A Lévy random bridge is a stochastic process {ξ_t} that transports between two probability distributions Φ and Ψ, with its internal dynamics governed by a Lévy process {Z_t} as the driving process. Formally, {ξ_t} is a (Φ, Ψ)-bridge under action of {Z_t} if: (1) the endpoints satisfy (ξ₀, ξ_T) ~ Γ(Φ, Ψ) for some coupling Γ, and (2) the conditional finite-dimensional distributions of ξ match those of Z conditioned on boundary values, for Φ-a.e. starting points and Ψ-a.e. endpoints.
The framework is very general. Because Lévy processes encompass Brownian motion, Poisson processes, and their combinations (via the Lévy-Khintchine representation with triplet (α, β, η)), (Φ, Ψ)-bridges can have purely continuous paths (Brownian random bridge), purely discontinuous paths (Poisson random bridge), or hybrid jump-diffusion paths. The choice of driving process controls the path regularity without altering the marginal transport property.
A key result is the non-anticipative semimartingale representation: for any Lévy driver with mutually independent coordinates and finite first moments, ξ_t^x = ∫₀ᵗ (E[Y | ξ_s^x] - ξ_s^x)/(T-s) ds + M_t, where M_t is a martingale. This drift has the identical functional form to the rectified flow velocity field, establishing a fundamental connection between random bridges and flow-based generative models. Moreover, E[ξ_t^{x,y}] = (t/T)y + ((T-t)/T)x for any Lévy driver — the expectation of the bridge equals the linear interpolation (i.e., the rectified flow path).
Key Details
- The (Φ, Ψ)-bridge framework does not enforce any specific dynamical form or time direction, encompassing both forward and backward constructs
- When the driving process is Markov, the bridge is also Markov (Proposition 2.3 in Shelley & Mengütürk)
- The transition densities follow Doob’s h-transform: P(ξ_t^{x,y} ∈ dz | ξ_s^{x,y} = r) = f_{t-s}(z-r) · h_{tT}(z;y) / h_{sT}(r;y)
- Bridges form collections of stochastic sub-transports: each {ξ_t^x}_{t∈[r,u]} defines a Φ_r-initialized bridge to Ψ_u for 0 ≤ r < u ≤ T
- The framework originates from Hoyle, Hughston, and Macrina (2011) in financial mathematics and was extended to generative modelling by Goria et al. (2025)