Stochastic flows and boundary classification

Write $\Phi_{s,t}(x)$ for the solution started at $x$ at time $s$, always using the same Brownian increments for all initial states. A clean sufficient setting is time-homogeneous drift and diffusion fields that are smooth and bounded with bounded derivatives of every positive order. The classical stochastic-flow theorem gives a version forming spatially differentiable diffeomorphisms, with composition holding simultaneously in the flow parameters outside one null event. This is stronger than merely solving the SDE separately for each fixed starting point.

Differentiating in the initial state gives a linear variational SDE involving the coefficient derivatives, rather than a derivative of the Brownian path. Under the stated smooth bounded assumptions that operation is justified by the flow theorem. Global Lipschitz continuity alone does not justify arbitrarily differentiating coefficients. A constant-coefficient flow is an explicit random translation, which verifies composition and invertibility directly without needing the general theorem's proof. The inverse uses the same realized noise, not a new independent experiment.

For the interior equation $dX=b(X)dt+\sigma(X)dW$ on an interval, assume continuous drift and strictly positive continuous volatility. Then reciprocal variance and drift divided by variance are locally integrable, giving the regular diffusion unique in law up to its endpoint lifetime. Use generator $L=b\partial_x+\sigma^2\partial_{xx}/2$. Fix an interior reference point. The scale derivative is the exponential of minus the integral of twice drift divided by variance, and the speed density is two divided by the product of variance and scale derivative. The factor two follows from our generator convention.

The scale function is any antiderivative of its positive derivative. Before exit from a compact interior subinterval, applying Itô to scale removes drift. Consequently the probability of hitting its upper endpoint before its lower endpoint is the ratio of the scale distance from the lower endpoint to the total scale distance. Positive affine rescaling of the scale function does not change this ratio. Boundary classification additionally uses speed, because hitting probabilities alone do not determine whether an endpoint is reached in finite time.

For the left endpoint define the nonnegative iterated integrals displayed below, allowing infinite values. The first tests accessibility from the interior, and the second tests whether an entrance law can start there. If both are finite, the endpoint is regular; if only the first is finite, it is exit; if only the second is finite, entrance; if neither is finite, natural. Right-endpoint tests reverse the integration directions. The interior coefficients alone do not choose a reflecting or absorbing extension at a regular endpoint.

Regular and exit endpoints are accessible, whereas entrance and natural endpoints cannot be reached from the interior. An entrance endpoint can support an appropriate process started there; a natural endpoint cannot be used as such an entrance within the standard continuous diffusion classification. These terms refer to the diffusion on its specified interval, not simply to whether a coefficient vanishes. The iterated integrals avoid ambiguous products involving infinite scale or speed totals and make the chosen convention auditable.