Clarification: Dimensionality of Brownian Motion \(w(t)\)

Key Point

\(w(t)\) and \(x(t)\) live in the same space: \(\mathbb{R}^d\)

Details

In the SDE formulation:

\[ dx(t) = f(x(t), t)\,dt + g(t)\,dw(t) \]

• \(x(t) \in \mathbb{R}^d\): The state vector (e.g., flattened image, gene expression vector)
• \(w(t) \in \mathbb{R}^d\): A d-dimensional Wiener process (Brownian motion)

Why This Matters

The Brownian motion \(w(t)\) must be d-dimensional because:

1. The differential \(dw(t)\) behaves like:

$$

dw(t) \sim \sqrt{dt} \cdot \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, I_d) $$

where \(I_d\) is the \(d \times d\) identity matrix, so \(\varepsilon \in \mathbb{R}^d\).

1. The noise term \(g(t) \, dw(t)\) is added directly to \(x(t)\):
2. Both must have the same dimension for the addition to be valid

3. The actual noise added to the image is \(g(t) \, dw(t)\), not just \(dw(t)\)

4. For an image diffusion model:

5. If the image is \(H \times W \times C\) pixels, then \(d = H \times W \times C\)
6. \(w(t)\) is a \(d\)-dimensional random walk
7. Each component of \(w(t)\) is an independent 1D Brownian motion

Intuition

• \(w(t)\): The underlying d-dimensional random walk (source of randomness)
• \(g(t)\): A scalar (or matrix) that scales the noise magnitude
• \(g(t) \, dw(t)\): The actual noise increment added to the image at time \(t\)

Think of it as: \(w(t)\) provides the "direction and magnitude" of randomness in d-dimensional space, and \(g(t)\) controls how much of that randomness gets injected into the image.

Connection to the Document

This clarification relates to section 2.3 (Brownian Motion) in sde_formulation.md. The document could be more explicit that \(w(t)\) is d-dimensional to match \(x(t)\).