Uniform concentration inequality for ergodic diffusion processes

We consider the deviation function in the ergodic theorem for an ergodic diffusion process (yt)(yt)ΔT(φ)=T−1/2∫0T(φ(yt)−m(φ))dt, where φφ is some function, m(φ)m(φ) is the integral of φφ with respect to the ergodic distribution of (yt)(yt). We prove a concentration inequality for ΔT(φ)ΔT(φ) which is uniform with respect to φφ and T≥1T≥1.