Symmetric jump processes

Silverstein, Martin L.

doi:10.1090/S0002-9947-1974-0348849-7

Symmetric jump processes
HTML articles powered by AMS MathViewer

by Martin L. Silverstein PDF

Trans. Amer. Math. Soc. 192 (1974), 79-96 Request permission

Abstract:

We use the theory of Dirichlet spaces to construct symmetric Markov processes of pure jump type and to identify the Lévy measures for these processes. Particular attention is paid to lattice and hard sphere systems which interact through speed change and exclusion.

References

R. L. Dobrušin, Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Verojatnost. i Primenen 13 (1968), 201–229 (Russian, with English summary). MR 0231434
R. L. Dobrušin, Gibbsian random fields. General case, Funkcional. Anal. i Priložen 3 (1969), no. 1, 27–35 (Russian). MR 0239794
Masatoshi Fukushima, Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc. 162 (1971), 185–224. MR 295435, DOI 10.1090/S0002-9947-1971-0295435-0
Masatoshi Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka Math. J. 4 (1967), 183–215. MR 231444
T. E. Harris, Nearest-neighbor Markov interaction processes on multidimensional lattices, Advances in Math. 9 (1972), 66–89. MR 307392, DOI 10.1016/0001-8708(72)90030-8
Richard Holley, A class of interactions in an infinite particle system, Advances in Math. 5 (1970), 291–309 (1970). MR 268960, DOI 10.1016/0001-8708(70)90035-6
Thomas M. Liggett, Existence theorems for infinite particle systems, Trans. Amer. Math. Soc. 165 (1972), 471–481. MR 309218, DOI 10.1090/S0002-9947-1972-0309218-7
Thomas M. Liggett, A characterization of the invariant measures for an infinite particle system with interactions, Trans. Amer. Math. Soc. 179 (1973), 433–453. MR 326867, DOI 10.1090/S0002-9947-1973-0326867-1
Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
P.-A. Meyer, Intégrales stochastiques. I, II, III, IV, Séminaire de Probabilités (Univ. Strasbourg, Strasbourg, 1966/67) Springer, Berlin, 1967, pp. 72–94, 95–117, 118–141, 142–162 (French). MR 0231445
Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
Tokuzo Shiga and Takesi Watanabe, On Markov chains similar to the reflecting barrier Brownian motion, Osaka Math. J. 5 (1968), 1–33. MR 246372
Martin L. Silverstein, Dirichlet spaces and random time change, Illinois J. Math. 17 (1973), 1–72. MR 348850
Martin L. Silverstein, The reflected Dirichlet space, Illinois J. Math. 18 (1974), 310–355. MR 356253
Frank Spitzer, Interaction of Markov processes, Advances in Math. 5 (1970), 246–290 (1970). MR 268959, DOI 10.1016/0001-8708(70)90034-4
Shinzo Watanabe, On discontinuous additive functionals and Lévy measures of a Markov process, Jpn. J. Math. 34 (1964), 53–70. MR 185675, DOI 10.4099/jjm1924.34.0_{5}3