On bounded solutions of a balanced pantograph equation

A functional-differential equation with rescaled argument of the form $y'(x) = a y(qx) + b y(x)$ ("pantograph equation") was introduced by J. Ockendon et al. (1971) in connection with the dynamics of a current collection system on an electric locomotive. Soon thereafter, T. Kato posed a problem of existence and characterization of bounded solutions of such equations. In the talk, this problem is addressed for a "balanced"' pantograph equation $y'(x)+y(x)=E [y(\alpha x)]$ with a random $\alpha>0$, and show that any bounded solution is constant if and only if $E [\ln\alpha] \le 0$. The result in the critical case $(E [\ln\alpha] = 0)$ settles a long-standing problem due to G. Derfel (1989). The proof exploits a link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with "multiplication" jumps. The talk is based on joint work with G. Derfel, S. Molchanov and J. Ockendon.