Stochastic nonlinear Schrödinger equation. I: A priori estimates (Q2736079)

A nonlinear stochastic Schrödinger equation NEWLINE\[NEWLINE \dot v = \delta\Delta v - i|v|^2 v + \zeta, \;v(0,\cdot) = \xi, \quad t\geq 0, \;x\in\mathbb R^{n}, \tag{1} NEWLINE\]NEWLINE is considered, the solution being sought in the space of complex-valued odd functions 2-periodic in \(x\), \(v(t,x_1,\ldots, x_{n}) = v(t,\ldots,x_{j}+2,\ldots)\), \(j=1,\ldots,n\). It is supposed that \(n\leq 3\), \(\delta\in\mathopen]0,1\mathclose]\), and that the (generalized) random field \(\zeta\) is of the form \(\zeta(t,x) = \eta(t,x)\dot w(t)\), where \(w\) is a standard Wiener process and the random field \(\eta\) is such that \(|\eta(t,x) |\leq 1\) for all \((\omega,t,x)\), \(\eta\) is continuous in \((t,x)\), odd and 2-periodic in \(x\) for almost all \(\omega\), and \(\eta(\cdot,x)\) is adapted for all \(x\). Furthermore, denote by \(\|\cdot\|_{m}\) the norm of the Sobolev space \(H^{m}_{\roman{op}}\) of complex-valued odd 2-periodic functions on \(\mathbb R^{n}\) and assume that \(\mathbf E\|\eta(t,\cdot)\|^{p}_{m}\leq C(m,p)\) for all \(t\geq 0\) and \(p,m\in\mathbb N\). Let the initial datum \(\xi\) satisfy \(\mathbf E\text{ ess sup} |\xi|^{p} \leq C_{p}\delta^{-p/2}\), \(\mathbf E \|\xi\|^{2}_{m} \leq C_{m}\delta^{-2m-1}\) for all \(m\in\mathbb N\), \(p\geq 1\). Then it is proven that there exists a unique (mild) solution \(v\) to (1), which satisfies NEWLINE\[NEWLINE \mathbf E\sup_{t\leq s\leq t+1/\delta}\sup_{x} |v(s,x)|^{q} \leq C_{q}\delta^{-q/2}, \quad \mathbf E\|v(t)\|^{q} _{m} \leq C_{q,m}\delta^{-qm-q/2} NEWLINE\]NEWLINE for every integer \(m\geq 0\) and all \(t\geq 0\), \(q\geq 1\). Moreover, if \(\eta\) is a deterministic function independent of \(t\), then it is shown that (1) defines a Markov process in each space \(H^{m}_{\roman{op}}\), \(m\geq 2\), and there exists an invariant probability measure for this process.NEWLINENEWLINEFor the entire collection see [Zbl 0967.00102].