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Every diffusion model (DDPM, DDIM, score SDEs) has Brownian motion at its core. The discrete forward chain you saw in the DDPM tutorial, clusters dissolving into Gaussian noise, is the discretization of a continuous-time Wiener process. Building intuition for the Wiener process makes the rest of the diffusion-model machinery click.

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Math recap: the Wiener process

A standard Wiener process WtW_t on [0,T][0, T] is a continuous-time stochastic process satisfying:
  • W0=0W_0 = 0
  • Independent increments: for 0≤s<t0 \le s < t, Wt−Ws∼N(0,t−s)W_t - W_s \sim \mathcal{N}(0, t - s), independent of the path up to time ss
  • Continuous paths
A discrete-time approximation with step Δt\Delta t is the random walk Wt+Δt=Wt+Δt  ξ,ξ∼N(0,1)W_{t + \Delta t} = W_t + \sqrt{\Delta t}\; \xi, \quad \xi \sim \mathcal{N}(0, 1) Multiplied by a diffusion coefficient σ\sigma this gives Brownian motion with variance σ2t\sigma^2 t. The DDPM forward chain q(xt∣xt−1)=N(xt;1−βt xt−1,βtI)q(x_t \mid x_{t-1}) = \mathcal{N}(x_t; \sqrt{1-\beta_t}\, x_{t-1}, \beta_t I) is a variance-preserving discrete-time analogue: it adds Brownian-style noise and shrinks the signal each step so the marginal stays bounded.

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Brownian motion in motion

Eight particles released from the origin, each tracing an independent random walk. The trail behind each particle is its history, short trajectories on either side of the origin diverge into the wider variance the longer the chain runs.

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Eight Brownian trajectories

Eight particles released from the origin, each undergoing an independent random walk over 500 steps with σ=0.7\sigma = 0.7 and Δt=1/60\Delta t = 1/60. The colored end-dots mark each trajectory’s final position; the black dot at the origin marks the common start. 
import numpy as np
import matplotlib.pyplot as plt

rng = np.random.default_rng(42)
n_particles = 8
n_steps = 500
dt = 1.0 / 60
sigma = 0.7

increments = sigma * np.sqrt(dt) * rng.standard_normal((n_steps + 1, n_particles, 2))
increments[0] = 0.0
paths = np.cumsum(increments, axis=0)
Output from cell 2

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Snapshot grid at the same timesteps as the DDPM forward process

200 fresh particles released from the origin, snapshotted at t=0,10,50,100,200,500t = 0, 10, 50, 100, 200, 500, the same timesteps the DDPM tutorial uses for its forward-noising visualization. The cloud spreads with variance proportional to tt, the hallmark of a Wiener process. 
rng = np.random.default_rng(0)
n = 200
T = 500
snapshots = [0, 10, 50, 100, 200, 500]
sigma_grid = 0.5

increments = sigma_grid * np.sqrt(dt) * rng.standard_normal((T + 1, n, 2))
increments[0] = 0.0
grid_paths = np.cumsum(increments, axis=0)
Output from cell 4

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Connection back to diffusion

Continuous-time diffusion frames the forward process as the SDE dxt=f(xt,t) dt+g(t) dWtdx_t = f(x_t, t)\, dt + g(t)\, dW_t where WtW_t is exactly the Brownian motion above. The variance-preserving SDE used by DDPM picks f(xt,t)=−12β(t) xtf(x_t, t) = -\tfrac{1}{2}\beta(t)\, x_t and g(t)=β(t)g(t) = \sqrt{\beta(t)}, so the marginal stays Gaussian and the variance is bounded. The discrete chain you trained earlier is the Euler-Maruyama discretization of this SDE. Two takeaways:
  • Forward = Brownian motion + drift. Everything diffusion-model-related sits on this foundation.
  • Reverse = score-based ODE/SDE. Anderson’s reverse-time SDE is what your trained ϵθ\epsilon_\theta approximates; deterministic DDIM is the probability-flow ODE limit.

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References

  1. Anderson. Reverse-time diffusion equation models. Stochastic Processes Appl. 1982.
  2. Song et al. Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021. arxiv.org/abs/2011.13456
  3. Karatzas & Shreve. Brownian Motion and Stochastic Calculus. Springer 1991. (textbook)
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