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STS517 Jan Rosinski´
measure formula (4), see (5) and (6). Still, relatively easy to handle is case =
2, where

The infinite divisibility of squared Bessel processes was not known until
Shiga, T., & Watanabe, S. (1973) paper. It still took several years until the Lévy
measures of such processes were found by Pitman, J.W., & Yor, M. (1982). The
description of such measures is in terms of Itô measure of the Brownian
positive excursions and the total accumulated local time of such excursions.
Therefore, it seems that Lévy measures here are relatively more complicated
than the infinitely divisible processes. This is completely opposite to the case
of Lévy processes, where Lévy measures are simple but processes can get
involved.
Squared Bessel processes and Feller processes belong to the class of
permanental processes. Permanental distributions were first considered in
statistics as an extension of gamma distributions to the multivariate case.
Vere-Jones, D. (1967) established the infinite divisibility of bivariate
permanental distributions and found their Lévy measures. Significant progress
to characterize the infinite divisibility of multivariate permanental distributions
was made by Bapat, R.B. (1989). For a complete discussion of infinite divisibility
of permanental distributions see Eisenbaum, N. and Kaspi, H. (2009) and
references therein. The celebrated Dynkin Isomorphism Theorem (Dynkin, E.B.
(1984)) can now be viewed in the framework of admissible translations (2), see
Rosinski, J. (2018).

References
1. Bapat, R.B. (1989). Infinite divisibility of multivariate gamma distribution
and Mmatrices, Sankhya 51 73-78.
2. Barndorff-Nielsen, O.E., Sauri, O., and Szozda, B. (2015)
Selfdecomposable Fields. J. Theor. Probab., Online First, Springer.
3. Dynkin, E.B. (1984). Gaussian and non-Gaussian random fields associated
with Markov processes, J. Funct. Anal. 55 344–376.
4. Eisenbaum, N. (2003). On the infinite divisibility of squared Gaussian
processes, Probab. Theory Related Fields 125 381–392.
5. Eisenbaum, N. (2008). A Cox Process Involved in the Bose–Einstein
Condensation, Ann. Henri Poincaré 9 1123–1140.
6. Eisenbaum, N. and Kaspi, H. (2006). A characterization of the infinitely
divisible squared Gaussian processes, Ann. Probab. 34(2) 728–742.
7. Eisenbaum, N. and Kaspi, H. (2009). On permanental processes,
Stochastic Process. Appl. 119(5) 1401–1764.
8. Kabluchko, Z. and Stoev, S. (2016). Stochastic integral representations
and classification of sum– and max-infinitely divisible processes, Bernoulli
22(1), 107–142.

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