ifc-0267

18.1 Diffusion is one infinitesimal direction

In the continuous-time formulation, a forward diffusion satisfies

\[ dX_t=f(X_t,t)\, dt+g(t)\, dW_t, \]

and gradually transports a data distribution toward a tractable noise distribution. Its reverse-time SDE depends on the time-indexed score \(\nabla _x\log p_t(x)\). A text-conditioned model replaces this by a conditional score such as \(\nabla _x\log p_t(x\mid \ell )\), where \(\ell \) is a prompt or encoded semantic state.

The associated probability-flow ODE has velocity

\[ v_t^\ell (x) =f(x,t)-\frac{1}{2}g(t)^2\nabla _x\log p_t(x\mid \ell ). \]

Under suitable regularity, its flow maps \(\Phi _{s,t}^\ell \) are invertible and satisfy

\[ \Phi _{t,u}^\ell \circ \Phi _{s,t}^\ell =\Phi _{s,u}^\ell , \qquad (\Phi _{s,t}^\ell )^{-1}=\Phi _{t,s}^\ell . \]

This is the cleanest entry point for groupoid semantics: the pair groupoid of noise times acts through a family of conditioned transport maps on image states. The construction is conditional on existence, uniqueness, and regularity of the flow; a well-defined evolution of densities alone does not guarantee a pointwise invertible transport.

The stochastic statement is different. Markov transition kernels generally have no inverse and therefore do not form a groupoid. Under stronger smoothness conditions an SDE may generate a pathwise stochastic flow of diffeomorphisms, but its infinitesimal generator is a second-order differential operator rather than an ordinary vector field. ARTISTIC consequently begins with the probability-flow ODE as its classical Lie-theoretic realization and treats a fully stochastic groupoid or algebroid semantics as an open extension.

. The reverse-time SDE reconstructs a data distribution from a noise distribution; it is not generally a sample-by-sample inverse of the particular Brownian path that destroyed an image. The deterministic probability-flow ODE and a pathwise stochastic flow are stronger, distinct constructions.