Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form

Amano, Kazuo

doi:10.1090/S0002-9947-1984-0756041-3

Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form

Author: Kazuo Amano
Journal: Trans. Amer. Math. Soc. 286 (1984), 295-312
MSC: Primary 35H05; Secondary 35J70, 60J60
DOI: https://doi.org/10.1090/S0002-9947-1984-0756041-3
MathSciNet review: 756041
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Abstract: In the theory of partial differential equations, there is no explicit representation of solutions for general degenerate elliptic-parabolic equations. However, Stroock and Varadhan [15] have obtained a stochastic representation for such a wider class of equations in ${L^\infty }$ space. In this paper we establish, by using Stroock and Varadhan's stochastic representation, a method which enables us to construct solutions with singularities of second order equations with semidefinite characteristic form. Our theorems are not probabilistic paraphrases of the results obtained in the theory of partial differential equations. In fact, each assumption of the theorems is much weaker than any assumption of corresponding known results.

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