A SYMMETRIC RANDOM WALK DEFINED BY THE TIME-ONE MAP OF A GEODESIC FLOW

The paper proves the equivalence by (i) reducing existence of an absolutely continuous P-stationary measure to the cohomological equation log(p/(1−p)) = φ∘g − φ (via Conze–Guivarc’h) and (ii) invoking the periodic-cycle functional criterion applied to log ϕ (not ϕ) to obtain a continuous (indeed smooth, under mild hypotheses) solution. The candidate solution correctly flags that the periodic-cycle obstruction must be applied to log ϕ, but then makes a substantive error constructing the density: defining ρ := q e^u and deriving the “detailed balance” ρ(x)p(x) = ρ(gx)q(gx) is algebraically incorrect; the correct choice is ρ = e^u so that p/q = ρ∘g/ρ and ν = ρ µ is stationary. The model also asserts detailed balance from stationarity without justification. Aside from minor presentational slips in the paper (a typographical use of F(C)(ϕ) instead of F(C)(log ϕ) and an implicit “g preserves µ” hypothesis in one statement that is made explicit later), the paper’s argument is correct.