Journal
ANNALS OF PROBABILITY
Volume 32, Issue 2, Pages 1247-1315Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/009117904000000207
Keywords
stochastic differential equations; strong solution; stochastic flow; stochastic flow of kernels; Sobolev flow; isotropic Brownian flow; coalescing flow; noise; Feller convolution semigroup
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We are interested in stationary fluid random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.
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