Evolution of measure governed by differential equations (Q2706033)

The author investigates the initial problem \(u' (t)=F(t,u(t))\), \(u(0)=a\). Every solution \(u: t\to u(t)\), \(t\in [0,T]\), to the above stated problem is called an individual solution which may be regarded as the time-evolution of \(a\). This problem is generalized here in the following way. If \(X\) is a Banach space of all admissible initial conditions endowed with the probabilistic measure \(\mu_0\) then the time evolution of \(\mu_0\) imposed by the considered differential equation leads to a family of probabilistic measures \(\{\mu (t,.)\}_{t\in [0,T]}\) which may be interpreted as a statistical solution to the same equation with the initial condition \(\mu_0\). The author introduces a theory of statistical solutions to the considered differential equations in Banach space without uniqueness of the individual solution.