Diffusion Bridges and Functional Diffusion Processes

1. Introduction

Diffusion processes form the theoretical backbone of many generative modeling techniques, including diffusion probabilistic models and score-based generative models. Two advanced extensions of diffusion processes are explored in this post:

• Diffusion Bridges: Diffusion processes conditioned to start and end at specific states.
• Functional Diffusion Processes: Diffusion processes that evolve not in finite-dimensional vector spaces but in infinite-dimensional function spaces.

2. Diffusion Bridges

2.1 Definition

A diffusion bridge is a stochastic process \( \{X_t\}_{t \in [0,T]} \) that behaves like a diffusion but is conditioned on reaching a fixed terminal point. Formally, for a diffusion process \( X_t \) governed by the SDE

\[ dX_t = b(t, X_t) dt + \sigma(t, X_t) dW_t, \]

a diffusion bridge from \( x_0 \) to \( x_T \) is the process \( X_t \) conditioned on \( X_0 = x_0 \) and \( X_T = x_T \).

2.2 Modified Drift for Bridge Construction

To simulate such a bridge, one can use the Girsanov theorem to modify the drift of the original SDE. Let \( p(t, x; T, y) \) be the transition density. Then, the modified drift becomes:

\[ \tilde{b}(t, x) = b(t, x) + \sigma(t, x)^2 \nabla_x \log p(t, x; T, x_T). \]

This correction term steers the process towards the terminal point \( x_T \).

2.3 Applications in Generative Models

• Schrödinger Bridge as a stochastic optimal transport problem.
• Diffusion bridges appear in methods like guided generation and stochastic interpolants.

3. Functional Diffusion Processes

3.1 Motivation

In many modern applications (e.g., image, shape, or waveform generation), the data lives in infinite-dimensional spaces like function spaces. Modeling uncertainty or evolution in such spaces calls for functional diffusion processes.

3.2 Brownian Motion in Function Spaces

Let \( \mathcal{H} \) be a separable Hilbert space. A process \( \{X_t\}_{t \in [0,1]} \subset \mathcal{H} \) is said to be a cylindrical Brownian motion if:

• For each \( h \in \mathcal{H} \), the process \( \langle X_t, h \rangle \) is a standard Brownian motion in \( \mathbb{R} \).
• Paths are continuous with probability 1 in the weak topology.

Functional SDEs can be expressed as:

\[ dX_t = \mathcal{A}(X_t) dt + \mathcal{B}(X_t) dW_t, \]

where \( \mathcal{A}, \mathcal{B} \) are (possibly nonlinear) operators on \( \mathcal{H} \).

3.3 Example: Heat Equation as a Functional Diffusion

The stochastic heat equation on \( [0,1] \) with noise:

\[ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \xi(t, x), \]

where \( \xi \) is space-time white noise, can be viewed as a diffusion process in a function space. It admits solutions in Sobolev spaces or spaces of continuous functions.

3.4 Functional Score-Based Models

In generative modeling, score-based methods can be extended to infinite-dimensional settings where the score is a functional derivative:

\[ \nabla_{\mathcal{H}} \log p(u), \]

where \( u \in \mathcal{H} \) and the score guides the evolution in function space.

4. Connections and Open Problems

• Functional SDEs underlie many PDE-driven generative models (e.g., wave or fluid data).
• Understanding bridge processes in function spaces is important for stochastic control and guided sampling.
• Schrödinger bridge in function spaces relates to regularized Wasserstein flows and entropy minimization over paths.