Deterministic applicability of first-exit-time variation technique

Determine how the compact-set, first-exit-time-based variation technique—where one fixes a compact set K containing the initial condition a, sets the final time to the first exit time T_K of the trajectory from K, defines a vector field X vanishing on {a} ∪ ∂K, and constructs a variational family yielding the variation δT = X(T) so that δT = 0 at t = 0 and t = T_K—can be applied in the deterministic variational setting on manifolds.

Background

The paper discusses two approaches to constructing fixed-endpoint and adapted variations for stochastic processes. The first approach fixes a compact set K containing the initial condition and chooses the final time as the first exit time T_K from K, then selects a vector field X that vanishes on {a} ∪ ∂K, leading to variations δT = X(T) that vanish at t = 0 and at t = T_K. This method is well-suited to stochastic settings because it preserves adaptedness and uses random stopping times.

The authors highlight that, although this approach is effective for stochastic problems, its extension or analogue in deterministic calculus of variations is not evident. Clarifying how this first-exit-time construction applies in the deterministic setting would bridge methodological gaps between stochastic and classical variational frameworks.