Geodesic
Stochastic Nonlinear Schr\"odinger Equation. 1.~A~priori Estimates
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 232-256.

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We consider a nonlinear Schrödinger equation with a small real coefficient $\delta$ in front of the Laplacian. The equation is forced by a random forcing that is a white noise in time and is smooth in the space-variable $x$ from a unit cube; Dirichlet boundary conditions are assumed on the cube's boundary. We prove that the equation has a unique solution that vanishes at $t=0$. This solution is almost certainly smooth in $x$, and the $k$th moment of its $m$th Sobolev norm in $x$ is bounded by $C_{m,k}\delta^{-km-k/2}$. The proof is based on a lemma that can be treated as a stochastic maximum principle. 
@article{TM_1999_225_a14,
     author = {S. B. Kuksin},
     title = {Stochastic {Nonlinear} {Schr\"odinger} {Equation.} {1.~A~priori} {Estimates}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {232--256},
     publisher = {mathdoc},
     volume = {225},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_225_a14/}
}
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S. B. Kuksin. Stochastic Nonlinear Schr\"odinger Equation. 1.~A~priori Estimates. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Solitons, geometry, and topology: on the crossroads, Tome 225 (1999), pp. 232-256. http://geodesic.mathdoc.fr/item/TM_1999_225_a14/

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