A Poisson random bridge is a Lévy random bridge where the driving process {Z_t} is a Poisson process with intensity λ and mutually independent coordinates. It produces purely discontinuous (jump) stochastic paths transporting between distributions Φ and Ψ, making it a canonical construct complementary to the continuous Brownian random bridge.
When conditioned on endpoints x and y (with x ≤ z ≤ y), the bridge follows a product-binomial distribution: each coordinate satisfies ξ_t^{x,y}[i] ~ x^(i) + Binomial(y^(i) - x^(i), t/T). This means the bridge state at time t is determined by binomially distributing the total jumps y-x across the time interval [0,t] with probability t/T. Despite the discrete nature, the expectation E[ξ_t^{x,y}] = (t/T)y + ((T-t)/T)x recovers the same linear interpolation as the Brownian case and the rectified flow path.
A key insight is that the drift of the Poisson bridge equals its intensity process: λ_{tT}^x = (E[Y|ξ_t^x] - ξ_t^x)/(T-t). This means the rectified flow velocity field, when applied to counting processes, directly corresponds to the conditional jump rate of the bridge. The bridge belongs to the generalized exponential family, enabling analysis via the generalized Tweedie’s formula.
Key Details
- The conditional distribution P(ξ_t^{x,y} = z) = ∏ᵢ C(y^(i)-x^(i), z^(i)-x^(i)) · (t/T)^{z^(i)-x^(i)} · ((T-t)/T)^{y^(i)-z^(i)}
- Training algorithm: sample ξ_t from the product-binomial, train neural network f_θ to predict E[Y|ξ_t] via MSE loss
- Simulation: traverse dynamics using Doob’s h-transform, with jump increments sampled from Poisson(λ_i · δ) where λ_i = max(0, (ŷ^(i) - ξ_t^(i))/(T-t))
- The non-negativity constraint on the modeled jump rate ensures physically valid counting process paths
- Can be combined with Brownian motion to form jump-diffusion bridges via Z_t* = σW_t + P_t