Existence of solutions to the time-dependent gradient-flow equation defining Santaló curves

Establish the existence of at least one sufficiently smooth curve s:R^+→R^n that solves the time-dependent gradient-flow ODE s'(t) = -(p/(2q)) ∇_z Q(t, s(t)) associated with Q(t,z) = log ∫_{R^n} (τ_z f_t)(x) dx, where f_t is the evolution of a fixed nonnegative function f under the heat or Fokker–Planck semi-group, and such that lim_{t→∞} s(t) = 0.

Background

To analyze monotonicity of M_p(f_t) without explicitly invoking the Santaló point at each time, the authors consider the family of admissible centering curves s(t) for which t ↦ ∫ (τ_{s(t)} f_t) increases. They propose a natural candidate given by a gradient-flow ODE driven by the time-dependent log-Laplace transform Q(t,z).

The existence of such a curve (with appropriate regularity and boundary condition s(t)→0 as t→∞) remains an open question; proving existence would provide a principled centering procedure along the flow and potentially simplify monotonicity proofs.