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Let's suppose that we want to consider a reversible diffusion process, which has transition density and a unique stationary distribution, for instance, O-U process with generator $$ L=\frac{d^{2}}{dx^{2}}+\theta(\mu-x)\frac{d}{dx}. $$ If $\theta\rightarrow\infty$, the O-U process $X_{t}^{\theta}$ will approach to $\mu$. My question is: will the spectral gap of O-U process go to infinity as $\theta\rightarrow\infty$? If so, Can this be generalized to other similar diffusion models? What's the scale of spectral gap?

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