Uniform concentration inequality for ergodic diffusion processes

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