Local existence of solutions to randomized Gross-Pitaevskii hierarchies (Q2787957)

The author proves the existence of local-in-time solutions to independently or dependently Gross-Pitaevskii hierarchies. Taking \(\Lambda =\mathbb{R}^{d} \) or \(\Lambda =\mathbb{T}^{d}\), the Gross-Pitaevskii hierarchy is defined as the infinite system of linear parabolic equations \(i\partial _{t}\gamma ^{(k)}+(\Delta _{\overrightarrow{x}_{k}}-\Delta _{\overrightarrow{x} _{k}^{\prime }})\gamma ^{(k)}=\sum_{j=1}^{k}B_{j,k+1}\gamma ^{(k+1)}\) with the initial conditions \(\gamma ^{(k)}\mid _{t=0}=\gamma _{0}^{(k)}\in \Lambda ^{k}\times \Lambda ^{k}\rightarrow \mathbb{C}\). \(\Delta _{ \overrightarrow{x}_{k}}\) is the Laplace operator and \(B_{j,k+1}\) is a collision operator. Following a previous paper, the author introduces the randomized collision operator \(\left[ B_{j,k+1}\right] ^{\omega }\) for a given parameter \(\omega \) taken in an appropriate probability space \(\Omega \). He then defines the depently or independently randomized Gross-Pitaevskii hierarchies replacing the collision operator \(B_{j,k+1}\) by \(\left[ B_{j,k+1} \right] ^{\omega }\) or by \(\left[ B_{j,k+1}\right] ^{\omega _{k+1}}\), respectively. The main purpose of the paper is to prove the existence of local-in-time solutions to these independently or dependently Gross-Pitaevskii hierarchies, the author here extending existence results obtained in an earlier paper for the randomized Gross-Pitaevskii hierarchy [\textit{V. Sohinger} and \textit{G. Staffilani}, Arch. Ration. Mech. Anal. 218, No. 1, 417--485 (2015; Zbl 1372.35287)]. For the proof, the author applies the strategy developed by \textit{T. Chen} and \textit{N. Pavlović} [Proc. Am. Math. Soc. 141, No. 1, 279--293 (2013; Zbl 1260.35197)], that he adapts in this randomized case and he takes care of the random parameters.