Source
Nadler, Lafon, Coifman and Kevrekidis (2006; arXiv 2005), LIT-1250, read in full in NOTE-986. The paper supplies the setting: data sampled from the equilibrium of the Langevin equation (2), ẋ = −∇U(x) + √2ẇ, and an α = 1/2 diffusion map whose small-ε generator is that equation’s backward operator (30), Δψ − ∇ψ·∇U, up to the factor ½ that NOTE-986 finds missing. It proposes the leading eigenfunctions as reaction coordinates and as the coarse description “at medium to long times, at which the fast coordinates have equilibrated”. The statement and proof below are the record’s.
What was actually shown
Setting. U: ℝⁿ → ℝ is smooth with ∫e^{−U} = 1, and μ = e^{−U}dx. X solves dX = −∇U(X)dt + √2 dW, has a unique non-exploding strong solution and is started in equilibrium, X₀ ~ μ, so it is stationary. F_t is the filtration of (X₀, W). The generator is L = Δ − ∇U·∇, with Δ = ∇². Let ψ ∈ C² satisfy Lψ = −λψ with λ > 0, and assume ψ ∈ L⁴(μ) and |∇ψ| ∈ L⁴(μ). Write Y_t = ψ(X_t), h(x) = |∇ψ(x)|², and G_t = σ(Y_r : r ≤ t).
Step 1. The conditional mean given the full state. By Itô’s formula, d(e^{λt}Y_t) = e^{λt}(λψ + Lψ)(X_t)dt + e^{λt}√2∇ψ(X_t)·dW_t = e^{λt}√2∇ψ(X_t)·dW_t. The integrand is square-integrable on [0, T]: by stationarity, E∫₀ᵀ e^{2λs}·2h(X_s)ds = 2∫h dμ·∫₀ᵀe^{2λs}ds < ∞. So e^{λt}Y_t is a martingale, and
E[Y_{t+s} | F_t] = e
Step 2. No memory given ψ’s own past. G_t ⊆ F_t, so by the tower property E[Y_{t+s} | G_t] = E[e^{−λs}Y_t | G_t] = e^{−λs}Y_t. The best mean-square predictor of ψ’s future from its own history uses only its present value, linearly. Nothing is gained by adding the rest of the state: the two predictors are equal. In Mori’s form of the Mori–Zwanzig decomposition, with the projection Pf = ⟨f, ψ⟩_μ ψ/‖ψ‖²_μ applied to the evolution of observables, Lψ = −λψ lies in the range of P. So QLψ = 0, and the memory and orthogonal-dynamics terms for ψ vanish identically.
Step 3. The path of Y. Let n(x) = ∇ψ(x)/|∇ψ(x)| where ∇ψ ≠ 0, and a fixed unit vector elsewhere, and B_t = ∫₀ᵗ n(X_s)·dW_s. B is a continuous martingale with ⟨B⟩_t = t, so it is a Brownian motion by Lévy’s characterization. Then
dY_t = −λY_t dt + √(2h(X_t)) dB_t, ⟨Y⟩_t = 2∫₀ᵗ h(X_s)ds.
Step 4. When Y is Markov. If h = g∘ψ μ-almost everywhere for some g, then Y is a weak solution of the one-dimensional equation dY = −λY dt + √(2g(Y)) dB. Where that equation has uniqueness in law (for instance when √g is locally Lipschitz and non-explosive), its solutions are strong Markov processes, so Y is Markov.
Step 5. When it is not. The quadratic variation ⟨Y⟩ is a functional of Y’s path, so h(X_t) = lim_{δ↓0}(⟨Y⟩t − ⟨Y⟩{t−δ})/2δ is G_t-measurable. Now expand the squared increment: E[(Y_{t+s} − Y_t)² | G_t] = 2E[∫_t^{t+s}h(X_r)dr | G_t] + R_s, where R_s collects the drift terms. E|R_s| ≤ E[(λ∫Y)²] + 2(E[(λ∫Y)²]E[M²])^{1/2} = O(s^{3/2}), using ψ ∈ L²(μ) and h ∈ L¹(μ). By stationarity and path continuity, with h ∈ L²(μ) for uniform integrability, (1/s)E[∫_t^{t+s}h dr | G_t] → h(X_t) in L¹. So
(1/s)E[(Y_{t+s} − Y_t)² | G_t] → 2h(X_t) in L¹.
If Y were Markov in its own filtration, then since ψ ∈ L⁴(μ) and E[Y_{t+s} | G_t] = e^{−λs}Y_t, the left side would be a function of Y_t for each s, and so would its L¹ limit. Hence h(X_t) would be σ(Y_t)-measurable, i.e. h = g∘ψ μ-a.e. Contrapositively, if |∇ψ|² is not a.e. a function of ψ, then Y is not Markov. Its own past predicts the size of its next fluctuation better than its present does.
Checked numerically. Take the 2D Ornstein–Uhlenbeck process U = x₁²/2τ₁ + x₂²/2τ₂ with τ₁ = 1 and τ₂ = 1/4, the paper’s §5.2 with these values. It was simulated by Euler–Maruyama, dt = 10⁻³, for 2 × 10⁶ steps, and three eigenfunctions were examined at lag s = 0.2:
| ψ | λ | |∇ψ|² | slope of E[Y_{t+s} | Y_t] vs e^{−λs} | coefficient on Y_{t−s} in the mean | R² for the squared residual: present, + past |
|---|---|---|---|---|---|
| x₁ | 1 | 1 | 0.821 vs 0.819 | −0.008 | 0.000, 0.001 |
| x₁² − τ₁ | 2 | 4(ψ + τ₁) | 0.660 vs 0.670 | +0.017 | 0.118, 0.118 |
| x₁x₂ | 5 | x₁² + x₂² | 0.370 vs 0.368 | −0.003 | 0.025, 0.032 |
The mean is closed in all three cases. For x₁x₂ only, the past adds to the prediction of the squared residual, as Step 5 requires.
What could have failed: an eigenfunction’s conditional mean could have depended on unresolved coordinates through the noise. Step 1 shows it cannot, because the noise term is a martingale. And closure in the mean could have implied closure in law. x₁x₂ shows that it does not.
What this does not say
- Not that diffusion maps recover the eigenfunctions. That rests on the α = 1/2 limit (THEORY-744, Proposed; Nadler et al.’s expansion is formal) and on finite-sample convergence, which neither paper proves. NOTE-986 found that, from equilibrium samples of a double well, the eigenfunction was recovered well and the slow eigenvalue was not.
- Not that the slowest eigenfunction is the best coarse variable. Exact closure of the mean holds for every eigenfunction, slow or fast. Slowness matters only for how much of the dynamics the variable carries at long times.
- Not for non-reversible or non-equilibrium dynamics. Steps 1–5 use only Lψ = −λψ, the martingale property and stationarity. For a non-reversible generator, eigenfunctions are complex and may lack a real eigenbasis, which this entry does not treat. The data-driven link needs reversibility as well.
- Not that a closed mean makes a level. Under THEORY-257’s dynamical independence the coarse variable’s own past must exhaust what the micro past says about its future in law, not only in mean. Step 5 shows the two come apart.
- Not anything about finite collections of eigenfunctions. The argument covers one ψ. The vector (ψ₁, …, ψ_k) is closed in mean componentwise. It is Markov when the matrix ⟨∇ψ_i, ∇ψ_j⟩ is a function of the vector, by the same argument; that matrix version is not written out here.
Connections
- QUESTION-222: answers in part. For coarse variables that are generator eigenfunctions of a reversible diffusion, the memory term of the mean dynamics is exactly zero, so no augmentation is needed for the mean. What can carry memory is the second moment, through the noise intensity |∇ψ|². The natural augmentation is then the single coordinate h(X) = |∇ψ(X)|², which ψ’s own path reveals. This is the converse shape of CLAIM-389: there zero memory did not imply closure, and here closure of the mean does not imply closure in law.
- THEORY-257. The criterion conditions on the coarse variable’s own history, in law. An eigenfunction that fails Step 4’s condition fails it. Even its own past is not summarized by its present, though its mean is.
- CLAIM-331. In a reversible system the slow coarse variables that diffusion maps find are generator eigenfunctions, and they are closed in mean at every timescale, not only at long times. What separates the timescales is the decay rate λ, not any degree of closure.
- CLAIM-388. An eigenfunction is a function of the whole state, not of a block, and its mean closes exactly. That is another instance of closure without a block, in the mean.
- CLAIM-157 and CLAIM-164. Nothing here uses a gap. A gap decides which eigenfunctions dominate after time t. It does not decide which are closed.