A spectral-based numerical method for Kolmogorov equations in Hilbert spaces.

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The method is based on the spectral decomposition of the Ornstein鈥揢hlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener鈥揅haos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating 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鈥揔PP stochastic equation and a stochastic Burgers equation in dimension 1.


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The method is based on the spectral decomposition of the Ornstein鈥揢hlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener鈥揅haos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating 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鈥揔PP stochastic equation and a stochastic Burgers equation in dimension 1. Read More: http://www.worldscientific.com/doi/abs/10.1142/S021902571650020X</p> <div class="row"> <div class="col-lg-12 col-xl-9"> <div class="card"> <div class="card-body"> <div> <h3 class="mb-1 text-secondary">A spectral-based numerical method for Kolmogorov equations in Hilbert spaces.</h3> <h5> <em><span class="contributor externalContributor externalContributore3aace00-8f61-4c98-e053-3705fe0acb7e m-1 disabled" aria-disabled="true" style="display:none;">Delgado-Vences, Francisco;</span><span class="contributor internalContributor internalContributore3aace00-8f61-4c98-e053-3705fe0acb7e m-1"><a href="/cris/rp/rp02728" class="authority author">Flandoli, Franco</a></span></em> <span onclick="JQ('.externalContributore3aace00-8f61-4c98-e053-3705fe0acb7e').toggle();JQ('.showExternalTexte3aace00-8f61-4c98-e053-3705fe0acb7e').toggle();JQ('.hideExternalTexte3aace00-8f61-4c98-e053-3705fe0acb7e').toggle();" class="btn btn-sm btn-outline-secondary m-1"> <span class="showExternalTexte3aace00-8f61-4c98-e053-3705fe0acb7e"><span class="fas fa-users" title="mostra contributor esterni"></span></span> <span class="hideExternalTexte3aace00-8f61-4c98-e053-3705fe0acb7e" style="display:none;"><span class="fas fa-user" title="nascondi contributor esterni"></span></span> </span> </h5> <h5>2016</h5> <hr/> <h4 class="mb-1">Abstract</h4> We propose a numerical solution for the solution of the Fokker鈥揚lanck鈥揔olmogorov (FPK) equations associated with stochastic partial differential equations in Hilbert spaces. The method is based on the spectral decomposition of the Ornstein鈥揢hlenbeck semigroup associated to the Kolmogorov equation. This allows us to write the solution of the Kolmogorov equation as a deterministic version of the Wiener鈥揅haos Expansion. By using this expansion we reformulate the Kolmogorov equation as an infinite system of ordinary differential equations, and by truncating 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鈥揔PP stochastic equation and a stochastic Burgers equation in dimension 1. 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A spectral-based numerical method for Kolmogorov equations in Hilbert spaces.