A Brownian random bridge is a Lévy random bridge where the driving process is a scaled Brownian motion {Z_t} = {σW_t}. It produces purely continuous stochastic paths transporting between distributions Φ and Ψ. When conditioned on endpoints x and y, the bridge ξ_t^{x,y} ~ N(E_t, V_t) is Gaussian with mean E_t = x + (t/T)(y-x) and variance V_t = t(T-t)σ²/T.
The Brownian bridge admits two key representations. The anticipative form is ξ_t = (t/T)Y + σ(W_t - (t/T)W_T) + ((T-t)/T)X, which reveals the bridge as a rectified flow interpolation plus a standard Brownian bridge perturbation: ξ_t = R_t + σB̂_t where B̂_t is a Brownian bridge with B̂₀ = B̂_T = 0. The non-anticipative form is ξ_t^x = ∫₀ᵗ (E[Y|ξ_s^x] - ξ_s^x)/(T-s) ds + σW_t, showing the drift is the same target-pulling velocity as rectified flow.
A central result is the drift decomposition (Proposition 3.3): the target-pull velocity (E[Y|ξ_t] - ξ_t)/(T-t) decomposes into a source-push term (ξ_t - x)/t plus a score term σ²∇log p_t(ξ_t). This decomposition connects the bridge framework to score-based generative modelling and yields the probability flow ODE as the arithmetic mean of source-push and target-pull velocities. Setting σ = 0 collapses the bridge to deterministic rectified flow.
Key Details
- The probability flow ODE is dξ_t = ½[(ξ_t - x)/t + (E[Y|ξ_t] - ξ_t)/(T-t)]dt
- The reverse-time SDE simplifies to dξ̄_t = (ξ̄_t - x)/t dt̄ + σdW̄_t due to exact cancellation of score terms, a manifestation of Nelson’s osmotic velocity
- Training: sample ξ_t^{x,y} ~ N(x + (t/T)(y-x), t(T-t)σ²/T), regress neural network on E[Y|ξ_t] with MSE loss
- The bridge variance V_t = t(T-t)σ²/T is maximal at t = T/2, creating maximum stochasticity at the midpoint
- Gaussian random bridges have been studied in quantum measurement, financial derivative pricing, and stochastic filtering