Solution Existence and Uniqueness for Degenerate SDEs with Application to Schrödinger-Equation 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 high-dimensional manifold. The second consists of singularities that may occur on a smoothly parameterized curve.
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NSF-PAR ID:
10288233
Journal Name:
Comms. communications in Information and Systems.
Volume:
14
Issue:
4
Page Range or eLocation-ID:
213 - 231
3. We study the focusing NLS equation in $R\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical ) regime, with $H^1$ data at the mass-energy threshold $\mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where Q is the ground state. Previously, Duyckaerts–Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li–Zhang for higher dimensions. In the intercritical case, Duyckaerts–Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts–Roudenko for any dimension and any power of the nonlinearity for the entire intercritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving an alternative proof of the Li–Zhang result and unifying the critical and intercritical cases. These results are obtained by studying the linearized equation around themore »