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Under the Bakry–Emery's \(\Gamma_{2}\)-minoration condition, we establish the logarithmic Sobolev inequality for the Brownian motion with drift in the metric \(L^2\) instead of the usual Cameron–Martin metric. The involved constant is sharp and does not explode for large time. This inequality with respect to the \(L^2\)-metric provides us the gaussian concentration inequalities for the large time behavior of the diffusion.

Logarithmic Sobolev Inequalities of Diffusions for the L 2 Metric