Branching processes and partial differential equations Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/70795b63d

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  • The recursive and stochastic representation of solutions to the Fourier transformed Navier-Stokes equations, as introduced by [34], is extended in several ways. First, associated families of functions known as majorizing kernels are analyzed, in light of their apparently essential role in the representation. Second, the theory is put on a more comprehensive foundation by constructing the basic recursive object, the multiplicative functional or its successor, the random field, without invoking the strong Markov property. This allows the theory to embrace a wider class of evolutionary equations. Third, this methodology, that has delivered global existence and uniqueness theorems for the Navier- Stokes equations given suitably small initial datum, is extended to obtain local in time existence and uniqueness results when the initial datum is arbitrarily large. Fourth, the theory is applied to the semi-linear KPP equation linking it with, and extending previously known results on the representation of solutions with branching Brownian motion.
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