Appl Math Optim 46:89–96 (2002)
2002 Springer-Verlag New York Inc.
Stochastic Vorticity and Associated Filtering Theory
and Gopinath Kallianpur
Department of Statistics, University of Michigan,
Ann Arbor, MI 48109-1092, USA
Department of Statistics, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Abstract. The focus of this work is on a two-dimensional stochastic vorticity
equation for an incompressible homogeneous viscous ﬂuid. We consider a signed
measure-valued stochastic partial differential equation for a vorticity process based
on the Skorohod–Ito evolution of a system of N randomly moving point vortices. A
nonlinear ﬁltering problem associated with the evolution of the vorticity is consid-
ered and a corresponding Fujisaki–Kallianpur–Kunita stochastic differential equa-
tion for the optimal ﬁlter is derived.
Key Words. Nonlinear ﬁltering, Stochastic vorticity, Systems of stochastic dif-
ferential equations, Signed measure-valued SPDE.
AMS Classiﬁcation. 60G35, 60H20, 60H15.
Experimental results often suggest that the nature of certain hydrodynamical phenomena
calls for their stochastic formulation. High sensitivity to initial conditions and to pertur-
bations, interplay of large numbers of degrees of freedom, and presence of conditions,
under which existing microscopic perturbations get ampliﬁed to macroscopic scales, give
rise to unsteady and chaotic ﬂows. Thus, in many cases a natural approach to modeling
of chaotic behavior in ﬂuids is given via stochastic partial differential equations (SPDEs)
The focus of the current paper is on the stochastic modeling of the motion of
a homogeneous viscous incompressible ﬂow in R
and the solution of an associated
nonlinear ﬁltering problem.