Flows, coalescence and noise

By Yves Le Jan, Olivier Raimond. 2002

195 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/503b5a05b160ee84c9272178ffe306b10b05a5bb

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Citations in the corpus

Stochastic equations, flows and measure-valued processes (2010, background): “We shall see that {Yt : t ≥ 0} is actually an immigration superprocess in the sense of Li (2011) with trivial underlying spatial motion.”
Stochastic flows associated to coalescent processes II: Stochastic differential equations (2005, unknown): “These flows, or more precisely the dual flows B̂s,t = B−t,−s, fit in the general framework of Le Jan and Raimond [9].”
Stochastic flows associated to coalescent processes (2003, background): “Further, in the continuous case (i.e. when the map r → B−10,t (r) is continuous a.s. for every t ≥ 0), the Feller property still holds if we restrict the state space to the open interval ]0, 1[, and the family ( B−1s,t ,−∞ s ≤ t ∞ ) then induces a coalescing stochastic flow on ]0, 1[, in the term”
On the convergence of stochastic transport equations to a deterministic parabolic one (2019, methodology): “From the mathematical point of view, it has been treated in a very complete but rather technical way in [26], [27]; in Section 4.”