Existence and uniqueness results for solutions of stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coeffcient is nondegenerate. Here, existence and uniqueness of strong solutions is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solutions of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coeffcient is exactly half that of the state. In addition to this degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential. The first type consists of discontinuities that may occur on a possibly highdimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve.
Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations
Existence and uniqueness results for stochastic differential equations (SDEs) under exceptionally weak conditions are well known in the case where the diffusion coefficient is nondegenerate. Here, existence and uniqueness of a strong solution is obtained in the case of degenerate SDEs in a class that is motivated by diffusion representations for solution of Schrödinger initial value problems. In such examples, the dimension of the range of the diffusion coefficient is exactly half that of the state. In addition to the degeneracy, two types of discontinuities and singularities in the drift are allowed, where these are motivated by the structure of the Coulomb potential and the resulting solutions to the dequantized Schrödinger equation. The first type consists of discontinuities that may occur on a possibly highdimensional manifold (up to codimension one). The second consists of singularities that may occur on a lowerdimensional manifold (up to codimension two).
 Award ID(s):
 1908918
 Publication Date:
 NSFPAR ID:
 10171205
 Journal Name:
 21st IFAC World Congress
 Page Range or eLocationID:
 16
 Sponsoring Org:
 National Science Foundation
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