 }$$

Hence, while for the practically viable one-loop approximation of the 1PI effective action employed in [3], the influence of the fluctuations on the steady-state of the mean is provided by the bare correlation function, the 2PI effective action takes the corresponding corrections to the mean into account automatically when second-order diagrams are included.

To first order in the interaction parameter β, the statistical propagator obeys the equation of motion

$$\begin{align}\hfill 0& =\left({\partial }_{t}-\alpha -2\beta \bar{x}\right)F(t,{t}^{\prime })-\mathrm{i}D{G}^{A}(t,{t}^{\prime }),\hfill \end{align} \tag{ 65 }$$

which in the equal-time limit gives

$$\begin{align}\hfill {\partial }_{t}F(t,t)& =\left(2\alpha +4\beta \bar{x}\right)F(t,t)+D.\hfill \end{align} \tag{ 66 }$$

Hence, we find the steady-state solution ${F}^{\ast }=-D/\left(2\alpha +4\beta {\bar{x}}^{\ast }\right)$. Inserting this into equation (64) reproduces the result from reference [3]. Note also that since F* > 0, the parameters α and β need to be chosen such that the denominator of F* is always negative, which coincides with the stability condition found earlier. For completeness, we note that to this order, the retarded Green function is determined by

$$\begin{equation}-\mathrm{i}\delta (t-{t}^{\prime })=\left({\partial }_{t}-\alpha -2\beta \bar{x}\right){G}^{R}(t,{t}^{\prime }).\end{equation} \tag{ 67 }$$

An approximation beyond mean-field plus quadratic fluctuations is obtained upon including the second-order diagrams shown in figures 7(b) and (c). Writing down the corresponding contributions to the 2PI effective action, we find

$$\begin{align}\hfill {{\Gamma}}_{2}& ={\beta }^{2}\int \mathrm{d}t\,\mathrm{d}{t}^{\prime }\,\left[2{G}^{R}(t,{t}^{\prime }){G}^{A}(t,{t}^{\prime })F(t,{t}^{\prime })+{F}^{2}(t,{t}^{\prime })Q(t,{t}^{\prime })\right]+\mathcal{O}({\beta }^{4}).\hfill \end{align} \tag{ 68 }$$

Again taking derivatives with respect to the Green functions, we obtain the one-loop self-energy

$$\begin{align}\hfill \mathbf{\Sigma }(t,{t}^{\prime })& =-{\beta }^{2}\left(\begin{matrix}\hfill 0\hfill & \hfill 4{G}^{A}(t,{t}^{\prime })\hfill \\ \hfill 4{G}^{R}(t,{t}^{\prime })\hfill & \hfill 2F(t,{t}^{\prime })\hfill \end{matrix}\right)F(t,{t}^{\prime }).\hfill \end{align} \tag{ 69 }$$

Equation (67) now turns into an integro-differential equation

$$\begin{equation}-\mathrm{i}\delta (t-{t}^{\prime })=\left({\partial }_{t}-\alpha -2\beta \bar{x}\right){G}^{R}(t,{t}^{\prime })-4\mathrm{i}{\beta }^{2}{\int }_{0}^{t}\mathrm{d}s\,\,F(t,s){G}^{R}(t,s){G}^{R}(s,{t}^{\prime }),\end{equation} \tag{ 70 }$$

while equation (65) becomes accordingly

$$\begin{align}\hfill 0& =\left({\partial }_{t}-\alpha -2\beta \bar{x}\right)F(t,{t}^{\prime })-\mathrm{i}D{G}^{A}(t,{t}^{\prime })\hfill \\ \hfill & \quad -\mathrm{i}{\beta }^{2}{\int }_{0}^{t}\mathrm{d}s\,\left\{4{G}^{R}(t,s)F(t,s)F(s,{t}^{\prime })+2{F}^{2}(t,s){G}^{A}(s,{t}^{\prime })\right\}.\hfill \end{align} \tag{ 71 }$$

Observe that the second term under the integral only needs to be evaluated up to t' (from causality of G^A , and assuming t ⩾ t'). In the equal-time limit, equation (71) leads to

$$\begin{align}\hfill {\partial }_{t}F(t,t)& =\left(2\alpha +4\beta \bar{x}\right)F(t,t)+D+12{\beta }^{2}{\int }_{0}^{t}\mathrm{d}s\,\,\mathcal{K}(t,s){F}^{2}(t,s),\hfill \end{align} \tag{ 72 }$$

where the symmetric integral kernel $\mathcal{K}(t,{t}^{\prime })$ obeys the equation

$$\begin{align}\hfill 0& =\left({\partial }_{t}-\alpha -2\beta \bar{x}\right)\mathcal{K}(t,{t}^{\prime })-4{\beta }^{2}{\int }_{{t}^{\prime }}^{t}\mathrm{d}s\,\,F(t,s)\mathcal{K}(t,s)\mathcal{K}(s,{t}^{\prime })\hfill \end{align} \tag{ 73 }$$

for t ⩾ t', such that ${G}^{R}(t,{t}^{\prime })=-\mathrm{i}\theta (t-{t}^{\prime })\mathcal{K}(t,{t}^{\prime })$ and ${G}^{A}(t,{t}^{\prime })=-\mathrm{i}\theta ({t}^{\prime }-t)\mathcal{K}({t}^{\prime },t)$, where $\mathcal{K}({t}^{\prime },t)=\mathcal{K}(t,{t}^{\prime })$. Note that the integral term in equation (73) is expected to be small because the overlap of the two kernels tends to be negligible. This kernel may also be employed in the solution of equation (71) since

$$\begin{equation}{\int }_{0}^{t}\mathrm{d}s\,\,\mathrm{i}{G}^{R}(t,s)={\int }_{0}^{t}\mathrm{d}s\,\,\mathcal{K}(t,s),\hspace{25.0pt}{\int }_{0}^{t}\mathrm{d}s\,\,\mathrm{i}{G}^{A}(s,{t}^{\prime })={\int }_{0}^{{t}^{\prime }}\mathrm{d}s\,\,\mathcal{K}(s,{t}^{\prime }),\end{equation} \tag{ 74 }$$

again assuming w.l.o.g. that t ⩾ t'. On the equal-time diagonal, the kernel is constant and unity, i.e. $\mathcal{K}(t,t)=1$.

The coupled set of non-linear equations (63) and (71)–(73) needs to be solved self-consistently. This can be achieved efficiently and accurately by employing, for instance, a multi-step predictor–corrector method on a fixed-step two-dimensional grid [37, 38]. Further discussions of numerical methods for equations of this type (in many-body physics usually referred to as Kadanoff–Baym equations [31]) can be found in references [39–42].

As a comparison for our numerical solutions of the above integro-differential equations, we employ state-of-the-art Runge–Kutta methods for SDEs [43, 44]. Specifically, we use implementations [34, 45] of the algorithms SRI1W1 and SRI2W1 derived in reference [43]. Both algorithms converge with strong order 1.5, while their weak orders are 2.0 and 3.0, respectively. Roughly speaking, in the strong case, one is concerned with the convergence of an approximate stochastic process toward the true process, while weak convergence addresses the convergence of expectation values of functionals of such processes [46].

As an analytical comparison, we furthermore use the stationary solutions for the mean and the variance derived in one-loop approximation from the 1PI effective action in reference [3], which we denote by ${\bar{x}}_{1\text{PI}}^{\ast }$ and ${F}_{1\text{PI}}^{\ast }$, respectively. Note that the 1PI one-loop approximation for the propagators is also of second order in β.

Exemplary results are presented in figure 8 for parameters D/α = 1.5 and β/α = 0.15. The integro-differential equations were solved on a two-dimensional grid with fixed step size αΔt = 2⁻¹⁰, while the SDE solvers implemented in reference [34] work with adaptive step sizes. Our step size was chosen small to ensure convergence given that the integrals in the equations of motion are evaluated with the trapezoidal rule only. However, the steady-states shown in figure 8 in fact converge reasonably for a step size of αΔt = 2⁻⁸. Advanced numerical methods for integro-differential equations, employing higher-order integrators and thus enabling larger step sizes, are discussed in reference [42].

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For the SDE solvers SRI1W1 and SRI2W1, we fixed the ensemble size to 2es10⁶ and 4es10⁶, respectively. Divergent trajectories escaping to x → ∞ were discarded and replaced. In this way, in order to reach two (four) million trajectories in total, 57 (102) divergent trajectories were discarded. Since these numbers correspond to negligible fractions, this is equivalent to only considering non-escaping trajectories [3]. Note that we used a greater ensemble size in the comparison with SRI2W1 because it serves to confirm one of our main results.

Figures 8(a) and (b) show our results for the mean and variance from the second-order 2PI effective action (2PI-2), starting from initial conditions $\bar{x}(0)=-10$ and F(0, 0) = 2. The insets show the absolute difference to the statistics obtained via the SDE solver with weak order 3.0 (WO3), thus confirming the overall accuracy of the solutions obtained from equations (63) and (71)–(73). Figure 8(c) summarizes the results for the mean $\bar{x}(t)$, where we also show how the first-order approximation of the 2PI effective action (2PI-1) recovers ${\bar{x}}_{1\text{PI}}^{\ast }$ in the long-time limit. Here, both of the SDE solvers reproduce our solution 2PI-2. This is not true once we move to figure 8(d), where only the algorithm of weak order 3.0 (SRI2W1) captures the second-order result from the 2PI effective action. The algorithm of weak order 2.0 (SRI1W1) instead reproduces the first-order approximation 2PI-1. In comparison to the one-loop 1PI result ${F}_{1\text{PI}}^{\ast }$, we see that the additional diagrams resummed by the 2PI effective action lead to a systematic improvement of about half a percent, which is expected to become more pronounced at larger non-linearity. The different effective actions are thus found to converge to different stationary statistics at large final times.