---
title: Starobinsky's Slow-Roll Fokker–Planck Equation
url: https://www.emergentmind.com/topics/starobinsky-s-slow-roll-fokker-planck-equation
type: topic
---

# Starobinsky's Slow-Roll Fokker–Planck Equation

Starobinsky’s slow-roll Fokker–Planck equation is the probability-density evolution equation associated with stochastic inflation, in which the long-wavelength inflaton is treated as a drift-plus-noise process: classical slow roll supplies a deterministic force down the potential, while horizon-crossing quantum fluctuations generate diffusion. In the one-field, leading slow-roll limit, the equation takes the drift–diffusion form
$$
\partial_t P(\phi,t)=\frac{1}{3H}\,\partial_\phi\!\big(V'(\phi)P\big)+\frac{H^3}{8\pi^2}\,\partial_\phi^2 P,
$$
and later work has generalized, corrected, or reinterpreted this structure in multi-field models, accelerating FLRW backgrounds, explicit open-system derivations, and boundary random-walk dualities [2507.02070, 1508.07874, 2310.00375].

## 1. Stochastic origin and canonical form

The equation arises from the stochastic-inflation split between sub-horizon and long-wavelength modes. In this framework, the coarse-grained inflaton obeys a Langevin equation in which the classical slow-roll force competes with a white-noise source generated by horizon crossing. The direct multi-field analogue studied in a recent numerical investigation is
$$
\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad
\dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),
$$
with independent noises satisfying
$$
\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad
\langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad
\langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.
$$
This is the same physical structure identified with Starobinsky’s stochastic description: classical drift competes with quantum diffusion [2310.00375].

For exact de Sitter as a reference case, the corresponding probability density \(p(t,\varphi)\) satisfies
$$
\frac{\partial p}{\partial t}
=
\frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right]
+
\frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},
$$
while an equivalent compact notation writes
$$
\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big),
\qquad
A(\phi)\sim-\frac{V'(\phi)}{3H},\quad
B(\phi)\sim\frac{H^3}{4\pi^2}.
$$
In this form, the first term is the slow-roll drift and the second is the diffusion induced by quantum kicks [1508.07874, 2507.02070].

## 2. Stationary distributions, time variables, and stochastic conventions

A central use of the slow-roll Fokker–Planck equation is the study of stationary or late-time probability distributions. In the Starobinsky–Vilenkin case with no environmental noise, one stationary solution is
$$
P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),
$$
with equivalent variants depending on the stochastic interpretation: Stratonovich gives \(P\sim (V+V_0)^{-1}\exp(\cdots)\), Ito gives \(P\sim (V+V_0)^{-2}\exp(\cdots)\), and the e-fold-time formulation gives another power prefactor. The same work explicitly notes that this stationary distribution may be non-integrable for many potentials [1802.00841].

The time variable is not unique. One formulation uses cosmic time \(t\), while another uses e-fold time
$$
v=\int H\,ds=\ln\!\left(\frac{a}{a_0}\right).
$$
The constant-\(\epsilon\) FLRW analysis rewrites the stochastic dynamics in terms of \(N=\ln a\), whereas the two-field numerical study works in cosmic time \(t\) and states that although \(N\) is often preferred in the literature, the two are treated as interchangeable for the numerical purposes at issue there [1508.07874, 2310.00375, 1802.00841].

The stochastic-calculus convention also matters because the noise is multiplicative. In the two-field generalization, a parameter \(\beta\) encodes the convention: \(\beta=\tfrac12\) corresponds to Stratonovich and \(\beta=1\) to Ito. The main analysis adopts the Stratonovich version and checks that the Ito version gives qualitatively similar outcomes. In the environmental-noise treatment, the stochastic wave equation is likewise interpreted in the Stratonovich sense because the chain rule is preserved [2310.00375, 1802.00841].

## 3. Slow-roll parameters, multi-field structure, and volume weighting

A recurrent misconception is that the normalized stochastic distribution simply tracks the potential. The two-dimensional generalization makes the dependence more precise. For two fields \((\phi,\chi)\), the normalized probability satisfies
$$
\frac{\partial P_{\rm FP}}{\partial t}
=
\frac{\partial}{\partial \phi}\!\left[ \frac{V_{,\phi}}{3H}P_{\rm FP}
+\frac{H^{3(1-\beta)}}{8\pi^2}\frac{\partial}{\partial\phi}\!\left(H^{3\beta}P_{\rm FP}\right) \right]
+
\frac{\partial}{\partial \chi}\!\left[ \frac{V_{,\chi}}{3H}P_{\rm FP}
+\frac{H^{3(1-\beta)}}{8\pi^2}\frac{\partial}{\partial\chi}\!\left(H^{3\beta}P_{\rm FP}\right) \right].
$$
Introducing
$$
\epsilon_{\phi\phi}=\frac{M_{\rm Pl}^2}{2}\left(\frac{V_{,\phi}}{V}\right)^2,\qquad
\epsilon_{\chi\chi}=\frac{M_{\rm Pl}^2}{2}\left(\frac{V_{,\chi}}{V}\right)^2,
$$
and
$$
\eta_{\phi\phi}=M_{\rm Pl}^2\frac{V_{,\phi\phi}}{V},\qquad
\eta_{\chi\chi}=M_{\rm Pl}^2\frac{V_{,\chi\chi}}{V},
$$
the PDE can be rewritten as
$$
P_{{\rm FP},t}=c_{1_{\rm FP}}P_{\rm FP}+c_2P_{{\rm FP},\phi}+c_3P_{{\rm FP},\chi}+c_4\left(P_{{\rm FP},\phi\phi}+P_{{\rm FP},\chi\chi}\right),
$$
with coefficients built from \(V\), \(\epsilon\), and \(\eta\). The paper’s key message is that the distribution is controlled by this combined slow-roll structure, especially \(c_1\), rather than by the bare potential alone [2310.00375].

The same study introduces the volume-weighted probability \(P_{\rm VW}\) through \(P_{\rm FP}\propto e^{-3Ht}P_{\rm VW}\), leading to an additional \(3HP_{\rm VW}\) term in the evolution equation and a modified coefficient \(c_{1_{\rm VW}}\). That extra term biases the distribution toward regions of larger Hubble rate and therefore typically larger potential when classical drift dominates. In the classical regime, the normalized \(P_{\rm FP}\) tends to align with local maxima of \(c_{1_{\rm FP}}\), while the volume-weighted \(P_{\rm VW}\) peaks where \(c_{1_{\rm VW}}\) is largest, which in that regime coincides with the maximum of the potential. In the quantum-dominated regime, both \(P_{\rm FP}\) and \(P_{\rm VW}\) match the relevant \(c_1\)-profile rather than the potential minimum alone [2310.00375].

The numerical analysis uses a spectral method for spatial derivatives, a Crank–Nicolson method for time evolution, and periodic boundary conditions. These choices conserve integrated probability,
$$
\int \frac{\partial P_{\rm FP}}{\partial t}\,d\phi\,d\chi=0,
$$
and allow both smooth periodic potentials and piecewise-defined non-smooth potentials to be studied without boundary artifacts. In the non-smooth case, discontinuities in \(c_1\) propagate directly into the probability distribution, so \(P_{\rm FP}\) and \(P_{\rm VW}\) inherit the blocky or raised structure of the piecewise potential [2310.00375].

## 4. Beyond exact de Sitter: constant-\(\epsilon\) accelerating FLRW

The slow-roll Fokker–Planck framework extends beyond exact de Sitter to spatially homogeneous accelerating FLRW backgrounds with constant principal slow-roll parameter
$$
\epsilon\equiv-\frac{\dot H}{H^2},\qquad 0\le \epsilon<1.
$$
In this setting, \(H\propto a^{-\epsilon}\), the stochastic equation is naturally written in e-fold time \(N=\ln a\), and the Fokker–Planck equation becomes
$$
\frac{\partial}{\partial t}p(N,\varphi)
=
\frac{1}{(3-\epsilon)H^2}\,\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right]
+
\frac{H^2}{(3-\epsilon)^2}\,A(\epsilon,\nu,\mu)\,\frac{\partial^2 p}{\partial\varphi^2}.
$$
Relative to exact de Sitter, both the friction and diffusion sectors are modified by factors involving \(3-\epsilon\) [1508.07874].

The decisive step is the rescaling
$$
\tilde\varphi\equiv \frac{\varphi}{H},
$$
after which the Fokker–Planck equation acquires an extra transport term \(\epsilon\,\partial_{\tilde\varphi}(\tilde\varphi\,p)\). A time-independent late-time PDF exists only if the rescaled potential
$$
\tilde V(\tilde\varphi)\equiv \frac{V(H\tilde\varphi)}{H^4}
$$
is itself time-independent. This occurs for scale-invariant potentials, in practice the quartic interaction \(V(\varphi)=\lambda\varphi^4/4!\), possibly with nonminimal coupling. Quadratic or cubic terms break this scaling and spoil the exact late-time solution in this form [1508.07874].

At late times one obtains
$$
p_\infty(\tilde\varphi)=p_0\,\exp\!\left[-v_{\rm eff}(\tilde\varphi)\right],
$$
with
$$
v_{\rm eff}(\tilde\varphi)=\frac{3-\epsilon}{A}\,\tilde V(\tilde\varphi)+\frac{(3-\epsilon)\epsilon}{2A}\,\tilde\varphi^2.
$$
For the quartic theory this becomes
$$
p_\infty(\tilde\varphi)
=
p_0\exp\!\left[
-\frac{3-\epsilon}{A}\frac{\lambda}{4!}\tilde\varphi^4
-\frac{(3-\epsilon)\epsilon}{2A}\tilde\varphi^2
\right].
$$
The extra quadratic term is the order-\(\epsilon\) correction to the Starobinsky–Yokoyama equilibrium distribution and is written as an induced mass-squared shift \(\Delta m^2=-(3-\epsilon)\epsilon H^2\). The resulting late-time PDF can then be used to compute coincident \(n\)-point functions by ordinary statistical averaging [1508.07874].

## 5. Environmental noise, stabilization, and open-system derivations

One extension of the slow-roll equation adds an environmental noise to the usual Starobinsky–Vilenkin quantum noise. In that construction the inflaton interacts with an infinite number of fields treated as an environment, and after a Markovian approximation the noise term becomes
$$
n=\gamma a^{-3/2}\dot B + H^{5/2}\dot W,
$$
with \(B\) and \(W\) independent Gaussian white noises. The corresponding Fokker–Planck equation contains two diffusion sectors, one from the environment and one from quantum noise:
$$
\dot P
=
\frac{\gamma^2}{2}a^{-3}H^{-2}\,\partial_\phi^2 P
+\frac12 H^3\,\partial_\phi^2 P
+\text{drift terms},
$$
with the drift term supplied by the slow-roll force \((3H)^{-1}V'(\phi)\partial_\phi P\). The stationary asymptotics are governed by the behavior of \(a(\phi)^3H(\phi)^5\): if \(a^3H^5\to\infty\), the classic Starobinsky form is recovered, whereas if \(a^3H^5\) stays finite or tends to zero, the environmental noise can restore integrability and remove the need for boundary conditions introduced solely to eliminate infinite inflation [1802.00841].

A more microscopic derivation reinterprets stochastic inflation as an open quantum system. In that approach, short-wavelength modes are traced out with the Schwinger–Keldysh formalism, the reduced density matrix obeys a Lindblad-type master equation, and the diagonal part of the reduced state becomes a Fokker–Planck equation after a slow-roll saddle eliminates the momentum variable:
$$
\partial_t P(\phi,t)
=
-\partial_\phi\!\Big(a^{-3}\Pi_c\,P(\phi,t)\Big)
+\partial_\phi^2\!\Big(C_1(\phi)\,P(\phi,t)\Big).
$$
At leading order in slow roll and in exact de Sitter, this reproduces Starobinsky’s equation with
$$
C_1=\frac{H^3}{8\pi^2}.
$$
The full phase-space evolution is more general: it is written for the Wigner function \(W(\phi,\Pi,t)\), retains mixed diffusion \(C_2\) and momentum diffusion \(C_3\), and is explicitly Lindbladian. In global de Sitter, the associated Fokker–Planck equation has no equilibrium solution until the late-time regime \(aH\gg 1\) [2507.02070].

These developments sharpen two points. First, normalizability of the stationary distribution is not automatic. Second, the standard field-space Fokker–Planck equation is a reduced description: a more complete phase-space or open-system equation exists even when the diagonal, slow-roll limit reproduces the familiar Starobinsky form [1802.00841, 2507.02070].

## 6. Boundary random walks and analytic reductions

A distinct line of work reinterprets the Fokker–Planck equation as a boundary random-walk problem. In one version, the relevant stochastic variable is the curvature perturbation \(\zeta\), whose distribution obeys
$$
\frac{\partial}{\partial N}\rho_N(\zeta)=2g\,\frac{\partial^2}{\partial \zeta^2}\rho_N(\zeta),
$$
with Gaussian solution
$$
\rho_N(\zeta)=\frac{1}{\sqrt{8\pi g(t)N(t)}}\,
\exp\!\left[-\frac{\zeta^2}{8g(t)N(t)}\right],
\qquad
\langle \zeta^2\rangle=4gN.
$$
In another version, the stochastic variable is the conformal zero mode \(w\), with
$$
\partial_N P(w,N)=\hat g(N)\,\partial_w^2 P(w,N).
$$
Both approaches emphasize a pure diffusion equation with drift neglected in the low-energy approximation, interpret the Brownian motion of a boundary degree of freedom as dual to bulk slow roll, and connect the \(O(N)\) growth of two-point functions to slow-roll scaling [2406.00886, 2303.07676].

These boundary formulations do not reproduce the field-space drift term of the standard slow-roll equation. This suggests that they function primarily as reinterpretations of stochastic inflation—through de Sitter duality, entropy arguments, or random-walk universality—rather than as direct replacements of the original drift–diffusion equation [2406.00886, 2303.07676].

At a more formal level, linear drift–diffusion versions of the Fokker–Planck equation admit a rich symmetry structure. The equation
$$
u_t=-a_2u-(a_2x+a_1)u_x+u_{xx}
$$
has constant diffusion, linear drift, a conserved form
$$
u_t + D_x\!\left(-(a_2x+a_1)u+u_x\right)=0,
$$
an auxiliary potential system, a six-dimensional finite symmetry algebra, and additional potential symmetries. The analysis explicitly states that this is the Ornstein–Uhlenbeck or linear slow-roll type after suitable normalization and identification of coefficients. In the context of stochastic inflation, that result identifies the linearized Starobinsky-type Fokker–Planck operator as a symmetry-rich special case amenable to similarity reduction and exact solution generation [1503.02209].

Source: https://www.emergentmind.com/topics/starobinsky-s-slow-roll-fokker-planck-equation