436 research outputs found
A spectral-based numerical method for Kolmogorov equations in Hilbert spaces
We propose a numerical solution for the solution of the
Fokker-Planck-Kolmogorov (FPK) equations associated with stochastic partial
differential equations in Hilbert spaces.
The method is based on the spectral decomposition of the Ornstein-Uhlenbeck
semigroup associated to the Kolmogorov equation. This allows us to write the
solution of the Kolmogorov equation as a deterministic version of the
Wiener-Chaos Expansion. By using this expansion we reformulate the Kolmogorov
equation as a infinite system of ordinary differential equations, and by
truncation it we set a linear finite system of differential equations. The
solution of such system allow us to build an approximation to the solution of
the Kolmogorov equations. We test the numerical method with the Kolmogorov
equations associated with a stochastic diffusion equation, a Fisher-KPP
stochastic equation and a stochastic Burgers Eq. in dimension 1.Comment: 28 pages, 10 figure
Logarithmic Gradient Transformation and Chaos Expansion of Ito Processes
Since the seminal work of Wiener, the chaos expansion has evolved to a
powerful methodology for studying a broad range of stochastic differential
equations. Yet its complexity for systems subject to the white noise remains
significant. The issue appears due to the fact that the random increments
generated by the Brownian motion, result in a growing set of random variables
with respect to which the process could be measured. In order to cope with this
high dimensionality, we present a novel transformation of stochastic processes
driven by the white noise. In particular, we show that under suitable
assumptions, the diffusion arising from white noise can be cast into a
logarithmic gradient induced by the measure of the process. Through this
transformation, the resulting equation describes a stochastic process whose
randomness depends only upon the initial condition. Therefore the stochasticity
of the transformed system lives in the initial condition and thereby it can be
treated conveniently with the chaos expansion tools
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