Existence of analytic solutions of linear differential equations for measures on a Hilbert space
A Cauchy problem is considered for a linear differential equation in an unknown measure \(\mu\) on a Hilbert space: \[ (1)\quad \partial^ n_ t\mu (t,A)=\sum^{n}_{i=0}L_ i(\mu^{(i)}(t,A))+{\bar \mu}(t,A), \] \[ (2)\quad \partial_ t^{\ell}\mu (t,A)|_{t=0}={\bar \mu}_{\ell}(A),\quad 0\leq \ell \leq n-1, \] where \(L_ i\) (1\(\leq i\leq n)\) is a linear differential operator in t of order n-i and \(L_ 0\) is a linear differential operator in t of order n-1. The paper proves an analogue of Kovalevsky's theorem for the Cauchy problem (1)-(2). Theorem 1. Assume that the coefficients of the operators \(L_ i\) are analytic with neighborhood w and with respect to the parameter on \(S=[- t_ 0,t_ 0]\), where \(t_ 0>0\) and the measures \(\mu\) and \({\bar \mu}{}_{\ell}\), where \(0\leq \ell \leq n-1\), are analytic with neighborhood W and in the parameter on \([-t_ 0,t_ 0]\). There exists then a unique solution in the class of analytic measures of problem (1)- (2) defined with respect to the parameter in a certain neighborhood of zero.
linear differential equation in an unknown measure
scientific article; zbMATH DE number 4204847