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We study the asymptotic behaviour of a real-valued diffusion whosenon-regular drift is given as a sum of a dissipative term and a boundedmeasurable one. We prove that two trajectories of that diffusion converge a.s.to one another at an exponential explicit rate as soon as the dissipativecoefficient is large enough. A similar result in $L_p$ is obtained.

Exponential a.s. synchronization of one-dimensional diffusions with non-regular coefficients