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Flows, coalescence and noise. (English) Zbl 1065.60066

Ann. Probab. 32, No. 2, 1247-1315 (2004); correction ibid. 48, No. 3, 1592-1595 (2020).
This is a systematic presentation of the description of stationary random motion with independent increments, with a particular attention on the “coalescent” case, that is, the probability of hitting of two moving points starting from two different points is positive and these two moving points after hitting cannot separate. To describe the motion of \(n\) moving points on a metric space \(M\), the authors introduce a compatible family of Feller semigroups acting on the continuous functions on \(M^n\) \((n= 1,2,\dots)\), convolution semigroups of probability measures on the mapping on \(M\), stochastic flow of mappings and stochastic flow of kernels. Thus, a coalescing flow also can be described by a compatible family of Feller semigroups (Th. 4.1). They observe also that the probability measure defined by the mapping of a coalesing flow is atomic (Prop. 4.1). The relations between the motion of \(n\) moving points described by a compatible family of Feller semigroups and the stochastic differential equations and the associated diffusion are described. Finally, they study a Brownian motion on a two-point symmetric space with an isotropic covariant funcion, for which various types of flow (with respect to the probability of hitting and that of separation after hitting) can be observed.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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