Formulation of Stochastic Contact Hamiltonian Systems

Both the paper and the model establish that the stochastic flow φ_t of the Stratonovich contact Hamiltonian SDE preserves the contact structure conformally, with φ_t^*η = λ_t η and λ_t = exp(−∫_{t0}^t ∂H0/∂z dτ − Σ_k ∫_{t0}^t ∂Hk/∂z ∘ dB_τ^k), exactly as stated in Theorem 3.2 of the uploaded paper . The model achieves this via Kunita’s pullback formula for Stratonovich flows and the computation L_{X_H}η = −R(H)η in Darboux coordinates, yielding the same exponential factor. The paper’s proof proceeds by differentiating the flow with respect to initial data and solving a scalar linear Stratonovich SDE for λ_t (equations (3.10)–(3.13)) . Minor internal sign inconsistencies exist between the intrinsic relations (2.1) and the stated local expression (2.4) as well as Remark 3.1 , but these do not affect the main theorem’s statement nor the conformal factor used there.