Abstract
The Cauchy problem for a class of diffusion equations in a Hilbert space is studied. It is proved that the Cauchy problem in well posed in the class of uniform limits of infinitely smooth bounded cylindrical functions on the Hilbert space, and the solution is presented in the form of the so-called Feynman formula, i.e., a limit of multiple integrals against a gaussian measure as the multiplicity tends to infinity. It is also proved that the solution of the Cauchy problem depends continuously on the diffusion coefficient. A process reducing an approximate solution of an infinite-dimensional diffusion equation to finding a multiple integral of a real function of finitely many real variables is indicated.
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Remizov, I.D. Solution of a cauchy problem for a diffusion equation in a Hilbert space by a Feynman formula. Russ. J. Math. Phys. 19, 360–372 (2012). https://doi.org/10.1134/S1061920812030089
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DOI: https://doi.org/10.1134/S1061920812030089