Regular Diffusions on [0,∞)

On Markov Intertwining Relations and Primal Conditioning

“…The goal of this appendix is to recall that r P is the transition kernel of the Markov chain conditioned to stay in A. This result is classical and the proof is given here for the convenience of the reader, since we did not find a suitable reference (for the continuous-time setting, see Section 3.2 of Collet, Martínez and San Martín [2]).…”

“…, if y 1 1 " y 1 ą 1 and y 1 2 P Ypy 2 q 1 ´y1 ´1 N , if y 1 1 " y 1 ´1 and y 1 2 " y 2 2 0 , otherwise where y 2 2 is obtained from y 2 by removing its first value (with the convention that y 2 " H if y 1 " 1) and where Ypy 2 q is the set obtained from y 2 by inserting the first value of y 2 somewhere in y 2 2 except in last position. Furthermore, Qpp0, Hq, ¨q is the Dirac mass at p0, Hq.…”

Generalized scale functions of standard processes with no positive jumps

“…Standard processes with no negative jumps include several important classes of Markov processes such as one-dimensional diffusions, spectrally positive Lévy processes and continuousstate branching processes. For each class of the processes mentioned, there have already been detailed studies; e.g., [6], [18] for one-dimensional diffusions, [2], [33] for spectrally positive Lévy processes, and [16] for continuous-state branching processes. Our approach serves as a unified way to treat these processes.…”

On metastability

“…By Perron-Frobenius' theorem, Λ contains an eigenvalue λ 0 ě 0 which is strictly smaller than the real parts of all the other elements of Λ. It is sometimes called the first Dirichlet eigenvalue or the exponential survival rate of L, see for instance the book [3] of Collet, Martínez and San Martín. In particular, the algebraic multiplicity of λ 0 is 1.…”

“…For more details about the eigenmeasure ν, which gives positive weights to all the elements of V , we refer again to the book of Collet, Martínez and San Martín [3].…”