Stationary Distributions and First-Passage Observables

A stationary density and a first-passage distribution answer different questions. The first describes probability after relaxation under a specified current condition; the second describes when a trajectory first reaches a declared boundary. Both depend on the field domain and boundary conditions, and neither by itself defines a volume-weighted measure for eternal inflation.

Required background. Langevin and Fokker–Planck dynamics fixes the stochastic calculus and probability current; stochastic coarse-graining fixes the long field and noise normalization; and Fokker–Planck evolution supplies forward and backward generators. Helpful background. Open-system noise and dissipation explains when the Markov reduction is licensed; detailed balance and fluctuation–dissipation clarifies zero-current equilibrium; and Hawking–Moss and stochastic crossover supplies the gravitational comparison.

Probability current and stationary measures

Consider one light spectator field in fixed de Sitter space, with negligible gravitational backreaction, a coarse-graining prescription that gives additive white noise, and cosmic time $t$. At leading order its Itô equation is

$$d\phi=-\frac{V'(\phi)}{3H}\,dt +\frac{H^{3/2}}{2\pi}\,dW_t, \qquad \mathbb E[dW_t^2]=dt.$$

Write $\mu=1/(3H)$ for the mobility, $D_\phi=H^3/(8\pi^2)$ for the diffusion coefficient, and $\Theta=D_\phi/\mu=3H^4/(8\pi^2)$ for the effective stochastic scale. $\Theta$ has the dimensions of a potential-energy density; it is not the de Sitter temperature $H/(2\pi)$. The probability density satisfies $\partial_tP=-\partial_\phi J$ with

$$J(\phi,t) =-\frac{V'}{3H}P -\frac{H^3}{8\pi^2}\partial_\phi P.$$

Setting $J=0$ gives

$$P_{\rm eq}(\phi)=Z^{-1}e^{-V(\phi)/\Theta}, \qquad Z=\int_D e^{-V(\phi)/\Theta}\,d\phi.$$

On a finite interval with reflecting endpoints and a smooth potential, $Z$ is finite and this is the unique normalized stationary density. On the line, normalizability and the boundary behavior must be checked separately. On a circle, a single-valued periodic potential with this constant diffusion admits the periodic zero-current density. A nonzero stationary current instead requires a different steady-flow problem, such as a driven system. These statements concern comoving histories; physical-volume reproduction changes the evolution equation.

Existence of a stationary density does not say that a finite-duration state has reached it. On the reflecting finite interval, deviations decay in eigenmodes of the Fokker–Planck operator; the smallest positive decay eigenvalue $\Lambda_1$ gives the longest relaxation time $\Lambda_1^{-1}$. A high barrier makes interwell equilibration much slower than relaxation inside either well. For the fixed-background scalar problem, see Starobinsky and Yokoyama 1994, §III, Eqs. (12), (14)–(21).

First-passage boundary problem

Now change the experiment: reflect at $a$ and stop at $b>a$. Define $\tau_b=\inf\{t\geq0:\phi_t=b\}$ for the reflected process started at $\phi_0\in[a,b)$. The survival probability and hitting-time density are

$$S(t\mid\phi_0)=\Pr_{\phi_0}(\tau_b>t), \qquad f_\tau(t\mid\phi_0)=-\partial_tS,$$

The backward generator is $\mathcal L_{\rm b}=-\mu V'\partial_\phi+D_\phi\partial_\phi^2$. Thus $\partial_tS=\mathcal L_{\rm b}S$, with $S(0\mid\phi)=1$ in the interior, $S(t\mid b)=0$, and $\partial_\phi S(t\mid a)=0$. Integrating over time gives $T(\phi)=\mathbb E_\phi[\tau_b]=\int_0^\infty S(t\mid\phi)\,dt$ and

$$D_\phi T''-\mu V'T'=-1, \qquad T'(a)=0,\quad T(b)=0.$$

The reflecting condition prevents probability loss at $a$; the absorbing condition stops the clock at $b$. Two absorbing endpoints define a different exit problem. In particular, a normalized equilibrium density on a reflecting domain is not a stationary density for a killed process: survival mass decreases through its absorber.

Multiplying the backward equation by $e^{-V/\Theta}$ makes it an exact derivative,

$$\frac{d}{d\phi}\left(e^{-V/\Theta}T'\right) =-D_\phi^{-1}e^{-V/\Theta}.$$

Integrate from $a$ to $y$, use $T'(a)=0$, then integrate from $\phi$ to $b$ using $T(b)=0$. This yields the exact one-dimensional answer

$$T(\phi;b)=\frac1{D_\phi} \int_\phi^b dy\,e^{V(y)/\Theta} \int_a^y dz\,e^{-V(z)/\Theta}.$$

Both integration limits matter. This derivation also gives $T'(\phi)<0$ in the interior and $T\geq0$, useful sign checks. The same overdamped first-passage integral and its dependence on the exit surface appear in Hofmann and Ivanyuk 2003, Eq. (2) and Fig. 3, with their friction and temperature replacing $\mu^{-1}$ and $\Theta$.

First application: escape from a bounded spectator well

Take a confining double-well potential and bound the sampled field domain explicitly:

$$V(\phi)=\frac{\lambda}{4}(\phi^2-v^2)^2, \qquad \lambda>0,$$

For equilibrium use $[-2v,2v]$ with both endpoints reflecting. For first passage start at $-v$, reflect at $-2v$, and absorb either at the saddle $b=0$ or at $b=v/2$. The latter stops a trajectory on the descending side, before the right minimum. The potential itself is bounded below and grows quartically on the line; “bounded” here specifies the field domain.

Let $k_{\rm w}=V''(-v)=2\lambda v^2$, $k_{\rm s}=\lvert V''(0)\rvert=\lambda v^2$, and $\Delta V=\lambda v^4/4$. For $\Delta V/\Theta\gg1$, the inner integral in $T$ samples a full Gaussian around the left minimum. With an absorber exactly at the saddle, the outer integral samples only the left half of its Gaussian. Consequently,

$$\begin{aligned} T(-v;0) &\simeq \frac{\pi}{\mu\sqrt{k_{\rm w}k_{\rm s}}} e^{\Delta V/\Theta},\\ \Gamma_{\rm hit}\equiv T(-v;0)^{-1} &\simeq \frac{\sqrt{k_{\rm w}k_{\rm s}}}{3\pi H} e^{-\Delta V/\Theta}. \end{aligned}$$

If the absorber lies many saddle widths $\sqrt{\Theta/k_{\rm s}}$ beyond the top, while still preceding the next well, the outer Gaussian is complete. The leading time doubles and its inverse becomes the usual overdamped crossing estimate

$$T_{\rm K}=\frac{2\pi}{\mu\sqrt{k_{\rm w}k_{\rm s}}} e^{\Delta V/\Theta}, \qquad \Gamma_{\rm cross}\simeq \frac{\sqrt{k_{\rm w}k_{\rm s}}}{6\pi H} e^{-\Delta V/\Theta}.$$

Within the locally quadratic saddle region the interpolation is

$$T(-v;b)\simeq\frac{T_{\rm K}}2 \left[1+\operatorname{erf}\!\left( b\sqrt{\frac{k_{\rm s}}{2\Theta}}\right)\right].$$

This boundary effect changes the leading prefactor. A trajectory that first touches the saddle can return to the original well; reaching a surface beyond the saddle includes such repeated attempts. The difference is therefore not merely downhill travel time. The half-time at the saddle is also explicitly identified in Hofmann and Ivanyuk 2003, discussion of Fig. 3. Interpreting $1/T$ as a constant decay rate further requires a metastable regime with intrawell relaxation much faster than escape; in general the hazard $-\partial_t\log S$ depends on time.

Numerical comparison

Rescale the field and time to

$$\begin{gathered} x=\phi/v,\qquad s=\frac{\lambda v^2}{3H}t,\qquad U(x)=\frac{(x^2-1)^2}{4},\\ dX_s=-U'(X_s)\,ds+\sqrt{2D}\,dW_s,\qquad D=\frac{3H^4}{8\pi^2\lambda v^4},\qquad B=\frac{\Delta U}{D}=\frac1{4D}. \end{gathered}$$

All times below are measured in $s$; multiply them by $3H/(\lambda v^2)$ for cosmic time. The stationary density on $[-2,2]$ is $P_x=e^{-U/D}/Z_x$ and $P_\phi=P_x/v$. Direct quadrature gives

Barrier-to-noise ratio $B$	$Z_x$	$\langle x^2\rangle_{\rm eq}$
2	1.410915	0.852136
4	0.947838	0.917671

The mean is zero by reflection symmetry; normalization on a finite reflecting interval is exact after division by $Z_x$. Halving the integration spacing from $0.001$ to $0.0005$ changes the second moments by less than $2\times10^{-13}$.

For first passage, two independent numerical constructions can now be compared. The continuum mean comes from the nested integral above. The trajectory construction approximates the diffusion by a continuous-time nearest-neighbor jump process on $x_j=-2+jh$ with rates

$$q_{j,\pm}=\frac{D}{h^2} \exp\!\left[-\frac{U(x_j\pm h)-U(x_j)}{2D}\right].$$

At the reflecting node set $q_{0,-}=0$ and double $q_{0,+}$; the absorbing node has no outgoing transitions. The interior generator tends to $-U'\partial_x+D\partial_x^2$ with error $O(h^2)$ on smooth functions. Adjacent interior rates obey detailed balance with $e^{-U/D}$; the reflecting endpoint has half the interior quadrature weight. This fixes the boundary discretization independently of the trajectory data.

At each visited node, draw independent uniforms $u_1,u_2$ in $(0,1)$. Advance time by $-\log u_1/(q_{j,+}+q_{j,-})$ and jump right if $u_2<q_{j,+}/(q_{j,+}+q_{j,-})$, otherwise left. The exponential waiting law and rate-weighted channel selection are the direct stochastic simulation algorithm of Gillespie 1977, §IIIC, Eqs. (21a)–(21b). This simulation is exact for the chosen grid generator; approximation to a continuous field still requires spatial refinement.

An independent deterministic check solves that grid generator’s backward equation. If $d_j=T_j-T_{j-1}$, then

$$d_1=-\frac1{q_{0,+}},\qquad d_{j+1}=\frac{q_{j,-}d_j-1}{q_{j,+}},\qquad T_j=-\sum_{k=j+1}^{n}d_k,$$

where $x_n=b$. Starting at $x=-1$, 6,000 completed trajectories for each case at $h=0.05$ give:

$B$	Absorber $b$	Continuum mean	Exact grid mean	Trajectory mean ± approximate 95% sampling interval half-width
2	0	18.4262	18.4007	18.3730 ± 0.4493
2	0.5	35.0270	34.9739	34.6119 ± 0.8496
4	0	135.7830	135.2711	136.3560 ± 3.4340
4	0.5	263.6999	262.6920	262.6902 ± 6.5188

The reported half-width is $1.96$ times the sample standard error; it is an approximate normal interval, not a rigorous finite-sample bound. No trajectory was discarded or censored. Every sample mean is within $0.84$ standard errors of its independently solved grid mean. Sampling error and grid bias are different: across these four cases the maximum relative bias falls from $1.52\%$ at $h=0.1$ to $0.383\%$ at $h=0.05$ and $0.096\%$ at $h=0.025$. Moving the continuum reflector from $-2$ to $-2.5$ changes these means by less than $5\times10^{-10}$ relative, so the chosen cutoff is immaterial at the shown precision.

The same trajectories estimate finite-duration survival. For example, at $B=4$, $b=0.5$, and $s=262.692$ the surviving fraction is $0.3662$ with binomial standard error $0.0062$. This is close to $e^{-1}$ for this example; it does not assume that the entire hitting-time distribution is exponential. The accompanying data give survival at half, one, and twice each grid mean, together with its sampling error.

Finally, high barriers test the asymptotic prefactor without the exponentially expensive task of waiting for ordinary trajectories to escape. With $T_{\rm K}=(2\pi/\sqrt2)e^B$ in dimensionless time, direct continuum quadrature gives:

$B$	$T(-1;0)/T_{\rm K}$	$T(-1;0.5)/T_{\rm K}$
20	0.510000	1.019982
40	0.504834	1.009669
80	0.502379	1.004759

The limiting values are $1/2$ and $1$. For the fixed absorber $0.5$, the number of saddle widths grows as $\sqrt B$, so it enters the full-Gaussian regime as $B$ increases. The finite-$B$ departures from those limits remain after quadrature refinement; they are corrections to the asymptotic approximation.

The calculation record contains the parameters, numerical results, grid refinements, fixed random seeds, completed trajectory counts, and uncertainty estimates. From a checkout of the site source, reproduce it with node scripts/benchmark-stationary-first-passage.mjs --check. The dependency-free Node script recomputes the record, checks discrete detailed balance, verifies refinement and trajectory agreement, and compares all stored results with the recomputation. Use --write to regenerate the data, or --quick for a smaller exploratory ensemble. The high-barrier integration evaluates $e^{-B}T$ to keep exponentials within numerical range.

The structure map separates normalization and zero-current relaxation from absorbing-boundary observables. Inspect where the boundary data enter before interpreting a rate.

Stationarity requires normalizable zero-current evolution, whereas a hitting distribution requires declared absorbing and reflecting surfaces. Schematic; not to scale.

Domain and failure conditions

Use the chapter’s domain table. The formulas assume a single Markov coordinate, fixed $H$, additive leading noise, negligible energy transfer to the geometry, and a comoving probability density. In a physical spectator application also check that $\lvert V''\rvert\ll H^2$ over the sampled field range and that its energy density is small compared with the background. A large $B$ by itself establishes neither condition: barrier height relative to noise and lightness relative to $H$ are separate parameters. With field-dependent $H$, multiplicative noise, curved field space, or memory, return to the appropriate generator and measure before imposing boundaries.

The benchmark checks the stated diffusion model, not its derivation from a full interacting quantum field theory. It also tests neither volume weighting nor a changing inflationary background. If the available de Sitter duration is short compared with equilibration or escape, report finite-time $S(t\mid\phi_0)$ for the specified initial state. For very rare events, direct trajectories require many completed escapes to estimate the mean; a collection of trajectories that has not escaped is censored data and cannot be averaged as though its stopping times were known.

The failure map rejects equilibrium claims with nonnormalizable densities or hidden flux, and it rejects decay rates whose boundary or duration is unspecified. Time-dependent backgrounds pass to quasi-de Sitter validity; genuine gravitational saddle comparisons pass to Hawking–Moss and stochastic crossover.

Equilibrium, hitting probabilities, and escape rates have distinct normalization, boundary, duration, and backreaction tests. Schematic; not to scale.

Exercises

A flat-potential check. Set $V'=0$ on a reflecting–absorbing interval $[a,b]$. Evaluate $T(x;b)$ and check its dimensions and both boundary conditions.

Solution

The inner integral is $y-a$, so

$$T(x;b)=\frac{(b-a)^2-(x-a)^2}{2D_\phi}.$$

It obeys $D_\phi T''=-1$, $T'(a)=0$, and $T(b)=0$. Since $D_\phi$ has units of field squared per time, the result has units of time. There is no barrier and no justified Kramers exponential.

A boundary within the saddle region. For an absorber at $b=c\sqrt{\Theta/k_{\rm s}}$, what fraction of $T_{\rm K}$ is obtained at leading order? Explain why “slightly beyond the top” is not a sufficient specification.

Solution

The fraction is $[1+\operatorname{erf}(c/\sqrt2)]/2$. At $c=0$ it is $1/2$; it tends to $1$ only for $c\gg1$. The relevant displacement is measured in noise-dependent saddle widths. An absorber that approaches the saddle while $\Theta$ tends to zero can therefore retain an intermediate prefactor.

References

• Gillespie, D. T., “Exact Stochastic Simulation of Coupled Chemical Reactions,” The Journal of Physical Chemistry 81(25), 2340–2361 (1977), doi:10.1021/j100540a008.
• Hofmann, H., and F. A. Ivanyuk, “Mean First Passage Time for Nuclear Fission and the Emission of Light Particles,” Physical Review Letters 90, 132701 (2003), doi:10.1103/PhysRevLett.90.132701. Open PDF.
• Starobinsky, A. A., and J. Yokoyama, “Equilibrium State of a Self-Interacting Scalar Field in the de Sitter Background,” Physical Review D 50, 6357–6368 (1994), doi:10.1103/PhysRevD.50.6357.