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We obtain the lower bounds for ergodic convergence rates, including spectralgaps and convergence rates in strong ergodicity for time-changed symmetricL\'{e}vy processes by using harmonic function and reversible measure. As directapplications, explicit sufficient conditions for exponential and strongergodicity are given. Some examples are also presented.

Ergodic convergence rates for time-changed symmetric L\'{e}vy processes in dimension one