Initial value problem for white noise operators and quantum stochastic processes (Q2751517)

White noise nonlinear differential equations are studied. Especially, the initial problem NEWLINE\[NEWLINE{d\Xi \over dt}= F(t,\Xi), \quad \Xi |_{ t=0} =\Xi_0,\quad 0\leq t\leq T,\tag{1}NEWLINE\]NEWLINE is considered. The function \(F: [0,T] \times {\mathcal L}({\mathcal W},{\mathcal W}^*) \to{\mathcal L}({\mathcal W},{\mathcal W}^*)\) is a continuous function and \({\mathcal L}({\mathcal W},{\mathcal W}^*)\) is the space of white noise operators. The main result is the followingNEWLINENEWLINENEWLINETheorem 10. Let \(\alpha= \{\alpha (n)\}\) and \(\omega= \{\omega(n)\}\) be two weight sequences with conditions (A1)--(A5) such that their generating functions are related by \(G_\alpha (t)= \exp\gamma \{G_\omega(t) -1\}\). Let \(F:[0,T] \times {\mathcal L}( {\mathcal W}_\alpha, {\mathcal W}^*_\alpha) \to{\mathcal L}({\mathcal W}_\alpha, {\mathcal W}^*_\alpha)\) be a continuous function and assume that there exist \(p\geq 0\) and a nonnegative function \(K\in L^1[0,T]\) such that NEWLINE\[NEWLINE\bigl|\widehat F(s,\Xi_1) (\xi, \eta) -\widehat F(s,\Xi_2) (\xi,\eta)2\bigr |^2\leq K(s)G_\omega \bigl( |\xi |^2_p\bigr) G_\omega\bigl( |\eta |^2_p \bigr) \bigl |\widehat \Xi_1(\xi, \eta)-\widehat\Xi_2(\xi,\eta) \bigr|^2,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\bigl|\widehat F(s,\Xi)(\xi,\eta) \bigr|^2\leq K(s)G_\omega \bigl(|\xi |^2_p \bigr)G_\omega \bigl(|\eta|^2_p \bigr) \biggl(1+ \bigl|\widehat\Xi (\xi,\eta) \bigr|^2 \biggr)NEWLINE\]NEWLINE for all \(\xi,\eta\in E_C\), \(\Xi\in {\mathcal L}({\mathcal W}_\omega, {\mathcal W}^*_\omega)\), and \(s\in [0,T]\). Then, for any \(\Xi_0\in {\mathcal L}({\mathcal W}_\omega, {\mathcal W}^*_\omega)\) the initial value problem (1) has a unique solution in \({\mathcal L}({\mathcal W}_\alpha, {\mathcal W}^*_\alpha)\).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].