Let $X_t$ represent a continuous-time Markov process on $\mathbb{R}^d$, say a diffusion with locally Lipschitz coefficients. Suppose that there exists a unique invariant measure $\mu$ on the space, and $X_0$ is distributed as a point mass at $x_0 \in \mathbb{R}^d$, how might one compute or bound above the total variation between the measure $\mu_t$ defined by $\mu_t(A) = P(X_t \in A)$ and the invariant measure, $\mu$?
For a concrete example, if $dX_t = -X_t dt + dW_t$ (that is, $X_t$ is an Ornstein-Uhlenbeck process) one finds that the unique invariant distribution is absolutely continuous with respect to Lebesgue measure and coincides with the distribution of a standard normal variable: $$\mu(dx) = {\pi}^{-1/2}e^{-x^2} dx. $$ If $X_0 = 1$ a.s. one 'knows' that $\mu_t$ will obey the forward equation and will rapidly approach $\mu$, but I haven't found many useful tools to prove this or convergence in the general case.
Looking at contraction semigroup $T_t$, for example, one knows $$ \| T_t \| \leq Me^{\beta t} ~~ {\rm for } ~~ \beta \in \mathbb{R}, M > 0. $$ But I cannot compute $\beta$ (to show $\beta < 0$) explicitly. As $T_t$ is strongly continuous we have a uniform bound, but I cannot rule out $\beta = 0$. Some very particular results have been found, for Hamiltonian systems for example, but I imagine that general methods must exist for such an elementary question.
Thanks in advance.