Starobinsky's Slow-Roll Fokker–Planck Equation

• Starobinsky’s slow-roll Fokker–Planck equation is a drift-diffusion PDE that models the probability evolution of the coarse-grained inflaton in stochastic inflation.
• The equation combines a deterministic slow-roll force from the inflaton potential with a diffusion term from horizon-crossing quantum fluctuations, with modifications for multi-field and accelerating FLRW backgrounds.
• Numerical methods such as spectral and Crank–Nicolson schemes reveal stationary solutions, boundary effects, and the impact of environmental noise on the inflaton’s dynamics.

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 (Li, 2 Jul 2025, Prokopec, 2015, Hong et al., 2023).

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 (Hong et al., 2023).

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 (Prokopec, 2015, Li, 2 Jul 2025).

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 (Haba, 2018).

The time variable is not unique. One formulation uses cosmic time $t$, while another uses e-fold time

$$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$0

The constant-$$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$1 FLRW analysis rewrites the stochastic dynamics in terms of $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$2, whereas the two-field numerical study works in cosmic time $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$3 and states that although $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$4 is often preferred in the literature, the two are treated as interchangeable for the numerical purposes at issue there (Prokopec, 2015, Hong et al., 2023, Haba, 2018).

The stochastic-calculus convention also matters because the noise is multiplicative. In the two-field generalization, a parameter $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$5 encodes the convention: $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$6 corresponds to Stratonovich and $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$7 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 (Hong et al., 2023, Haba, 2018).

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 $$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$8, the normalized probability satisfies

$$\dot{\phi}=-\frac{V_{,\phi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\phi(t),\qquad \dot{\chi}=-\frac{V_{,\chi}}{3H}+\frac{H^{3/2}}{2\pi}\Gamma_\chi(t),$$9

Introducing

$$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$0

and

$$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$1

the PDE can be rewritten as

$$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$2

with coefficients built from $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$3, $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$4, and $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$5. The paper’s key message is that the distribution is controlled by this combined slow-roll structure, especially $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$6, rather than by the bare potential alone (Hong et al., 2023).

The same study introduces the volume-weighted probability $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$7 through $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$8, leading to an additional $$\langle \Gamma_\phi(t)\Gamma_\phi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\chi(t)\Gamma_\chi(t')\rangle=\delta(t-t'),\quad \langle \Gamma_\phi(t)\Gamma_\chi(t')\rangle=0.$$9 term in the evolution equation and a modified coefficient $p(t,\varphi)$0. 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(t,\varphi)$1 tends to align with local maxima of $p(t,\varphi)$2, while the volume-weighted $p(t,\varphi)$3 peaks where $p(t,\varphi)$4 is largest, which in that regime coincides with the maximum of the potential. In the quantum-dominated regime, both $p(t,\varphi)$5 and $p(t,\varphi)$6 match the relevant $p(t,\varphi)$7-profile rather than the potential minimum alone (Hong et al., 2023).

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,

$p(t,\varphi)$8

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 $p(t,\varphi)$9 propagate directly into the probability distribution, so $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$0 and $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$1 inherit the blocky or raised structure of the piecewise potential (Hong et al., 2023).

4. Beyond exact de Sitter: constant-$$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$2 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

$$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$3

In this setting, $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$4, the stochastic equation is naturally written in e-fold time $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$5, and the Fokker–Planck equation becomes

$$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$6

Relative to exact de Sitter, both the friction and diffusion sectors are modified by factors involving $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$7 (Prokopec, 2015).

The decisive step is the rescaling

$$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$8

after which the Fokker–Planck equation acquires an extra transport term $$\frac{\partial p}{\partial t} = \frac{1}{3H}\frac{\partial}{\partial\varphi}\!\left[p\,V'(\varphi)\right] + \frac{H^3}{8\pi^2}\frac{\partial^2 p}{\partial\varphi^2},$$9. A time-independent late-time PDF exists only if the rescaled potential

$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$0

is itself time-independent. This occurs for scale-invariant potentials, in practice the quartic interaction $$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$1, possibly with nonminimal coupling. Quadratic or cubic terms break this scaling and spoil the exact late-time solution in this form (Prokopec, 2015).

At late times one obtains

$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$2

with

$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$3

For the quartic theory this becomes

$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$4

The extra quadratic term is the order-$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$5 correction to the Starobinsky–Yokoyama equilibrium distribution and is written as an induced mass-squared shift $$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$6. The resulting late-time PDF can then be used to compute coincident $$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$7-point functions by ordinary statistical averaging (Prokopec, 2015).

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

$$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$8

with $$\partial_t P=-\partial_\phi\!\big(A(\phi)P\big)+\frac12\partial_\phi^2\!\big(B(\phi)P\big), \qquad A(\phi)\sim-\frac{V'(\phi)}{3H},\quad B(\phi)\sim\frac{H^3}{4\pi^2}.$$9 and $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$0 independent Gaussian white noises. The corresponding Fokker–Planck equation contains two diffusion sectors, one from the environment and one from quantum noise:

$$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$1

with the drift term supplied by the slow-roll force $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$2. The stationary asymptotics are governed by the behavior of $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$3: if $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$4, the classic Starobinsky form is recovered, whereas if $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$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 (Haba, 2018).

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:

$$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$6

At leading order in slow roll and in exact de Sitter, this reproduces Starobinsky’s equation with

$$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$7

The full phase-space evolution is more general: it is written for the Wigner function $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$8, retains mixed diffusion $$P(\phi)\propto H^{-2}(\phi)\exp\!\left(\frac{3}{8G^2\,V(\phi)}\right),$$9 and momentum diffusion $P\sim (V+V_0)^{-1}\exp(\cdots)$0, and is explicitly Lindbladian. In global de Sitter, the associated Fokker–Planck equation has no equilibrium solution until the late-time regime $P\sim (V+V_0)^{-1}\exp(\cdots)$1 (Li, 2 Jul 2025).

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 (Haba, 2018, Li, 2 Jul 2025).

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 $P\sim (V+V_0)^{-1}\exp(\cdots)$2, whose distribution obeys

$P\sim (V+V_0)^{-1}\exp(\cdots)$3

with Gaussian solution

$P\sim (V+V_0)^{-1}\exp(\cdots)$4

In another version, the stochastic variable is the conformal zero mode $P\sim (V+V_0)^{-1}\exp(\cdots)$5, with

$P\sim (V+V_0)^{-1}\exp(\cdots)$6

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 $P\sim (V+V_0)^{-1}\exp(\cdots)$7 growth of two-point functions to slow-roll scaling (Kitazawa, 2024, Kitazawa, 2023).

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 (Kitazawa, 2024, Kitazawa, 2023).

At a more formal level, linear drift–diffusion versions of the Fokker–Planck equation admit a rich symmetry structure. The equation

$P\sim (V+V_0)^{-1}\exp(\cdots)$8

has constant diffusion, linear drift, a conserved form

$P\sim (V+V_0)^{-1}\exp(\cdots)$9

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 (Kamano et al., 2015).