Some Probabilistic Uses of Dirichlet Eigenvectors
summary
Proves quantitative rates of convergence to the quasi-stationary distribution for absorbing chains, gives the closed-form cosine law on a path and shows convergence requires order N^2 steps; develops a path congestion bound (Perron–Frobenius + Dirichlet form + Cauchy–Schwarz) to control sub-stochastic spectra; interprets left/right Perron vectors via the Doob transform; applies Fourier analysis on the Heisenberg group to reduce mixing to p×p Harper-type matrices and successfully bounds their extreme eigenvalues so N^2 is both necessary and sufficient.
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chapters (8)
- 0:00Spectral geometry and Markov chains
- 6:29Absorbing Markov chains
- 18:48Quasi-stationary distributions
- 26:29Quasi-stationarity via spectral geometry
- 37:28Path technology and eigenvalue bounds
- 40:46Matrix eigenvalue problem
- 42:31Heisenberg group and random walks
- 55:18Summary and spectral geometry
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