We conduct a preliminary study of the convexity of mutual information regarded as the function of time along the Fokker-Planck equation and generalize conclusions in the cases of heat flow and OU flow. We firstly prove the existence and uniqueness of the classical solution to a class of Fokker-Planck equations and then we obtain the second derivative of mutual information along the Fokker-Planck equation. We prove that if the initial distribution is sufficiently strongly log-concave compared to the steady state, then mutual information always preserves convexity under suitable conditions. In particular, if there exists a large time, such that the distribution at this time is sufficiently strongly log-concave compared to the steady state, then mutual information preserves convexity after this time under suitable conditions.

翻译：本文对沿福克-普朗克方程将互信息视为时间函数的凸性进行了初步研究，并推广了热流与OU流情形下的相关结论。我们首先证明了一类福克-普朗克方程经典解的存在唯一性，进而推导出沿该方程的互信息二阶导数。研究证明：若初始分布相对于稳态具有足够强的对数凹性，则在适当条件下互信息始终保持凸性。特别地，若存在某个时刻使得该时刻的分布相对于稳态具有足够强的对数凹性，则在适当条件下该时刻后互信息将保持凸性。