Abstract
The theory of diffusion processes with a nonnormable phase space (a nuclear Fréchet space) is developed and the Cauchy problem for parabolic equations relative to functions on this space is solved by probabilistic methods. A series of examples are given, demonstrating a significant difference between the theory of stochastic differential equations and parabolic equations in the case of locally convex spaces, on one hand, and the analogous theory in the case of Banach spaces, on the other hand. The difficulty which arises, when passing from a Banach space to a Fréchet space, involves basically a functional rather than a probabilistic character. There appears a sufficiently complex intertwinement of the theory of locally convex and pseudotopological spaces with probability theory.
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 190–209, 1986.
In conclusion, the author expresses his gratitude to O. G. Smolyanov for his constant interest in the paper and for useful advice.
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Khrennikov, A.Y. Diffusion processes in linear spaces and pseudotopologies. J Math Sci 45, 1523–1540 (1989). https://doi.org/10.1007/BF01097276
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DOI: https://doi.org/10.1007/BF01097276