diagrams. Meanwhile, there is also a quantum wavefunction coefficient
$\psi_{2}^{\rm 1-loop}$ at the $g^{2}$ order

$\tilde{W}^{(2)}_{I}=\int_{\bf k_{1},k_{2}}{\rm Re}\psi_{2}^{\rm
1-loop}\leavevmode\nobreak\ \phi_{\bf k_{1}}\phi_{\bf
k_{2}}\leavevmode\nobreak\ .$ (5.14)

Let’s first consider the semi-classical wavefunction from tree-level
processes. In perturbation theory, their $\Lambda$-dependences can be derived
by requiring that the corresponding correlators are independent of the cutoff
scales. As correlators of long wavelength modes are physical observables, the
final results should not be affected by the artificial UV cutoff scale that we
choose for smoothing out the short wavelength modes. At the linear order in
$g$, the three-point correlator is given by

$\langle\phi_{l}^{3}\rangle=\int D\phi\phi^{3}P_{\Lambda}=-\int_{\bf
k_{1},k_{2},k_{3}}\frac{\Omega_{\Lambda}(k_{1})\Omega_{\Lambda}(k_{2})\Omega_{\Lambda}(k_{3})}{8{\rm
Re}\psi_{2}(k_{1}){\rm Re}\psi_{2}(k_{2}){\rm Re}\psi_{2}(k_{3})}2{\rm
Re}\psi_{3}^{\Lambda}\leavevmode\nobreak\ .$ (5.15)

Then up to the $g$ order, its $\Lambda$-independence leads to

$\frac{d}{d\ln\Lambda}\langle\phi_{l}^{3}\rangle=0\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d{\rm
Re}\psi_{3}^{\Lambda}}{d\ln\Lambda}=-\frac{d}{d\ln\Lambda}\big{(}\ln\left[\Omega_{\Lambda}(k_{1})\Omega_{\Lambda}(k_{2})\Omega_{\Lambda}(k_{3})\right]\big{)}{\rm
Re}\psi_{3}^{\Lambda}+\mathcal{O}(g^{2})\leavevmode\nobreak\ ,$ (5.16)

which can be written as

$\frac{dW_{I}^{(1)}}{d\ln\Lambda}=-\frac{1}{2}\int_{\bf k}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\frac{\delta{W_{0}}}{\delta\phi_{\bf
k}}\frac{\delta{W_{I}}^{(1)}}{\delta\phi_{-\bf
k}}+\mathcal{O}(g^{2})\leavevmode\nobreak\ .$ (5.17)

This is the last term in (5.12). At this order, the first two terms in the
flow equation do not contribute, thus (5.12) is satisfied. Likewise, we can
check the $g^{2}$ order result using both the contact $\psi_{3}^{\Lambda}$ and
the single exchange $\psi_{4}^{\Lambda}$. Now by requiring that the four-point
function of long wavelength perturbations order $g^{2}$ is invariant with
respect to the change of the cutoff scale, we find

$\displaystyle\frac{d}{d\ln\Lambda}\langle\phi_{l}^{4}\rangle=0\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d{\rm Re}\psi_{4}^{\Lambda}}{d\ln\Lambda}$ $\displaystyle=$
$\displaystyle-\frac{d}{d\ln\Lambda}\bigg{(}\ln\left[\Omega_{\Lambda}(k_{1})\Omega_{\Lambda}(k_{2})\Omega_{\Lambda}(k_{3})\Omega_{\Lambda}(k_{4})\right]\bigg{)}{\rm
Re}\psi_{4}^{\Lambda}$ (5.18)
$\displaystyle-\frac{d\Omega_{\Lambda}(s)}{d\ln\Lambda}\frac{{\rm
Re}\psi_{3}^{\Lambda}\leavevmode\nobreak\ {\rm Re}\psi_{3}^{\Lambda}}{{\rm
Re}\psi_{2}(s)}+{\rm perms}.\leavevmode\nobreak\ $

where we have used the result in (5.16) for $\psi_{3}^{\Lambda}$’s cutoff
dependence. Thus at this order, we find

$\frac{dW_{I}^{(2)}}{d\ln\Lambda}=-\frac{1}{4}\int_{\bf k}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\left[\frac{\delta{W_{I}^{(1)}}}{\delta\phi_{\bf
k}}\frac{\delta{W_{I}^{(1)}}}{\delta\phi_{-\bf
k}}+2\frac{\delta{W_{0}}}{\delta\phi_{\bf
k}}\frac{\delta{W_{I}^{(2)}}}{\delta\phi_{-\bf
k}}\right]+\mathcal{O}(g^{3})\leavevmode\nobreak\ ,$ (5.19)

which is the last two terms in (5.12), while the first term there comes at
higher order in $g$. We can continue with this computation to check that the
higher-point tree-level wavefunction coefficients also satisfy this flow
equation, and thus the extra boundary term in (5.10) is justified.

However, the flow equation does not hold for loop-level wavefunction
coefficients. Let’s take a look at $\tilde{W}_{I}^{(2)}$ in (5.14) for
instance. Its $\Lambda$-dependence can be derived from the requirement that
$\langle\phi_{l}^{2}\rangle_{\rm 1-loop}$ is independent of the cutoff scale

$\displaystyle\frac{d}{d\ln\Lambda}\langle\phi_{l}^{2}\rangle=0\leavevmode\nobreak\
\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d}{d\ln\Lambda}{\rm Re}\psi_{2}^{\rm 1-loop}$ $\displaystyle=$
$\displaystyle-2\frac{d\ln\Omega_{\Lambda}}{d\ln\Lambda}{\rm Re}\psi_{2}^{\rm
1-loop}-\frac{1}{2}\int_{\bf
p}\frac{d\Omega_{\Lambda}(p)}{d\ln\Lambda}\frac{1}{{\rm
Re}\psi_{2}(p)}\left[{\rm Re}\psi_{4}^{\Lambda}\right.$ (5.20)
$\displaystyle\left.-\Omega_{\Lambda}\frac{{\rm Re}\psi_{3}^{\Lambda}{\rm
Re}\psi_{3}^{\Lambda}}{{\rm Re}\psi_{2}^{\Lambda}(|{\bf p}+{\bf k}_{1}|)}-{\rm
perm.}\right]$

which differs from the result of the flow equation at the $g^{2}$ order

$\frac{d\tilde{W}^{(2)}_{I}}{d\ln\Lambda}=-\frac{1}{4}\int_{\bf
k}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\left[\frac{\delta^{2}{W_{I}^{(2)}}}{\delta\phi_{\bf
k}\delta\phi_{-\bf k}}+2\frac{\delta{W_{0}}}{\delta\phi_{\bf
k}}\frac{\delta{\tilde{W}^{(2)}_{I}}}{\delta\phi_{-\bf
k}}\right]\leavevmode\nobreak\ ,$ (5.21)

Notice that the second term in (5.12) does not contribute because it contains
higher powers of $\phi$ at the $g^{2}$ order. This suggests that the modified
version of the Polchinski equation in (5.10) works for the semi-classical
wavefunction, but cannot capture the cutoff-dependence of loop-level
$\psi_{n}$. It remains interesting to see if there is a universal form of the
boundary term that works for loop-level wavefunction coefficients. In the end,
we notice that the above computation for $g\Phi^{3}$ can be easily extended to
other types of interactions, like $\lambda\Phi^{4}$.

Let’s summarize the logic of justifying the extra boundary term in (5.10).
Going back to the perturbation theory, we require that correlators of long
wavelength perturbations $\langle\phi_{l}^{n}\rangle$ should be independent of
the cutoff scale, which allows us to fix the $\Lambda$-dependence of
$\psi_{n}^{\Lambda}$ order by order. Then by explicitly comparing with (5.12),
we see that tree-level wavefunction coefficients satisfy the flow equation,
while loop-level ones do not. Thus in Polchinski’s equation (5.10), our choice
of the extra boundary term reflects the cutoff-dependence of the semi-
classical piece of the wavefunction. Accordingly we dub it the semi-classical
scheme.

Therefore, corrections to (5.10) are expected to be higher order in $\hbar$
from quantum loops. As we have seen, for theories with IR divergences in dS,
the quantum loop effects are always subdominant, and thus (5.10) gives us the
leading $\Lambda$-dependence of $W_{I}$. However, this may not be the case for
IR-finite theories where the quantum wavefunction can become equally important
at the loop order. It remains unclear if there is a flow equation that works
for loop-level wavefunction coefficients. Meanwhile, although the current
computation using the leading order perturbation theory is complete on its
own, it would be more satisfying to find a systematic derivation to take care
of all the corrections and understand better the physics of different schemes.
We leave these interesting open questions for future exploration.

#### Diffusion from RG flow

Now we are ready to derive the diffusion term in the Fokker-Planck equation.
Using the exact RG equation (5.10), we find the running of the probability
distribution with the cutoff scale

$\displaystyle\frac{d}{d\ln\Lambda}P_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{de^{W_{0}}}{d\ln\Lambda}e^{W_{I}}+\frac{de^{W_{I}}}{d\ln\Lambda}e^{W_{0}}=-\frac{1}{4}\int_{\bf
k}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\frac{\delta^{2}}{\delta\phi_{\bf
k}\delta\phi_{-\bf k}}P_{\Lambda}\leavevmode\nobreak\ .$ (5.22)

This already has the form of the diffusion equation, though it is in the
Fourier space and the functional derivatives on the right hand side are for
the Fourier mode $\phi_{\bf k}$. To make the connection with the stochastic
formalism more explicitly, we need to use the long wavelength perturbation in
coordinate space $\phi_{l}$ introduced in (4.1). Then the Fourier
transformation of the functional derivatives is given by

$\frac{\delta}{\delta\phi_{\bf k}}=\int d^{3}xe^{i{\bf k}\cdot{\bf
x}}\Omega_{\Lambda}\frac{\delta}{\delta\phi_{l}(\bf x)}\leavevmode\nobreak\ .$
(5.23)

By doing so, we rewrite (5.22) in the coordinate space

$\frac{d}{d\ln\Lambda}P_{\Lambda}=\int d^{3}x_{1}\int
d^{3}x_{2}\left(-\frac{1}{4}\int_{\bf k}e^{i{\bf k}\cdot({\bf x}_{1}-{\bf
x}_{2})}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}\right)\frac{\delta^{2}}{\delta\phi_{l}({\bf
x}_{1})\delta\phi_{l}({\bf x}_{2})}P_{\Lambda}\leavevmode\nobreak\ ,$ (5.24)

where $k$ is left in the bracket. By using $\mathrm{Re}\psi_{2}=-k^{3}/H^{2}$
and ${d\Omega_{\Lambda}}/{d\ln\Lambda}\simeq\Lambda\delta(k-\Lambda)$, we find
this momentum integration part becomes

$\displaystyle-\frac{1}{4}\int_{\bf k}e^{i{\bf k}\cdot({\bf x}_{1}-{\bf
x}_{2})}\frac{1}{{\rm
Re}\psi_{2}}\frac{d\Omega_{\Lambda}}{d\ln\Lambda}=\frac{H^{2}}{8\pi^{2}}j_{0}(\Lambda|{\bf
x}_{1}-{\bf x}_{2}|)\leavevmode\nobreak\ ,$ (5.25)

with $j_{0}(x)=\sin(x)/x$. Finally we restore the time-dependence of the
cutoff scale via $d\Lambda=\epsilon aH^{2}dt=\Lambda Hdt$. Then the second
term in (5.4) becomes

$\dot{\Lambda}\frac{\partial}{\partial\Lambda}P_{\Lambda}[\phi,t]=H\frac{dP_{\Lambda}}{d\ln\Lambda}=\int
d^{3}x_{1}\int d^{3}x_{2}\ \frac{H^{3}}{8\pi^{2}}j_{0}(\Lambda|{\bf
x}_{1}-{\bf x}_{2}|)\frac{\delta^{2}}{\delta\phi_{l}({\bf
x}_{1})\delta\phi_{l}({\bf x}_{2})}P_{\Lambda}\leavevmode\nobreak\ .$ (5.26)

Notice that $j_{0}(x)$ has a maximum at the origin and decays for $x>1$. Thus
only points with short separations $\Lambda|{\bf x}_{1}-{\bf x}_{2}|<1$
contribute to the spatial integrals on the right hand side. This concludes our
derivation of the diffusion term from the boundary RG flow. Interestingly,
with no input from the bulk, we find that the Polchinski equation for a
boundary quantum field theory at a fixed time $t$ is enough to deduce the
diffusion effects of long wavelength fluctuations. This derivation works for
quantum field theories in de Sitter beyond the perturbative regime, while the
only nontrivial assumption is that the semi-classical piece dominates the
wavefunction.

###  5.2 Fokker-Planck = Schrödinger + Polchinski

With the above derivation for the diffusion, now we look into the drift which
is supposed to be given by the first term in (5.4). The explicit time
derivative of the probability distribution is due to the bulk time flow of the
wavefunction. Using the Schrödinger equation, it is straightforward to show
that in the semi-classical regime we get the drift term in the Fokker-Planck
equation [12]. Here we recap the derivation using our coarse-grained
wavefunction.

We start with the Schrödinger equation at the late-time boundary with cosmic
time $t$

$\displaystyle i\frac{\partial}{\partial t}\Psi_{\Lambda}[\phi({\bf
x}),t]=\mathcal{H}\Psi_{\Lambda}[\phi({\bf x}),t]\leavevmode\nobreak\ .$
(5.27)

The form of the Hamiltonian $\mathcal{H}$ depends on the theory. In this
section, we shall just focus on the simplest single field case with
interactions in the potential $V(\Phi)$. The extension to the two-field
scenario with derivative interactions is presented in Appendix C. More
precisely, here we use

$\displaystyle\mathcal{H}=\int
d^{3}x\left[\frac{1}{2a(t)^{3}}\Pi_{\phi}^{2}+\frac{a(t)}{2}(\partial_{i}\phi)^{2}+\frac{a(t)^{3}}{2}V(\phi)\right]\leavevmode\nobreak\
,$ (5.28)

where the momentum operator is given as $\Pi_{\phi}({\bf
x})=-i\frac{\delta}{\delta\phi({\bf x})}$. Notice that everything is evaluated
at the boundary time $t$, and we have used $\Phi({\bf x},t)=\phi({\bf x})$.
Since the Hamiltonian is Hermitian, i.e. unitary, we can use the Schrödinger
equation to show that the probability distribution satisfies a continuity
equation

$\displaystyle\frac{\partial P_{\Lambda}[\phi,t]}{\partial t}$
$\displaystyle=-\frac{i}{2a^{3}}\int_{\bf
k}\left[\frac{\delta}{\delta\phi_{-\bf
k}}\left(\Psi_{\Lambda}^{*}\frac{\delta}{\delta\phi_{\bf
k}}\Psi_{\Lambda}-\Psi_{\Lambda}\frac{\delta}{\delta\phi_{\bf
k}}\Psi_{\Lambda}^{*}\right)\right]$ (5.29)

where we have performed the Fourier transformation on the right hand side.
Then to derive a closed equation for $P_{\Lambda}$, we notice that the
functional derivative of the wavefunction can be explicitly computed using the
saddle-point approximation (2.6)

$-i\frac{\delta}{\delta\phi_{\bf k}}\Psi_{\Lambda}\simeq\frac{\delta
S_{\Lambda}[\Phi_{\rm cl}]}{\delta\phi_{\bf k}}\Psi_{\Lambda}=\int
d^{3}xe^{i{\bf k}\cdot{\bf x}}\Omega_{\Lambda}(k)\frac{\delta
S_{\Lambda}[\Phi_{\rm cl}]}{\delta\phi_{l}({\bf x})}\Psi_{\Lambda}=-\int
d^{3}xe^{i{\bf k}\cdot{\bf x}}\Omega_{\Lambda}(k)\Pi_{l}({\bf
x},t)\Psi_{\Lambda}$ (5.30)

where we have used the Fourier transformation of functional derivatives in
(5.23) to introduce the derivative with respect to $\phi_{l}$ in coordinate
space. This simply shows that we can express $\delta\Psi/\delta\phi_{l}$ using
the corresponding field momentum at the boundary $\Pi_{l}\equiv
a^{3}\dot{\Phi}_{l}$, and then (5.29) becomes an equation for $P_{\Lambda}$
only. After performing the Fourier transformation for another functional
derivative, (5.29) becomes

$\displaystyle\frac{\partial P_{\Lambda}}{\partial t}$
$\displaystyle=-\frac{1}{a^{3}}\int d^{3}xd^{3}x^{\prime}\int_{\bf k}e^{i{\bf
k}\cdot({\bf x}-{\bf
x^{\prime}})}\Omega_{\Lambda}(k)\frac{\delta}{\delta\phi_{l}({\bf
x})}\Pi_{l}({\bf x},t)P_{\Lambda}=-\int
d^{3}x\frac{\delta}{\delta\phi_{l}({\bf x})}\left(\dot{\Phi}_{l}({\bf
x},t)P_{\Lambda}\right)\leavevmode\nobreak\ ,$ (5.31)

which is the drift term. Up to this point, again we have only used the saddle-
point approximation of the wavefunction. If we neglect the gradient terms, the
field momentum can be further expressed as $\dot{\Phi}_{l}\simeq
V^{\prime}(\phi_{l})/(3H)$ by using the approximated equation of motion for
perturbations on superhorizon scales. Then the drift term becomes the one that
we normally see in the stochastic formalism.

Now we can join two pieces of derivation together and describe the full time
evolution of the probability distribution. Then (5.4) becomes the Fokker-Plank
equation

$\displaystyle\frac{dP_{\Lambda}}{dt}=-\int
d^{3}x_{1}\frac{\delta}{\delta\phi(\mathbf{x}_{1})}\left(\dot{\Phi}_{l}\Lambda
P_{\Lambda}\right)+\int d^{3}x_{1}\int d^{3}x_{2}\
\frac{H^{3}}{8\pi^{2}}j_{0}(\Lambda|{\bf x}_{1}-{\bf
x}_{2}|)\frac{\delta^{2}}{\delta\phi_{l}({\bf x}_{1})\delta\phi_{l}({\bf
x}_{2})}P_{\Lambda}\leavevmode\nobreak\ .$ (5.32)

In summary, here the drift term corresponds to the bulk time flow given by the
Schrödinger equation, while the diffusion comes from the boundary RG flow
described by the Polchinski equation. This aligns with the equations derived
in previous works [11, 12], although we have arrived at this equation
exclusively through the consideration of boundary terms. It’s worth noting
that in interacting theories characterized by shift symmetries, the momentum
effectively becomes zero at the boundary, and the time evolution is
exclusively dictated by the diffusion term. In such cases, higher-order terms
in the action may become directly linked to higher-order derivatives in the
(RG) equation. If we are interested in the probability distribution of
$\phi_{l}$ at one spatial location $P_{\Lambda}[\phi_{l}({\bf x})]$, we get
$\delta$-functions from functional derivatives: ${\delta\phi_{l}({\bf
x})}/{\delta\phi_{l}({\bf x}_{1})}=\delta^{(3)}({\bf x}_{1}-{\bf x})$, which
remove the spatial integrals. In the end, the above equation simplifies to the
usual Fokker-Planck equation

$\displaystyle\frac{dP_{\Lambda}}{dt}=\frac{\partial}{\partial\phi_{l}}\left(\frac{V^{\prime}}{3H}P_{\Lambda}\right)+\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}P_{\Lambda}}{\partial\phi_{l}^{2}}$
(5.33)

where we have used that $\lim_{x\to 0}j_{0}(x)=1$.

We close the non-perturbative analysis with some concluding remarks. First of
all, the semi-classical wavefunction plays an important role in the above
derivation of the stochastic formalism. For the diffusion term, we used the
tree-level computation in perturbation theory to justify a modified version of
the Polchinski equation for the boundary field theory. For the drift term, we
simply applied the saddle-point approximation in the derivation. The final
equation (5.32) describes the evolution of the probability distribution that
comes from the semi-classical wavefunction, and thus can be seen as the
resummation of classical loops. This simplification works for theories with IR
divergences where the quantum corrections are always subdominant. However, we
may expect this semi-classical approximation breaks down for IR-finte
theories.

As we mentioned in the beginning of this section, our derivation of the
Fokker-Planck equation shares similarities with the approach in [12]. Now let
us clarify the differences. The starting point of Ref. [12] is the probability
distribution of the long modes (B.1), while for us it is the coarse-grained
wavefunction in (5.1). This choice does not lead to much difference in the
derivation of the drift term. But for the diffusion, (5.1) allows us to apply
the techniques of the Wilsonian effective field theory and establish the
connection with the exact RG flow on the boundary. This new perspective
provides one clear explanation for the physical origin of the diffusion
effects. In addition, Ref. [12] presented a quite systematic derivation, with
all the corrections carefully taken into account. Our approach is more based
on the insight from perturbation theory that the semi-classical wavefunction
and the Fokker-Planck equation are equivalent to each other. Therefore, we
used the saddle-point approximation as our starting point and focus on showing
the stochastic formalism as a consequence of that. It will be interesting to
understand how corrections to the Fokker-Planck equation arise from first
order corrections to the leading saddle-point.

Finally, we would like to briefly comment on possible connections with the
holographic RG in AdS/CFT. There it has been shown that the RG flow of the
boundary CFT is dual to the Hamilton-Jacobi equation in the bulk, and the
cutoff scale on the boundary is suspected to be related to the bulk radial
coordinate [87, 88, 89, 90]. Our computation suggests that something different
happened in dS. Here the “radial” direction becomes time, and its evolution is
governed by the Schrödinger equation, or equivalently the Hamilton-Jacobi
equation in the semi-classical regime. However, as we have shown, the bulk
time flow and the boundary RG flow lead to two distinct physical effects for
the probability distribution: the former is responsible for the drift, and the
later generates the diffusion. This distinction may have nontrivial
implications for the holographic description of de Sitter space, which may
deserve future exploration.

##  6 Summary and Outlook

Understanding the structure of the IR divergences in de Sitter not only has
conceptual interests in theoretical considerations but also can lead to new
phenomenological consequences of cosmic inflation. Previously, the problem was
investigated through two approaches: the conventional cosmological
perturbation theory, which demonstrate the IR-singular behaviour of tree-level
and loop-level correlators, and the stochastic formalism that describes an
equilibrium state in the non-perturbative regime. A link was expected between
these two approaches, and it has been argued for long that the stochastic
formalism is a resummation of IR divergences from higher order loops, though
it remained unclear how this resummation was achieved.

In this paper, we have established a precise connection between the semi-
classical piece of the wavefunction and the stochastic formalism, by
exploiting the wavefunction of the Universe and recent developments of the
cosmological bootstrap. Our analysis first made the explicit comparison
between two approaches in the perturbative regime, and then presented a non-
perturbative derivation of the Fokker-Planck equation with minimal
assumptions. More precisely, the main results are summarized as follows:

  * •

Within perturbation theory, we look into the loop-level cosmological
correlators with IR divergences using the wavefunction method and the
stochastic formalism.

    * –

Our starting point is to identify that in the perturbative computation of
wavefunction coefficients, while tree-level processes can be produced in semi-
classical approximations, loop diagrams there have a truly quantum origin. As
both of them contribute to the cosmological correlators, in general the loop-
level correlators are given by combinations of classical loops from tree-level
wavefunction coefficients, and quantum ones that cannot be generated in
classical theories.

    * –

The notion of classical loops becomes especially important for IR-divergences
in dS. We find loop-level correlators there are always dominated by the
classical loop contributions. This observation helps us to significantly
simplify the computation of leading IR-divergences in loop diagrams at any
order. Furthermore, it indicates that the leading IR behaviour of the system
is captured by the semi-classical wavefunction which contains all the tree-
level information.

    * –

Then we made the comparison with cosmological correlators from the stochastic
formalism. By solving the Fokker-Planck equation perturbatively, we identify
that at each order of coupling constants, the late-time divergent behaviour of
correlators matches the results from classical loops. Quantum loop
contributions are absent in the original stochastic computation, but can be
incorporated by considering the next-to-leading order corrections to the
Fokker-Planck equation. This simply shows that the stochastic formalism is
resumming classical loops from the perturbative wavefunction.

  * •

Beyond perturbation theory, we explore the direct consequence of the semi-
classical approximation. In particular, we apply the coarse-grained
wavefunction to study the time evolution of the probability distribution of
long wavelength perturbations. We show that the Fokker-Planck equation can be
understood as a combination of two distinct components: i) a drift term from
the bulk time evolution that is governed by the Schrödinger equation of the
semi-classical wavefunction; ii) a diffusion term that is associated with the
boundary RG flow and given by a revised version of the Polchinski equation. As
our analysis is based on the minimal assumption of the saddle-point
approximation, this non-perturbative derivation further demonstrates the
relation between the semi-classical wavefunction and the stochastic formalism.

While this work resolves a long-term puzzle about how the stochastic formalism
resums the IR loops in dS, it also initiated many future directions for
explorations. Here we list several obvious questions that would be interesting
for further considerations.

First, it will be interesting to explore the phenomenological consequences of
IR divergences in cosmological correlators. For theories of cosmic inflation,
it has been shown that the perturbative treatment of IR divergences leads to
local-type non-Gaussianities from multi-field models [27], Meanwhile, in past
several years, there has been a significant interest in extending the analysis
of inflationary fluctuations to the non-perturbative regime, which may
generate novel behaviour at the tail of the probability distribution [29, 28,
91, 92]. In particular, Ref. [93] showed that rare events on the tail can also
be captured by the semi-classical wavefunction in IR-finite theories. Possible
connections with our approach would be appealing, which might lead to a
systematical new understanding about non-Gaussian tails.

Second, the current analysis has been mainly focused on the semi-classical
wavefunction and the resulting classical loops, which capture the leading IR
divergences, while quantum corrections may become important at the next-to-
leading order, or in IR-finite theories. It will be interesting to compute
these corrections to the Fokker-Planck equation from our approach, and check
if a resummation of quantum loops is possible. Another possible correction
comes from gravitational interactions. The current analysis has been performed
in a fixed dS background, and ideally we also wish to consider a more
realistic setup with dynamical gravity.

Third, the renormalization group analysis for quantum field theories in de
Sitter is another new topic that needs to be better understood. In this work,
we have just focused on the semi-classical wavefunction and performed the
derivation for the Polchinski equation on the boundary, which can be seen as
the RG flow of an Euclidean classical field theory. In our current derivation,
there is an extra boundary term in addition to Polchinski’s original equation,
which we justify by returning to the leading order perturbation theory. It
would be more satisfying to perform a systematic analysis and understand
better the physics of this particular choice. Meanwhile, there could be other
derivations of the RG flow from a bulk perspective, and it remains curious to
see how the bulk version and the boundary one are related to each other. Last
but not least, it is tempting to explore possible connections with the
holographic RG in AdS/CFT. Excitingly, new insights along this direction may
shed light on the holographic description of de Sitter spacetime.

#### Acknowledgements

We acknowledge many inspiring discussions with Santiago Agui Salcedo, Carlos
Duaso Pueyo, Scott Melville, Enrico Pajer, Guilherme Pimentel, Arttu Rajantie
and Kostas Skenderis. SC is supported in by the STFC Consolidated Grants
ST/T000791/1 and ST/X000575/1. DGW is supported by a Rubicon Postdoctoral
Fellowship awarded by the Netherlands Organisation for Scientific Research
(NWO), and partially by the VIDI grant with Project No. 680-47-535 from NWO
and the STFC Consolidated Grants ST/T000694/1 and ST/X000664/1. ACD
acknowledges partial support from STFC Consolidated Grant ST/T000694/1.

##  Appendix A In-In Computation of One-Loop Corrections

In this section we apply the in-in formalism to compute the one-loop
corrections to the IR-divergent two-point correlators. The results here
confirm our computation in Section 3 using the wavefunction method. As we can
see, the classical and quantum contributions are mixed up in the in-in
computation. Although one can derive the leading IR divergences at one-loop
level, it is hard to identify their classical origin.

For the in-in computation of massless scalar correlators, we introduce the
bulk-to-boundary propagators as

$K_{+}(k,\eta)=\phi_{k}(\eta_{0})\phi^{*}_{k}(\eta),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ K_{-}(k,\eta)=\phi^{*}_{k}(\eta_{0})\phi_{k}(\eta),$
(A.1)

and the bulk-to-bulk propagators as

$\displaystyle G_{++}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\phi_{k}(\eta)\phi^{*}_{k}(\eta^{\prime})\theta(\eta-\eta^{\prime})+\phi^{*}_{k}(\eta)\phi_{k}(\eta^{\prime})\theta(\eta^{\prime}-\eta)$
$\displaystyle G_{+-}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\phi^{*}_{k}(\eta)\phi_{k}(\eta^{\prime})$ $\displaystyle
G_{-+}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\phi_{k}(\eta)\phi^{*}_{k}(\eta^{\prime})$ $\displaystyle
G_{--}(k,\eta,\eta^{\prime})$ $\displaystyle=$
$\displaystyle\phi_{k}(\eta)\phi_{k}^{*}(\eta^{\prime})\theta(\eta^{\prime}-\eta)+\phi^{*}_{k}(\eta)\phi_{k}(\eta^{\prime})\theta(\eta-\eta^{\prime})\leavevmode\nobreak\
,$ (A.2)

where the mode function of massless scalars is given by

$\displaystyle\phi_{k}(\eta)=\chi_{k}(\eta)=\frac{H}{\sqrt{2k^{3}}}(1+ik\eta)e^{-ik\eta}\leavevmode\nobreak\
.$ (A.3)

In the following we perform the in-in computation for the three types of
interactions that were considered in Section 3.

#### One-loop two-point function with one $\Phi^{4}$ vertex

We start with the one-loop correction to the two-point correlation function
$\langle\phi\phi\rangle$ from the quartic self-interaction
$\lambda{\Phi^{4}}/{4!}$. This is a well-studied example of the in-in
computation for IR divergent correlators.

$\displaystyle\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm 1-loop}$
$\displaystyle=$ $\displaystyle-\frac{i\lambda}{2}\int d\eta
a(\eta)^{4}\int_{\bf
p}\Big{[}K_{+}(k,\eta)K_{+}(k,\eta)G_{++}(p,\eta,\eta)-c.c.\Big{]}\leavevmode\nobreak\
,$

where we have a symmetry factor $2$ in the denominator of the overall
coefficient. Next, we put the momentum integrals outside and perform the time
integrals first and then the IR-divergent part of the correlator is given by

$\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm 1-loop}=\frac{\lambda
H^{2}}{12k^{3}}\int_{\bf
p}\frac{1}{p^{3}}\log(-2k\eta_{0})=\frac{H^{2}}{2k^{3}}\frac{\lambda}{12\pi^{2}}\log(\Lambda
L)\log(-2k\eta_{0})\leavevmode\nobreak\ .$ (A.4)

#### One-loop two-point function with two $\Phi^{3}$ vertices

Next, similarly let’s compute the one-loop correction to the
$\langle\phi\phi\rangle$ correlator from the cubic vertex $g\Phi^{3}/3!$

$\displaystyle\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm 1-loop}$
$\displaystyle=$ $\displaystyle-\frac{g^{2}}{2}\int d\eta
d\eta^{\prime}a(\eta)^{4}a(\eta^{\prime})^{4}\int_{\bf
p,q}\Big{[}K_{+}(k,\eta)K_{+}(k,\eta^{\prime})G_{++}(p,\eta,\eta^{\prime})G_{++}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-K_{+}(k,\eta)K_{-}(k,\eta^{\prime})G_{+-}(p,\eta,\eta^{\prime})G_{+-}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-K_{-}(k,\eta)K_{+}(k,\eta^{\prime})G_{-+}(p,\eta,\eta^{\prime})G_{-+}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+K_{-}(k,\eta)K_{-}(k,\eta^{\prime})G_{--}(p,\eta,\eta^{\prime})G_{--}(q,\eta,\eta^{\prime})\Big{]}(2\pi)^{3}\delta({\bf
p+q-k})\leavevmode\nobreak\ .$

Again, we put the momentum integrals outside and perform the time integrals
first

$\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm
1-loop}=-\frac{g^{2}}{2}\int_{\bf p,q}(2\pi)^{3}\delta({\bf
p+q-k})\mathcal{I}\leavevmode\nobreak\ .$ (A.6)

As we are mainly interested in the IR-divergent part, the secular growth piece
from bulk time integration is given by

$\lim_{\eta_{0}\rightarrow
0}\mathcal{I}=-\frac{1}{36p^{3}q^{3}}\Big{(}\log\left(-(k+p+q)\eta_{0}\right)\Big{)}^{2}+...$
(A.7)

Performing the loop integration, we find the IR-divergent correction to the
two-point function

$\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm
1-loop}=\frac{g^{2}}{2}\int_{\bf
p}\frac{H^{2}}{36p^{3}k^{3}}\Big{(}\log\left(-(2k+p)\eta_{0}\right)\Big{)}^{2}=\frac{H^{2}}{2k^{3}}\frac{g^{2}}{72\pi^{2}H^{2}}\log(kL)\log(-2k\eta_{0})^{2}\leavevmode\nobreak\
.$ (A.8)

#### One-loop two-point function with two $\dot{\Phi}\Sigma^{2}$ vertices

Now we compute the one-loop correction to $\langle\phi\phi\rangle$ from the
cubic vertex $\alpha\dot{\Phi}\Sigma^{2}/2$

$\displaystyle\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm 1-loop}$
$\displaystyle=$ $\displaystyle-\frac{\alpha^{2}}{2}\int d\eta
d\eta^{\prime}a(\eta)^{3}a(\eta^{\prime})^{3}\int_{\bf
p,q}\Big{[}\partial_{\eta}K_{+}(k,\eta)\partial_{\eta^{\prime}}K_{+}(k,\eta^{\prime})G_{++}(p,\eta,\eta^{\prime})G_{++}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\partial_{\eta}K_{+}(k,\eta)\partial_{\eta^{\prime}}K_{-}(k,\eta^{\prime})G_{+-}(p,\eta,\eta^{\prime})G_{+-}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\partial_{\eta}K_{-}(k,\eta)\partial_{\eta^{\prime}}K_{+}(k,\eta^{\prime})G_{-+}(p,\eta,\eta^{\prime})G_{-+}(q,\eta,\eta^{\prime})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\partial_{\eta}K_{-}(k,\eta)\partial_{\eta^{\prime}}K_{-}(k,\eta^{\prime})G_{--}(p,\eta,\eta^{\prime})G_{--}(q,\eta,\eta^{\prime})\Big{]}(2\pi)^{3}\delta({\bf
p+q-k})\leavevmode\nobreak\ .$

Next, we put the momentum integrals outside and perform the time integrals
first

$\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm
1-loop}=-\frac{\alpha^{2}}{2}\int_{\bf p,q}(2\pi)^{3}\delta({\bf
p+q-k})I\leavevmode\nobreak\ ,$ (A.10)

with

$\lim_{\eta_{0}\rightarrow
0}I=-\frac{H^{2}}{4p^{3}q^{3}}\Big{(}\log\left(-(k+p+q)\eta_{0}\right)\Big{)}^{2}...\leavevmode\nobreak\
.$ (A.11)

Then at one-loop level, the IR-divergent correction to the two-point
correlator becomes

$\langle\phi_{\bf k}\phi_{-\bf k}\rangle^{\prime}_{\rm
1-loop}=\frac{g^{2}}{2}\int_{\bf
p}\frac{H^{2}}{4p^{3}k^{3}}\Big{(}\log\left(-2k\eta_{0}\right)\Big{)}^{2}=\frac{H^{2}}{2k^{3}}\frac{\alpha^{2}}{8\pi^{2}}\log(kL)\log(-2k\eta_{0})^{2}\leavevmode\nobreak\
.$ (A.12)

While we can still perform the one-loop computation using in-in formalism
here, it becomes more complicated once we go beyond the lowest order for loop
corrections. Meanwhile, the IR-singular piece of the correlators in the end
look quite simple, with only two types of logarithmic functions: one from the
time integration; the other from the momentum integration. This indicates that
some simplification may be achieved if one is mainly interested in the leading
IR-divergent contributions. As we discussed in Section 3.4, the wavefunction
method provides a more intuitive understanding that identifies the origin of
dominant IR divergences as classical loops. Meanwhile, by looking at one
particular contribution from the product of contact wavefunction coefficients,
we can easily derive the leading logarithmic singularities.

##  Appendix B Probability Distribution from the Wavefunction

In this appendix, we compute the probability distribution of long wavelength
fluctuations from the coarse-grained wavefunction in perturbation theory. We
show that contributions from the saddle-point approximation satisfy the
Fokker-Planck equation, while the ones from loop-level wavefunction
coefficients are subdominant. This provides another justification for our
major conclusion that the stochastic formalism is resuming the classical loops
and thus is equivalently described by the semi-classical wavefunction.

This method, with a particular choice of $\Omega_{k}$ is similar to the path
integral used in [12] to derive the the Fokker-Planck. Our approach will be
different: instead of finding an equation for $P[\phi_{l}]$ we will do the
path integral show how the long wavelength physics appears and which diagrams
are considered. To be able to do this we will rely on knowing the perturbative
wavefunction $\Psi$, hence our results will be valid only while perturbation
theory is valid. Still these computations will be instructive when comparing
with the solutions of the Fokker-Planck equation.

We start by defining the probability for the long wavelength part of a scalar
fields as a functional integral

$\displaystyle P[\phi_{l}(\mathbf{x})]=\int\mathcal{D}\phi\
\delta\left(\phi_{l}(\mathbf{x})-\int_{\bf
k}\Omega_{k}e^{i\mathbf{k}\cdot\mathbf{x}}\phi_{\bf k}\right)|\Psi[\phi_{\bf
k}]|^{2}\leavevmode\nobreak\ .$ (B.1)

The idea is to use the already found wavefunction $\Psi[\phi_{\bf k}]$ and
then compute the path integral at leading order in a given coupling. The path
integral can be thought of as two consecutive Legendre transform. The first is
analogous to plugging a current with support only for the long modes. In the
end it computes a partition function composed only of connected correlators of
the long modes of the fields. The second Legendre transform rewrites the
partition function as a function of the long mode fields $\phi_{l}$ similarly
as the wavefunction.

#### Free Theory

To be more concrete let us start our discussion with an example. If the field
is free then we can write the probability as

$\displaystyle P[\phi_{k}]=\mathcal{N}\sqrt{\mathrm{Re\psi_{2}}({\bf
k})}\exp\left(-\frac{1}{2}\int_{\bf k}2\mathrm{Re}\psi_{2}({\bf k})\phi_{\bf
k}^{2}\right),$ (B.2)

where $\mathcal{N}$ is a real number such that the probability is normalized.
We want to now compute the probability of finding the long wavelength part of
the field $\phi_{l}$. As discussed previously this can be done through the
path integral,

$\displaystyle P[\phi_{l}({\bf x})]$
$\displaystyle=\mathcal{N}\sqrt{\mathrm{Re\psi_{2}(\mathbf{k})}}\int
D\phi_{\bf k}\ \delta\left(\phi_{l}(\mathbf{x})-\int_{\bf
k}e^{i\mathbf{k}\cdot\mathbf{x}}\Omega(k)\phi_{\bf
k}\right)\exp\left(-\frac{1}{2}\int_{\bf k}2\mathrm{Re}\psi_{2}({\bf
k})\phi_{\bf k}^{2}\right).$

Let us consider the case when we pick a point ${\bf x}$, by momentum
invariance the field $\phi_{l}$ has to have the same value on all space so it
can depends only on time $\phi_{l}(\mathbf{x},t)=\phi_{l}(t)$. Secondly, we
can see that the role of the $\delta$ function is to integrate over all modes
such that the average of the field $\phi_{\bf k}$ corresponds to a fixed value
$\phi_{l}$. The path integral can be done analytically by writing the $\delta$
function as an integral over a current $J$

$\displaystyle P(\phi_{l})$
$\displaystyle=\mathcal{N}\sqrt{\mathrm{Re\psi_{2}}({\bf k})}\int dJ\int
D\phi_{\bf k}\exp\left(+iJ\phi_{l}-\int_{\bf k}\left(\mathrm{Re}\psi_{2}({\bf
k})\phi_{\bf k}^{2}-i\Omega(k)J\phi_{\bf k}\right)\right)$ $\displaystyle=\int
dJ\exp\left(iJ\phi_{l}-\frac{1}{2}\int_{\bf
k}\frac{\Omega_{k}}{2\mathrm{Re\psi_{2}}({\bf k})}J^{2}\right)$
$\displaystyle=\sqrt{\frac{2\pi}{\int_{\bf
k}\frac{\Omega_{k}}{2\mathrm{Re\psi_{2}}({\bf
k})}}}\exp\left(-\frac{1}{2}\frac{\phi_{l}^{2}}{\int_{\bf
k}\frac{\Omega_{k}}{2\mathrm{Re\psi_{2}}({\bf k})}}\right).$ (B.4)

Now using that $1/2\mathrm{Re}\psi_{2}({\bf k})=\langle\phi_{\bf
k}^{2}\rangle^{\prime}$ we notice that the integral over the two point
function is nothing more than the variance of the field

$\displaystyle\sigma_{\phi}^{2}\equiv\int_{\bf
k}\frac{\Omega_{k}}{2\mathrm{Re\psi_{2}}({\bf k})}=\int_{\bf
k}\Theta(k-aH)\langle\phi_{k}^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log a,$ (B.5)

and so we can write that

$\displaystyle
P(\phi_{l})=\sqrt{\frac{2\pi}{\sigma_{\phi}^{2}}}\exp\left(-\frac{\phi_{l}^{2}}{2\sigma_{\phi}^{2}}\right),$
(B.6)

which is the usual result obtained by solving the Fokker-Planck equation for a
free field. It is possible to extend this results to higher order by using the
Born approximation.

For interacting theories, before doing this in detail let us understand what
is the role of the path integral. Given an action $S[\phi]$ for a interacting
scalar field $\phi$, let us call $S_{\mathrm{cl}}[\phi]$ the semi-classical
action, on-shell $S_{\mathrm{cl}}[\phi]$ is a polynomial in $\phi_{\bf k}$
that contains all possible tree-level interactions. Then we can write the
generator of connected correlators as [72]

$\displaystyle Z[J]=e^{W[J]}=\int D\phi e^{-i\int_{{\bf k}}\phi
J\Omega_{k}-2i\mathrm{Im}S_{\mathrm{cl}}[\phi_{k}]}.$ (B.7)

With this preparation, now let us take a look at some examples.

####  $\lambda\Phi^{4}/4!$ interaction

Let us generalise the free theory to the case with a quartic interaction. Up
to first order in $\lambda$ there are only two terms in the tree-level
wavefunction

$\displaystyle\Psi(\phi)\sim\exp\left[\frac{1}{2}\int_{\bf k}\psi_{2}({\bf
k})\phi_{\bf k}^{2}+\frac{1}{4!}\int_{{\bf k}_{1},{\bf k}_{2},{\bf k}_{3},{\bf
k}_{4}}\psi_{4}({\bf k}_{1},{\bf k}_{2},{\mathbf{k}}_{3},{\mathbf{k}}_{4})\
\phi_{{\bf k}_{1}}\phi_{{\bf k}_{2}}\phi_{{\bf k}_{3}}\phi_{{\bf
k}_{4}}\right],$ (B.8)

where $\psi_{2}$ and $\psi_{4}$ are derived in Section 3. The generating
functional for the connected correlators of long modes is

$\displaystyle e^{W[J]_{l}}=$ $\displaystyle\frac{1}{Z[0]}\int
D\phi\exp\left[-i\int_{\bf k}\Omega_{k}\phi_{\bf k}J-\frac{1}{2}\int_{\bf
k}2\mathrm{Re}\psi_{2}({\bf k})\phi_{\bf
k}^{2}-\frac{1}{12}\int_{\mathbf{k}_{1}\dots\mathbf{k}_{4}}2\mathrm{Re}\psi_{4}(\mathbf{k}_{1},\dots,\mathbf{k}_{4})\phi_{\mathbf{k}_{1}}\dots\phi_{\mathbf{k}_{4}}\right].$
(B.9)

We can remove the linear term in $\phi_{k}$ by shifting the field
$\phi_{k}\to\phi_{k}-i\frac{\Omega_{k}}{2\mathrm{Re\psi_{2}}}J$. Doing this
the measure does not change and we get,

$\displaystyle e^{W[J]_{l}}$
$\displaystyle=\frac{1}{Z[0]}\exp\left(-\frac{\sigma_{\phi}^{2}}{2}J^{2}-\frac{1}{16}\int_{\mathbf{k}_{1}\dots\mathbf{k}_{4}}\prod_{i=1}^{4}\frac{\Omega(k_{i})}{\mathrm{\mathop{\mathrm{Re}}}\psi_{2}({\bf
k}_{i})}\mathrm{Re}\psi_{4}(\mathbf{k}_{1},\dots,\mathbf{k}_{4})J^{4}\right)$
$\displaystyle\times\int D\phi_{\bf k}\exp\left(-\frac{1}{2}\int_{\bf
k}2\mathrm{Re}\psi_{2}({\bf k})\phi_{\bf
k}^{2}-\frac{1}{12}\int_{\mathbf{k}_{1}\dots\mathbf{k}_{4}}2\mathrm{Re}\psi_{4}(\mathbf{k}_{1},\dots,\mathbf{k}_{4})\varphi_{\mathbf{k}_{1}}\dots\varphi_{\mathbf{k}_{4}}\right.$
$\displaystyle\left.\hskip
62.59596pt+\frac{1}{4}\int_{\mathbf{k}_{1}\dots\mathbf{k}_{2}}\mathrm{Re}\psi_{4}(\mathbf{k}_{1},\dots,\mathbf{k}_{4})\frac{\Omega(k_{3})\Omega(k_{4})}{\mathrm{Re}\psi_{2}({\bf
k}_{3})\mathrm{Re}\psi_{2}({\bf
k}_{4})}\phi_{\mathbf{k}_{1}}\phi_{\mathbf{k}_{2}}J^{2}+5\
\mathrm{perm.}\right).$ (B.10)

Notice that the second line corresponds to $Z[0]$ so the path integral in the
second lines computes the correlation function of the term in the exponential.
Since the integral is still Gaussian in the fields it can be computed at first
order in $\lambda$ yielding,

$\displaystyle\left\langle
e^{\frac{1}{4}\int_{\mathbf{k}_{1}\dots\mathbf{k}_{4}}\frac{\Omega(k_{3})\Omega(k_{4})\mathrm{Re}\psi_{4}(\mathbf{k}_{1},\dots\mathbf{k}_{4})}{\mathrm{Re}\psi_{2}({\bf
k}_{3})\mathrm{Re}\psi_{2}({\bf
k}_{4})}\phi_{\mathbf{k}_{1}}\phi_{\mathbf{k}_{2}}J^{2}+5\
\mathrm{perm.}}\right\rangle=\left(1-\frac{J^{2}}{8}\int_{\mathbf{k},\mathbf{p},\mathbf{q}}\frac{\mathrm{Re}\psi_{2}(\mathbf{k})\mathrm{Re}\psi_{4}(\mathbf{k},\mathbf{p},\mathbf{q},-\mathbf{k})\Omega(p)\Omega(q)}{\mathrm{Re}\psi_{2}(\mathbf{p})\mathrm{Re}\psi_{2}(\mathbf{q})}\right)^{-1/2}.$
(B.11)

It is easy to see that is related to the loop correction we computed on the
first part. Indeed it appears as a loop correction to the determinant, which
is expected since we are truncating at first order in $\lambda$.

Using the notation from Eq. (4.2) we can write that

$\displaystyle\int_{\mathbf{k}_{1}\dots\mathbf{k}_{4}}\prod_{i=1}^{4}\frac{\Omega(k_{i})}{\mathrm{\mathop{\mathrm{Re}}}\psi_{2}(\mathbf{k}_{i})}\mathrm{Re}\psi_{4}({\bf
k}_{1},{\bf k}_{2},{\bf k}_{3},{\bf
k}_{4})=\langle\phi_{l}^{4}\rangle\leavevmode\nobreak\ .$ (B.12)

We then have that,

$\displaystyle P[\phi_{l}]=\int
dJ\frac{\exp\left(iJ\phi_{l}+\frac{\langle\phi_{l}^{2}\rangle_{0}}{2}J^{2}-\frac{\langle\phi_{l}^{4}\rangle_{0}}{16}J^{4}\right)}{\sqrt{1+J^{2}\langle\phi^{2}\rangle_{1-\mathrm{loop}}}}.$
(B.13)

and so we have checked explicitly that $\exp(W[J]_{l})$ is the generator of
connected correlator at long wavelengths. We can compute correlation functions
by taking functional derivatives of $\exp(W[J]_{l})$ and setting the current
to $0$ at the end of the computation. For the two-point function we get

$\displaystyle\langle\phi^{2}\rangle=\left.\frac{\delta^{2}W[J]}{\delta
J^{2}}\right|_{J=0}=\langle\phi_{l}^{2}\rangle_{0}+\langle\phi_{l}^{2}\rangle_{\mathrm{1-loop}}.$
(B.14)

#### Two-field model

The two-field interaction $\alpha\dot{\Phi}\Sigma^{2}/2$ can lead to further
simplifications in the computation of the probability distribution from the
perturbative wavefunction. Explicitly, we have

$\displaystyle P[\phi_{l},\chi_{l},t]$ $\displaystyle=\frac{1}{Z[0]}\int D\phi
D\chi\ \delta\left(\phi_{l}({\bf x})-\int_{{\bf k}}\Omega_{\Lambda}e^{i{\bf
k}\cdot{\bf x}}\phi_{\bf k}\right)\delta\left(\chi_{l}({\bf x})-\int_{\bf
k}\Omega_{\Lambda}e^{i{\bf k}\cdot{\bf x}}\chi_{\bf k}\right)P[\phi,\chi,t]$
$\displaystyle=\frac{1}{Z[0]}\int dJ_{\phi}dJ_{\chi}\int
D\phi_{k}D\chi_{k}\exp\left(iJ_{\phi}\phi_{l}+iJ_{\chi}\chi_{l}\right)$
$\displaystyle\times\exp\left[-\int_{\bf
k}\left(\mathrm{Re}\psi_{2}^{\phi}({\bf k})\phi_{\bf k}^{2}-ie^{i{\bf
k}\cdot{\bf x}}\Omega(k)J_{\phi}\phi_{\bf k}\right)-\int_{\bf
k}\left(\mathrm{Re}\psi_{2}^{\chi}({\bf k})\chi_{\bf k}^{2}-ie^{i{\bf
k}\cdot{\bf x}}\Omega(k)J_{\chi}\chi_{k}\right)\right.$
$\displaystyle\qquad\qquad\left.-\frac{1}{2}\int_{{\bf k}_{1},{\bf k}_{2},{\bf
k}_{3}}\mathrm{Re}\psi_{3}({\bf k}_{1},{\bf k}_{2},{\bf k}_{3})\phi_{{\bf
k}_{1}}\chi_{{\bf k}_{2}}\chi_{{\bf k}_{3}}\right]$
$\displaystyle=\frac{1}{Z[0]\sqrt{\mathrm{Re}\psi_{2}^{\phi}({\bf k})}}\int
dJ_{\phi}dJ_{\chi}\int
D\phi_{k}\exp\left[+iJ_{\phi}\phi_{l}+iJ_{\chi}\chi_{l}-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}+i\int_{{\bf
k},{\bf s}}\frac{\mathrm{Re}\psi_{3}({\bf k},-{\bf k},{\bf
s})}{2\mathrm{Re}\psi_{2}^{\phi}({\bf{s}})}\chi_{{\bf k}}^{2}J_{\phi}\right.$
$\displaystyle\left.\qquad+i\int_{\bf k}\mathrm{Re}\psi_{2}^{\chi}({\bf
k})\chi_{\bf k}^{2}+\frac{1}{4}\int_{{\bf k_{1}}\dots\bf
k_{4}}\frac{\mathrm{Re}\psi_{3}({\bf k}_{1},{\bf k}_{2},{\bf
s})\mathrm{Re}\psi_{2}(\mathbf{k}_{3},\mathbf{k}_{4},\mathbf{s})}{\mathrm{Re}\psi_{2}(\mathbf{s})}\chi_{{\bf
k}_{1}}\chi_{{\bf k}_{2}}\chi_{{\bf
k}_{3}}\chi_{\bf{k}_{4}}+\mathrm{perms}.\right]$ $\displaystyle=\int
dJ_{\phi}dJ_{\chi}\exp\left[-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}+\frac{\langle\chi^{2}\rangle}{1-i\frac{\langle\phi\chi^{2}\rangle}{\langle\phi^{2}\rangle}J\phi}\frac{J_{\chi}^{2}}{2}+\mathcal{O}(J_{\chi}^{4})\right]\leavevmode\nobreak\
,$ (B.15)

.To get the last line we have neglected the quartic term in $\chi$. We also
know the long wavelength limit of the three-point function

$\displaystyle\langle\phi\chi^{2}\rangle=-\int_{{\bf k}_{1},{\bf k}_{2},{\bf
k}_{3}}\frac{\mathrm{Re}\psi_{3}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})\Omega(k_{1})\Omega(k_{2})\Omega(k_{3})}{\mathrm{Re}\psi_{2}^{\phi}(\mathbf{k}_{1})\mathrm{Re}\mathbf{k}_{2})\mathrm{Re}\psi_{2}^{\chi}(\mathbf{k}_{3})}+\mathrm{perm}\leavevmode\nobreak\
.$ (B.16)

Notice that $W[J_{\phi},J_{\chi}]$ is non-local in $J_{\phi}$. This effect
comes from integrating out $J_{\chi}$ and it will be suppressed when $\chi$
has a large mass. Expanding in small $\alpha$ there will be a series of terms
quadratic in $J_{\chi}$ accounting for all possible exchange diagrams with two
external $\chi$ legs connecting internal $\chi$ propagators with external
$\phi$ legs. For the purpose of computing the probability we can expand in
$\alpha$ and solve the integrals over $J_{\chi}$ and $J_{\phi}$. We get

$\displaystyle P[\phi_{l},\chi_{l}]$ $\displaystyle=\int
dJ_{\phi}dJ_{\chi}\exp\left[iJ_{\phi}\phi_{l}+iJ_{\chi}\chi_{l}-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}+\frac{\langle\chi^{2}\rangle}{2}J_{\chi}^{2}-i\frac{\langle\phi\chi^{2}\rangle}{2}J_{\phi}J_{\chi}^{2}\right]$
$\displaystyle=\int
dJ_{\phi}\frac{\exp\left[iJ_{\phi}\phi_{l}-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}-\frac{1}{1+i\frac{\langle\phi\chi^{2}\rangle}{\langle\chi^{2}\rangle}J_{\phi}}\frac{\chi_{l}^{2}}{2\langle\chi^{2}\rangle}\right]}{\sqrt{\langle\chi^{2}\rangle+i\frac{\langle\phi\chi^{2}\rangle}{\langle\phi^{2}\rangle}J_{\phi}}}$
$\displaystyle\approx\sqrt{\frac{2\pi}{\langle\chi^{2}\rangle\langle\phi^{2}\rangle}}\exp\left[-\frac{\chi_{l}^{2}}{2\langle\chi^{2}\rangle}-\frac{1}{2\langle\phi^{2}\rangle}\left(\phi_{l}-\frac{\langle\phi\chi^{2}\rangle}{2\langle\chi^{2}\rangle^{2}}\chi_{l}^{2}\right)^{2}\right]\leavevmode\nobreak\
,$ (B.17)

where, in the second line, we have expanded in $\alpha$. This is the
probability distribution which was obtained in [28] as a solution of the
Fokker-Planck equation. The two-point functions $\langle\phi^{2}\rangle$ and
$\langle\chi^{2}\rangle$ are given by the variance in (4.4). For the three-
point function, we have

$\displaystyle\langle\phi\chi^{2}\rangle$ $\displaystyle=-\lambda
H^{3}\int_{{\bf k}_{1},{\bf k}_{2},{\bf
k}_{3}}\frac{\log(-(k_{1}+k_{2}+k_{3})\eta_{0})}{k_{1}^{3}k_{2}^{3}k_{3}^{3}}\Omega(k_{1})\Omega(k_{2})\Omega(k_{3})(2\pi)^{3}\delta^{(3)}({\bf
k}_{1}+{\bf k}_{2}+{\bf k}_{3})$ $\displaystyle=-\lambda H^{3}\int_{{\bf
k}_{1},{\bf
k}_{2}}\frac{\log(-2(k_{1}+k_{2})\eta_{0})}{k_{1}^{3}k_{2}^{3}(k_{1}+k_{2})^{3}}\Omega(k_{1})\Omega(k_{2})\leavevmode\nobreak\
.$ $\displaystyle=-\frac{\lambda
H^{3}}{2(4\pi^{2})^{2}}\log(a)^{3}\leavevmode\nobreak\ ,$ (B.18)

where we have used the delta function in the first line to eliminate the
$k_{3}$ integral. We can compute correlation functions by taking derivatives
from $W_{l}[J_{\phi},J_{\chi}]$ and setting the currents to zero. Let us check
the 1-loop corrections. First, integrating over $J_{\chi}$, we obtain

$\displaystyle
W_{l}[J_{\phi}]=-\frac{\langle\phi^{2}\rangle}{2}J_{\phi}^{2}-\frac{1}{2}\log\left(\frac{1}{\langle\chi^{2}\rangle}+\frac{i\langle\phi\chi^{2}\rangle}{\langle\chi^{2}\rangle}J_{\phi}\right)\leavevmode\nobreak\
.$ (B.19)

Then the 1-loop correction to the two-point function is given by

$\displaystyle\langle\phi^{2}\rangle_{1-\mathrm{loop}}$
$\displaystyle=\langle\phi^{2}\rangle-\frac{\langle\phi^{2}\chi\rangle^{2}}{2\langle\chi^{2}\rangle^{2}}=\frac{H^{2}}{4\pi^{2}}\log
a+\frac{\lambda^{2}H^{2}}{128\pi^{4}}\log(a)^{4}\leavevmode\nobreak\ ,$ (B.20)

which matches the perturbative computation we did in Section 3.

##  Appendix C Stochastic Formalism with Two Fields

In this Appendix, we present another version of the derivation for the Fokker-
Planck equation, especially focusing on the situation with two interacting
massless scalars.

To proceed let us first study the long wavelength dynamics. One of the
advantages of using wavefunctions is that it allow us to define a probability
through the relation $P[\phi,\chi;t]\sim|\Psi|^{2}$. The dynamics of this
probability distribution is given by the Schrodinger equation,

$\displaystyle\frac{d}{dt}\Psi=iH\Psi\leavevmode\nobreak\ ,$ (C.1)

which is strictly valid in the decoupling limit of the EFT (otherwise the
Hamiltonian is a constraint). Of course the equation for the probability
distribution is just the usual continuity equation from quantum mechanics. A
difference between our derivation and the analysis of Ref. [12] is that we
will include derivative interactions. Their main effect will on modifying the
classical dynamics. For this reason let us keep discussing the two field
scenario and let us consider the following interaction Lagrangian,

$\displaystyle S_{\mathrm{int}}=\int
d^{4}x\sqrt{-g}\left(V(\chi)+\dot{\phi}f(\chi)\right)\leavevmode\nobreak\ ,$
(C.2)

where $V(\phi)$ and $f(\chi)$ are polynomial in $\chi$. The effect of the
derivative interactions is twofold. Besides the usual interaction it modifies
the momenta which is now $a^{-3}\pi_{\phi}=\dot{\phi}-\frac{1}{2}f(\chi)$
which in turns implies an extra term in the interaction Hamiltonian
$f(\chi)^{2}/8$. The Hamiltonian is given by,

$\displaystyle
H=\frac{\pi_{\phi}}{2a^{3}}+\frac{\pi_{\chi}}{2a^{3}}+\frac{a^{2}}{2}((\nabla\chi^{2})-m^{2}a^{2}\chi^{2})+\frac{a^{2}}{2}(\nabla\phi)^{2}-\frac{1}{2}\pi_{\phi}f(\chi)+\frac{a^{3}}{8}(f(\chi)^{2}-4V(\chi))\leavevmode\nobreak\
.$ (C.3)

Notice that by Noether theorem the shift symmetry of $\phi$ implies a
conserved current in the continuity equation,

$\displaystyle
j^{0}=\dot{\phi}-\frac{1}{2}f(\chi)=\frac{\pi_{\phi}}{a^{3}}\leavevmode\nobreak\
,$ (C.4)

which is just a generalisation of the usual momentum conservation in models
with a single field. All this equation is saying is that in the long
wavelength limit $\dot{\phi}$ is not conserved, as in single field inflation,
but the more general combination given by $j_{0}$.

Assuming unitarity we can write a continuity equation for the probability
density by taking the time derivative of $P$ and using the Schrodinger
equation. Using this we can write the continuity equation as,

$\displaystyle\frac{\partial P[\phi,\chi,t]}{\partial t}$
$\displaystyle=[H,P]$ $\displaystyle=-\int
d^{3}x\left[\frac{i}{2a^{3}}\frac{\delta}{\delta\phi({\bf
x})}\left(\Psi^{*}\frac{\delta}{\delta\phi({\bf
x})}\Psi-\Psi\frac{\delta}{\delta\phi({\bf
x})}\Psi^{*}\right)+\frac{i}{2a^{3}}\frac{\delta}{\delta\chi({\bf
x})}\left(\Psi^{*}\frac{\delta}{\delta\chi({\bf
x})}\Psi-\Psi\frac{\delta}{\delta\phi({\bf x})}\Psi^{*}\right)\right.$
$\displaystyle\hskip 56.9055pt\left.-f(\chi)\frac{\delta}{\delta\chi({\bf
x})}\Psi\Psi_{*}\right]$ $\displaystyle=-\int
d^{3}x\frac{\delta}{\delta\phi({\bf
x})}\left(\left(\Pi_{\phi}+f(\chi)\right)P[\phi,\chi,t]\right)+\frac{\delta}{\delta\chi({\bf
x})}(\Pi_{\chi}P[\phi,\chi,t])\leavevmode\nobreak\ ,$ (C.5)

where we have used that $\pi_{\phi}\to-i\frac{\delta}{\delta\phi}$. If we
apply this definition over the wavefunction coefficients we can define a
classical momentum as,

$\displaystyle\Pi_{\phi^{i}}[\phi,{\bf
x},t]=-\frac{\delta}{\delta\phi^{i}({\bf x})}\mathrm{Im}\log\psi=\frac{\delta
S(\zeta,t)}{\delta\phi^{i}({\bf x})}\leavevmode\nobreak\ ,$ (C.6)

where by $S$ we mean that it is evaluated over the classical part of the
wavefunction. In this sense this definition can be thought of as the on-shell
long wavelength limit of the momentum. By using this we can write the
continuity equation as,

$\displaystyle\frac{\partial P[\phi,\chi,t]}{\partial t}$ $\displaystyle=-\int
d^{3}x\frac{\delta}{\delta\phi({\bf
x})}\left(\left(\frac{\Pi_{\phi}}{a^{3}}+f(\chi)\right)P[\phi,\chi,t]\right)+\frac{\delta}{\delta\chi({\bf
x})}\left(\frac{\Pi_{\chi}}{a^{3}}P[\phi,\chi,t]\right)\leavevmode\nobreak\ .$
(C.7)

The advantage of writing the equation in this form relies on the fact that we
can make use of the long wavelength limit of the equations of motion. In our
case, the shift symmetry of $\phi$ implies that the classical momentum
vanishes at late times. For the second field we can use the equations of
motion to get that $\Pi_{\chi}\to-(V^{\prime}(\chi)-f^{\prime}(\chi)/8)/3H$.
In order to incorporate the effect of short wavelengths we first coarse grain
the fields into long and short wavelength modes. This is achieved by writing,

$\displaystyle\phi({\bf
x})=\int_{0}^{\Lambda}\frac{d^{3}k}{(2\pi)^{3}}e^{i{\bf k}\cdot{\bf
x}}\phi({{\bf x}})+\int_{\Lambda(t)}^{\infty}\frac{d^{3}k}{(2\pi)^{3}}e^{i{\bf
k}\cdot{\bf x}}\phi({{\bf x}})\equiv\phi_{L}({\bf k})+\phi_{S}({\bf
k})\leavevmode\nobreak\ ,$ (C.8)

where $\Lambda(t)$ is a time dependent cut-off given by $\Lambda=\epsilon
a(t)H$ with $\epsilon$ a small number (not to be confused with the slow roll
parameter). In order to circumvent problems with the fact that the cut-off is
sharp let us define the smoothed window function,

$\displaystyle\Omega_{\Lambda}(k)=\begin{cases}1\qquad\mathrm{for}\
k\leq\Lambda(t)\\\ 0\qquad\mathrm{for}\
k\geq(1+\delta)\Lambda(t)\leavevmode\nobreak\ .\end{cases}$ (C.9)

such that in the limit $\delta\to 0$, the window function becomes a Heavyside
step-function, $\Omega_{\Lambda}=\Theta(k-\Lambda(t))$. An example of such
function is given in [81]. With the help of the window function (C.9) we can
define the smeared field as,

$\displaystyle\phi_{\Omega}({\bf
x})=\int_{\mathbf{k}}\Omega_{\Lambda}(k)e^{i{\bf k}\cdot{\bf x}}\phi({\bf
k})\leavevmode\nobreak\ ,$ (C.10)

and similarly for $\chi$. Notice that at leading order in $\delta$,
$\phi_{\Omega}$ is just the long wavelength part of the curvature modes. In
other words, expanding in $\delta$ we find that,
$\phi_{\Omega}=\zeta_{l}(1+\mathcal{O}(\delta))$. At the same time we can
define the PDF for the long wavelength modes as a path integral using the
window function,

$\displaystyle P_{\Omega}[\phi_{\Omega},\chi_{\Omega},t]$
$\displaystyle=\int\mathcal{D}\phi\mathcal{D}\chi\
\delta\left(\phi_{\Omega}({\bf
x})-\int\frac{d^{3}k}{(2\pi)^{3}}\Omega_{\Lambda}e^{i{\bf k}\cdot{\bf
x}}\phi({\bf k})\right)$
$\displaystyle\qquad\times\delta\left(\chi_{\Omega}({\bf
x})-\int\frac{d^{3}k}{(2\pi)^{3}}\Omega_{\Lambda}e^{i{\bf k}\cdot{\bf
x}}\chi({\bf k})\right)P[\phi,\chi,t]\leavevmode\nobreak\ .$ (C.11)

This expression generalises the notion of the long wavelength limit
probability we have discussed before. Notice that because the window function
also has a time dependence there will be another term when taking the
derivative of $P_{\Omega}$. One will be given by the classical solution
whereas the new one will take into account the short wavelength fluctuations.
Schematically we can write,

$\displaystyle\frac{\partial P_{\Omega}}{\partial t}=\mathrm{Drift}\
+\mathrm{Diff.}$ (C.12)

In analogy with the usual Fokker-Planck equation. Let us unpack this. For the
drift term we have already found an expression for the derivative of $P$ in
(C.5). Replacing it we find,

$\displaystyle\mathrm{Drift}$
$\displaystyle=\int\mathcal{D}\zeta\mathcal{D}\chi\ \delta[\dots]\left(-\int
d^{3}x\frac{\partial}{\partial\phi({\bf x})}\left(\Pi_{\phi}({\bf
x})-f(\chi)\right)P[\phi,\chi,t]\right.$
$\displaystyle\qquad\qquad\qquad\left.-\int
d^{3}x\frac{\partial}{\partial\chi({\bf x})}(\Pi_{\chi}({\bf
x})P[\phi,\chi,t]\right)\leavevmode\nobreak\ .$ (C.13)

Integrating by parts we find at leading order in $\delta$ that,

$\displaystyle\mathrm{Drift}$ $\displaystyle=\int
d^{3}x\int_{0}^{\Lambda(t)}\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\times\left(\frac{\partial}{\partial\phi({\bf
x})}\int\mathcal{D}\phi\mathcal{D}\chi\delta[\dots]\left(\Pi_{\phi}({\bf
k})+f(\chi)\right)P[\phi,\chi,t]\right.$
$\displaystyle\qquad\qquad\left.+\frac{\partial}{\partial\chi({\bf
x})}\int\mathcal{D}\phi\mathcal{D}\chi\delta[\dots]\Pi_{\chi}({\bf
k})P[\phi,\chi,t]\right)(1+\delta)\leavevmode\nobreak\ ,$ (C.14)

where $\Pi_{\phi,\chi}({\bf k})$ are the Fourier transform of the momenta
(C.6). After swapping the momentum integral inside the functional derivative
we see that we are basically taking the Fourier transform of the momenta until
$\Lambda(t)$. At leading order this, of course, corresponds to considering
only the long wavelength modes. Moreover the path integral considers only the
long wavelength probability. Hence we can write,

$\displaystyle\mathrm{Drift}=\int
d^{3}x\left(\frac{\partial}{\partial\phi({\bf x})}\left(\left(\Pi_{\phi}({\bf
x})+f(\chi)\right)_{\Lambda(t)}P_{\Omega}(t)\right)+\frac{\partial}{\partial\chi({\bf
x})}(\Pi_{\chi}({\bf x})_{\Lambda(t)}P_{\Omega}(t))\right)\leavevmode\nobreak\
,$ (C.15)

where the subscript in the momentum indicates that we are considering the long
wavelength part field.

The computation for the diffusion follows the same steps as in [12] which we
review in the appendix. After a straightforward computation we arrive at the
following,

$\displaystyle\mathrm{Diff}=\frac{1}{2}\frac{\partial^{2}}{\partial\phi^{2}}\left(\langle\dot{\Delta}\phi\Delta\phi\rangle
P\right)+\frac{1}{2}\frac{\partial^{2}}{\partial\chi^{2}}\left(\langle\dot{\Delta}\chi\Delta\chi\rangle
P\right)+\dots\leavevmode\nobreak\ ,$ (C.16)

where the $\Delta\phi$ is defined as,

$\displaystyle\Delta\phi({\bf
k})=\int_{\Lambda(t)}^{(1+\delta)\Lambda(t))}\frac{d^{3}k}{(2\pi)^{3}}e^{i{\bf
k}{\bf x}}\Omega_{\Lambda}(k)\phi({\bf k})$ (C.17)

such that we integrate over a thin layer of width $\delta$. The cross term
$\frac{\partial^{2}}{\partial\phi\partial\chi}\left(\langle\dot{\Delta}\phi\Delta\chi\rangle
P\right)$ gives a subdominant contribution. There will be higher order
correlations contributing to the diffusion terms. Notice that the time
derivative over $\Delta\phi$ only acts over the window function
$\Omega_{\Lambda}$. Our task then reduces to compute a regulated Fourier
transform of a two point function. Using the results from the previous section
we find

$\displaystyle\langle\dot{\Delta}\zeta\Delta\zeta\rangle$
$\displaystyle=\frac{H^{3}}{4\pi^{2}}$
$\displaystyle\langle\dot{\Delta}\chi\Delta\chi\rangle$
$\displaystyle=\frac{H^{3}}{4\pi^{2}}\leavevmode\nobreak\ ,$ (C.18)

Finally, the quadratic Fokker-Planck equation is,

$\displaystyle\frac{\partial P}{\partial t}$
$\displaystyle=\frac{\partial}{\partial\phi}\left(\left(f(\chi)\right)P\right)+\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}P}{\partial\phi^{2}}-\frac{\partial}{\partial\chi}\left(\frac{m^{2}}{3H}\chi
P\right)+\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\chi^{2}}P\leavevmode\nobreak\
.$ (C.19)

Notice that this equation generalises the results found in [28], where the
Fokker-Planck equation was obtained from a set of Langevin equations. In this
case with a minimal set of assumptions we have arrived to the same equation.

##  Appendix D Corrections to the Fokker-Planck equation

In this approach we show how to compute some corrections to the Fokker-Planck
equation (C.19).

### Diffusion terms

The effect from a quartic interaction can generate a mass term on long
distances $\lambda\phi_{l}^{2}$. To compute how this changes the diffusion
term let us consider the wavefunction coefficient of a massive scalar field

$\displaystyle\psi_{2}(k)=\frac{H}{(-H\eta)^{d}}\left(-\frac{d-2\nu}{2}+\frac{k\eta
H_{\nu-1}(-k\eta)}{H_{\nu}(-k\eta)}\right)\leavevmode\nobreak\ ,$ (D.1)

where $\nu=\sqrt{d^{2}/4-m^{2}/H^{2}}$. If we expand for small $m^{2}$ and
take the real part we get that,

$\displaystyle 2\mathrm{Re}\psi_{2}(k)=\frac{2k^{3}}{H^{2}}\left(1-(\log
k-\psi(3/2))\frac{2m^{2}}{3H^{2}}\right)\leavevmode\nobreak\ ,$ (D.2)

where $\psi(x)$ is the digamma function. As we mentioned, the mass term is
generated by the long modes and so we identify $m^{2}=\lambda\phi_{l}^{2}$ If
we plug this result back into the formula for the diffusion term Eq. (5.24) we
find the coefficient gets shifted to,

$\displaystyle\mathrm{Diff}=\frac{H^{3}}{8\pi^{2}}\frac{\partial^{2}}{\partial\phi^{2}}\left(\left(1+\frac{2\lambda}{H^{2}}(\log\Lambda-\psi(3/2))\phi^{2}\right)P(\phi)\right)\leavevmode\nobreak\
,$ (D.3)

which coincides with other results [12, 15, 18]. This new terms adds a
subleading correction to the correlation function computed using the Fokker-
Planck equation. Indeed we have that the recursion relations are now,

$\displaystyle\frac{d\langle\phi^{n}\rangle}{dt}=-\frac{n}{3H}\langle\phi^{n-1}V_{\phi}\rangle+\frac{n(n-1)H^{2}}{8\pi^{2}}\left(\langle\phi^{n-2}\rangle+\frac{2\lambda}{H^{2}}a\langle\phi^{n}\rangle\right)\leavevmode\nobreak\
,$ (D.4)

where $a=(\log\epsilon-\psi(3/2))$. If we solve for the variance up to first
order in $\lambda$ we get

$\displaystyle\langle\phi^{2}\rangle=\frac{H^{2}}{4\pi^{2}}\log
a-\frac{\lambda H^{2}}{144\pi^{4}}(\log a)^{3}+\frac{a\lambda
H^{2}}{32\pi^{4}}(\log a)^{2}\leavevmode\nobreak\ .$ (D.5)

Notice that the third term is a subleading loop contribution. Since it
contains less IR divergences we can trace it back to the 1-loop correction to
the wave function coefficient,

$\displaystyle\mathrm{Re}\psi_{2}^{\mathrm{1-loop}}(k)\sim\frac{\lambda
k^{3}}{H^{2}}\log(-k\eta)\leavevmode\nobreak\ ,$ (D.6)

which leads to a similar contribution to the two point function as the third
term of Eq.(D.5). Notice that the precise coefficient will depend on the
regularisation scheme. It will be interesting to explore this further.

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