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).
Solution Existence and Uniqueness for Degenerate SDEs with Application to SchrödingerEquation Representations
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.
 Award ID(s):
 1908918
 Publication Date:
 NSFPAR ID:
 10288233
 Journal Name:
 Comms. communications in Information and Systems.
 Volume:
 14
 Issue:
 4
 Page Range or eLocationID:
 213  231
 Sponsoring Org:
 National Science Foundation
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