A class of Markov processes with non-linear, bounded generators

Abstract

Symmetry is not necessary. Any AN can be replaced by a symmetric kernel (N !)-1 ‡"AN(xƒÐ(1),•c•c, xƒÐ(N)|x,), where a ranges over all permutations of {1, 2,•c•c, N}. *) The proof of Chapman-Kolmogorov equation for P(s, x, t,•)) is easy by (1.4). This model shows that for temporally inhomogeneous Markov processes it is not always natural to discuss transition probability and initial distribution separately, while this has been rather customary. **) We neglect the trivial case, where AN(x1,•c•c, xn|x,•E)•ß0 for all N_??_1. Proof of Lemma 3. Since for *) Note that this is also true for complex valued measures with finite total variation.

Purely Discontinuous Markov Processes with Nonlinear Generators and Their Propagation of Chaos

“…The key devices we use for constructing these semigroups will, on the one hand, naturally lead to a generalization of Wild's sum [8] ( 4,Theorem B), and on the other hand, can be used to prove a convergence theorem for certain linear semigroups ( 5, Theorem C), as a corollary of which the propagation of chaos will be established. In our arguments, a [7] yet unpublished at that time.…”

“…Based on Kac's idea (propagation of chaos), H. P. McKean, Jr. [2], [3] introduced a certain class of Markov processes with non-linear generators, and described such a process as the motion of tagged molecule in an infinite bath of like molecules. In this direction, D. P. Johnson [4] treated the case of 2-state Markov processes with non-linear generators in detail; T. Ueno [7], expanding the method of D. P. Johnson, proved that the propagation of chaos holds for the equation (1.1) under the following assumption (Ao)" (Ao) there exists a constant L such that nq, <_ p !L , p 1, 2, .... n=l In this paper, the propagation ofchaos will be established for the equation (1.1) under the following assumption (A)"…”

A connection between mixing and Kac's chaos

“…The "Boltzmann property" is commonly referred to as "Kac's chaos" and has attracted the interest of many people such as McKean [17], Johnson [12], Tanaka [24], Ueno [25], Grünbaum [9], Murata [20], Graham and Méléard [8], Sznitman [22], [23], Mischler [18], Carlen, Carvalho and Loss [4], Michler and Mouhot [19]. These authors considered a more general situation than a sequence f (n) of density functions defined on K n .…”

Quantum Kac’s chaos

“…Many authors including McKean [16], Johnson [14], Tanaka [24], Ueno [25], Grünbaum [13], Graham and Méléard [12], Sznitman [23], Mischler [18], Carlen, Carvalho and Loss [7], Mischler and Mouhot [19] have abstracted the idea of the "Boltzmann property" to a sequence of probability measures on a topological space. Instead of having the "Boltzmann property", the sequence of probability measures nowadays are said to be chaotic.…”