The relatively large-amplitude decaying regime of transverse oscillations of coronal loops has been known for two decades and interpreted in terms of MHD kink modes of cylindrical plasma waveguides. Recent observational analysis has revealed the so-called decay-less small-amplitude oscillations, with a multi-harmonic structure being detected. Several models have been proposed to explain them. In particular, decay-less oscillations have been described in terms of standing kink waves driven with continuous monoperiodic motions of loop footpoints, in terms of a simple oscillator model of forced oscillations due to harmonic external force, and as a self-oscillatory process due to the interaction of a loop with quasi-steady flows. However, an alternative mechanism is needed to explain the simultaneous excitation of several longitudinal harmonics of the oscillation. We study the mechanism of random excitation of decay-less transverse oscillations of coronal loops. With a spatially one-dimensional and time-dependent analytical model taking into account effects of the wave damping and kink speed variation along the loop, we consider transverse loop oscillations driven by random motions of footpoints. The footpoint motions are modelled by broad-band coloured noise. We have found the excitation of loop eigenmodes and analysed their frequency ratios as well as the spatial structure of the oscillations along the loop. The obtained results successfully reproduce the observed properties of decay-less oscillations. In particular, excitation of eigenmodes of a loop as a resonator can explain the observed quasi-monochromatic nature of decay-less oscillations and generation of multiple harmonics detected recently. We propose the mechanism that can interpret decay-less transverse oscillations of coronal loops in terms of kink waves randomly driven at the loop footpoints.
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