AI 中文总结
针对Hurst指数在(1/2,1)的一维分数布朗运动驱动的随机微分方程，证明了全变差意义下时间一致的Smoluchowski-Kramers近似收敛性，确定了收敛指数并构造了平稳解。
AI 中文摘要
设$B^H$为Hurst指数$H\in(1/2,1)$的一维分数布朗运动，研究动力学方程的小质量极限：$dX_t^\mu=Y_t^\mu dt$，$\mu dY_t^\mu=b(X_t^\mu)dt-Y_t^\mu dt +\sigma(X_t^\mu)dB_t^H$，其中噪声系数依赖于状态且一致非退化，极限方程为Young微分方程$dX_t=b(X_t)dt+\sigma(X_t)dB_t^H$。在变换漂移项满足一致椭圆性和严格耗散性的条件下，证明对任意$t_0>0$和任意$\rho<2H-1$，有$\sup_{t\ge t_0}d_{TV}(X_t^\mu,X_t) \le C_{t_0,\rho}\mu^\rho$，该估计在整个半直线上一致。还在双侧分数噪声空间上构造平稳解，得到其一时间边际的全变差收敛性，确定指数$2H-1$为自然端点：由Lamperti变换后的二次速度项产生，当$H\downarrow1/2$时退化，文中包含一个非常数一致椭圆性示例。
英文摘要
Let $B^H$ be a one-dimensional fractional Brownian motion with Hurst index $H\in(1/2,1)$. We study the small-mass limit of the kinetic equation \[ dX_t^μ=Y_t^μ\,dt,\qquad μ\,dY_t^μ=b(X_t^μ)\,dt-Y_t^μ\,dt +σ(X_t^μ)\,dB_t^H, \] where the noise coefficient is state dependent and uniformly nondegenerate. The limiting equation is the Young differential equation \[ dX_t=b(X_t)\,dt+σ(X_t)\,dB_t^H. \] Under uniform ellipticity and strict dissipativity of the transformed drift, we prove that, for every $t_0>0$ and every $ρ<2H-1$, \[ \sup_{t\ge t_0}d_{TV}\big(X_t^μ,X_t\big) \le C_{t_0,ρ}μ^ρ. \] The estimate is uniform on the entire half-line. We also construct stationary solutions on a two-sided fractional-noise space and obtain convergence of their one-time marginals in total variation. The exponent $2H-1$ is identified as the natural endpoint: it is generated by the quadratic velocity term after the Lamperti transform, and it degenerates as $H\downarrow1/2$. A nonconstant uniformly elliptic example is included.
