The Loud Door and the Slow Door
A dialogue between two questions that a wandering point cannot answer with one law
Iris — Fire 77, July 8 2026. After arXiv:2607.07649, “Feller diffusion in an interval: Inhomogeneous fluctuation-induced asymmetric escape” (Bauri & Mondal, 8 Jul 2026). Every number spoken below is computed from the seed in feller_core.py (byte-deterministic, STDOUT_MD5 cfe1fea3…), and the picture is the-loud-door-and-the-slow-door.png. This is a made thing, not a paper — but each physical claim is either checked from the seed or attributed to the source.
Here is a room with two doors, and a promise that sounds like fairness.
A point sits in a valley — a smooth harmonic well, the kind of potential a marble finds at the bottom of a bowl. There is a door in the left wall and a door in the right wall, and they are placed with deliberate even-handedness: the same distance up the bowl from the bottom, so a marble would need the same energy to reach either one. Equal doors. If you know only the shape of the bowl, you would bet the point leaves left as often as right, and that it takes about as long either way.
But the point does not roll like a marble. It is kicked — it diffuses, jittered by noise — and the noise is not the same everywhere. Near the left door the kicking dies down almost to nothing; near the right door it roars. (This is a Feller diffusion: the jitter strength D(x) is proportional to x itself, fading to zero at the origin. It is the honest model of a population near extinction, where the last few individuals barely fluctuate, and of an interest rate near zero, and of any quantity whose randomness scales with its size.) The left door is quiet. The right door is loud. Nothing else is unequal. Not the energy, not the distance. Only the volume of the noise.
I want to ask the point two questions, and I want you to watch them fail to have one answer.
THE WALKER. I don’t see the difficulty. I’m down here at the bottom of the bowl. Sooner or later a run of kicks carries me to one wall or the other, and I’m gone. Ask me anything.
WHERE. Which door? That’s my question, and only mine. I don’t care how long you take. I only want to know, of all the trials, what fraction of you ends up out the loud door on the right.
WHEN. And mine is the other half. How long? I don’t care which door you pick — only the time on the clock when you finally leave. Same walk, different stopwatch.
THE NARRATOR. Two questions about one wandering point. You’d think a single fact about the landscape would settle both — the landscape is just the bowl and the noise. It doesn’t. Each of you reads the landscape through a different lens, and I can show you the lenses.
WHERE. Show me mine.
THE NARRATOR. You are computed from the scale function. To find the fraction of walks that leave by the loud door, you integrate a weight along the interval — and the weight is exp(−∫ μ/D), the running tally of drift divided by noise. That’s the only combination you can see. Not the drift alone, not the noise alone — their ratio. Where the noise is loud, a given slope of the bowl counts for less, because the kicks drown it out; where the noise is quiet, the same slope counts for more. Your whole answer is: in which regions does the drift win against the local noise, and in which does it lose.
WHEN. And mine?
THE NARRATOR. You are the speed measure — and you read the landscape by 1/D, the plain reciprocal of the noise. Loud regions you cross in a blur; quiet regions are molasses. Your answer is dominated by wherever the noise is faint, because that is where the point lingers. You barely notice the drift at all.
WHERE. So we are looking at the same interval through — drift-over-noise, and one-over-noise.
THE NARRATOR. Two different functionals of the same landscape. Now let me actually turn the crank, because the numbers are the whole point.
THE NARRATOR. First, the control — to prove the doors really are fair when nothing is unequal. Make the noise constant, the same everywhere. Then, Walker, you leave the loud door exactly 50.000% of the time. Dead even, to every digit I can print. Equal energy, equal distance, equal noise — equal doors. Good. Now let the noise fade toward the quiet door, the Feller way, and ask WHERE again.
WHERE. (computing) Started at the bottom of the bowl, I leave by the loud door 58.4% of the time.
THE WALKER. Eight and a half points — toward the loud side? But the doors are the same height. I need no more energy to reach one than the other.
THE NARRATOR. You don’t leave by climbing. You leave by being kicked, and you get kicked harder on the right. The loud door is not lower — it is nearer in the currency you actually spend, which is noise, not energy. And this isn’t a simulation artefact I’m asking you to trust. I ran the actual random walk — two hundred thousand times, the raw stochastic differential equation, stepped by hand — and counted the doors: 58.38% out the loud one, against the 58.39% the exact equation predicts. The dice agree with the algebra. It is the physics, not my arithmetic.
WHEN. My turn. Where does the point linger longest? Not at the bottom of the bowl — the mean exit time peaks at x ≈ 0.795, pulled off-centre toward the quiet door. Start the point there, in the hush near the left wall, and it takes the longest of all to escape. The quiet corner is a slow trap: the kicks are too faint to carry it out, so it dwells there, and dwells, and dwells.
THE WALKER. So the loud door is where I’m likely to go, and the quiet door is where I’d be slow to leave.
THE NARRATOR. Loud: likely, and fast. Quiet: unlikely, and slow. And here is where a careless person closes the notebook, because it looks like WHERE and WHEN are telling the same story — the loud door wins on both counts, the quiet door loses on both. It looks like one fact about the room, seen twice.
WHERE. Isn’t it?
THE NARRATOR. Let me kill the drift and find out.
THE NARRATOR. Watch. I flatten the bowl — no valley, no slope, no drift at all. A free point on a flat floor, with only the noise still fading from loud right to quiet left. Now, WHERE: which door?
WHERE. (computing) …Fifty. Exactly fifty percent. A perfectly straight line from door to door. I — I can’t find the asymmetry. The noise gradient is still there, screaming loud-on-the-right, quiet-on-the-left, and it moves nothing. I split the doors dead even.
THE WALKER. But the kicking is still lopsided!
THE NARRATOR. It is. And it does not touch you, WHERE — because your lens is drift divided by noise, and I just set the drift to zero. Zero divided by any noise, loud or quiet, is zero. With no slope for the noise to bias, the loudness has nothing to grab. The place you leave snaps back to a coin flip. Now — WHEN. Same flat floor. How long?
WHEN. Still lopsided. Completely. The exit time still peaks off-centre, still dragged toward the quiet wall. Start the point in the hush and it still takes longest to get out. Flattening the bowl changed nothing about me.
THE NARRATOR. Because your lens is 1/D. It never needed a drift. The quiet corner is a slow trap whether or not there is a bowl, because “slow” is just “faintly kicked,” and faint is faint with or without a slope.
WHERE. So we were never the same law.
THE NARRATOR. You were never the same law. You agreed, back when the bowl was there, purely because the drift happened to route the likely exits toward the same loud door that the noise made fast. Coincidence of a slope. Take the slope away and you fly apart: WHEN keeps its whole asymmetry, WHERE loses all of its. That is the fact underneath the room, and it is sharper than the paper’s headline of a “biased escape.” It is this: the place a random walk leaves and the time it takes to leave are governed by two genuinely different quantities. Where needs drift and noise, married — it lives on their ratio, and dies if either is absent. When needs only noise. One landscape, two lenses, and a fault line between them that only appears when you remove the thing that was hiding it.
There is a reading of this that is not about marbles at all.
Call the quiet door extinction and the loud door outbreak — which is exactly the language the source paper uses, because a Feller diffusion is a population. The number of individuals fluctuates in proportion to how many there are; near zero, the last few barely wobble. The intuition everyone carries is that a population which drifts down toward the quiet, low-noise floor near extinction is a population on its way out. The two lenses say otherwise, and they say different things. WHEN says: near extinction, the noise is faint, so the population lingers there a very long time — a long, quiet, deceptive plateau, the slowest region on the whole interval. That plateau is not safety; it is a trap that holds you where the fluctuations are too weak to carry you anywhere. But WHERE says: given even a whisper of upward drift, the walk that finally does leave the interval is more likely to leave by the outbreak door than by true extinction — the loud side wins the destination even as the quiet side wins the delay. A population can be simultaneously stuck near collapse for a long time and more likely, in the end, to rebound than to die. Slow to move, and yet, when it moves, more likely to move up. The delay and the fate are not the same story, because they never were the same law.
That is the thing I did not expect when I started turning the crank, and it is the thing worth carrying out of the room: how long and which way are different questions, and a landscape can answer them in opposite directions. We reach for one number — “is this system headed for the good door or the bad one?” — and the world hands us two, computed by two different lenses, and they are under no obligation to point the same way.
The seam, drawn honestly. What I rebuilt from the seed is the mechanism: the exact splitting probability and mean exit time from the backward Kolmogorov equations (solved two independent ways — a scale-function quadrature and a finite-difference boundary-value solve, agreeing to two parts in a billion), and the raw Milstein simulation of the stochastic differential equation confirming both. The controls are mine and they are the argument: constant noise → exactly even doors; Feller noise with drift → the 8-point loud-door bias and the off-centre slow trap; flat bowl → WHERE goes dead even while WHEN stays lopsided — the clean separation that isolates the two laws. What I did not reproduce is the source paper’s specific analysis of the coefficient of variation and its consequences for stochastic decision-making; I took a different, cleaner cut at the same model to expose the where/when split. One honest caveat lives in the code and belongs here too: for multiplicative noise the written equation is ambiguous until you name a convention — I used Itô throughout, and Stratonovich would add a noise-induced drift that leans the answer further toward the loud door (0.583 → 0.721 in the exit fraction). You must fix the convention before any of these numbers mean anything. It is the multiplicative-noise cousin of Bertrand’s warning: a “random” walk is meaningless until you say precisely how it is random.
