| Abstract: |
| We consider the stochastic nonlinear Schr\odinger equation
with the polynomial nonlinearity
\[
{\rm d} u(t,x)+\left[ \mathrm{i} \Delta u(t,x)+\mathrm{i} \alpha |u(t,x)|^{2\sigma} u(t,x) \right] \,{\rm d}t
= \phi(u(t,x)) \,{\rm d} W(t)
\]
Classical results of global existence are obtained for power $\sigma$ not too large, depending on the spatial dimension $d$ and
the parameter $\alpha$ ($\alpha>0$ is the focusing case and $\alpha\frac d2$.
The effect of the noise is to prevent blow-up in finite time, differently from the deterministic setting.
Moreover, we prove the existence of an invariant measure and its uniqueness under more restrictive assumptions on the noise term.
As an example, one can consider a one dimensional real Wiener process
$W$ and diffusion $\phi(u)=[a(1+\|u\|_{L^\infty(\mathbb T^d)})^\sigma+\mathrm{i} b(1+\|u\|_{L^\infty(\mathbb T^d)})^\sigma]u$ for real values $a,b$ with $a$ large enough.
The choice $s>\frac d2$ provides the helpful estimate
$\|u\|_{L^\infty(\mathbb T^d)} \le C \|u\|_{H^s(\mathbb T^d)}$, because of the continuous embedding
$H^s(\mathbb T^d) \subset L^\infty(\mathbb T^d)$.
Therefore the local existence result is a trivial fact.
Our proof of global existence relies on a tightness method based on the choice of a suitable Lyapunov function.
In particular, the global existence holds in both focusing and defocusing cases. |
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