Nelson’s osmotic velocity is a concept from Edward Nelson’s stochastic mechanics (1967) that characterizes the relationship between the forward and backward drifts of a diffusion process through the score function of its marginal density. For any sufficiently regular diffusion process, the forward drift v₊(z,t) and backward drift v₋(z,t) satisfy:
v₊(z,t) - v₋(z,t) = σ²∇_z log p_t(z)
The difference — the osmotic velocity — measures the local tendency of the process to move toward regions of higher probability density. Nelson originally introduced this in the context of stochastic quantization, where it provides a probabilistic interpretation of quantum mechanical phenomena.
In the context of Brownian random bridge. This identity emerges naturally. The forward drift decomposes as (ξ_t - x)/t + σ²∇log p_t(ξ_t) (Proposition 3.3 in Shelley & Mengütürk 2025), while Anderson’s reverse-time SDE yields a backward drift of (ξ_t - x)/t. The score term σ²∇log p_t(ξ_t) cancels exactly, revealing that the discrepancy between the target-pulling velocity of the generative (forward) process and the source-retracting velocity of the reverse process is precisely the osmotic velocity. This cancellation is why the reverse-time bridge SDE has no score dependence.
Key Details
- The forward drift of the Gaussian bridge is: (E[Y|ξ_t] - ξ_t)/(T-t) = (ξ_t - x)/t + σ²∇log p_t(ξ_t)
- The backward drift is: (ξ_t - x)/t (pure source-retraction, no score term)
- Their difference is exactly σ²∇log p_t(ξ_t), confirming Nelson’s identity
- This identity underpins the duality between score-based and flow-based generative models
- The osmotic kinetic energy ½||u||² = ⅛||∇ln ρ||² is the Fisher information integrand — this is exactly the term separating Schrödinger bridge from OT in Chen et al. (2014)
- Originally from: Edward Nelson, “Dynamical Theories of Brownian Motion,” Princeton University Press, 1967