Abstract:
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.
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