A significant fraction of main-sequence stars are encircled by dusty debris discs, where the short-lived dust particles are replenished through collisions between planetesimals. Most destructive collisions occur when the orbits of smaller bodies are dynamically stirred up, either by the gravitational effect of locally formed Pluto-sized planetesimals (self-stirring scenario), or via secular perturbation caused by an inner giant planet (planetary stirring). The relative importance of these scenarios in debris systems is unknown. Here we present new Herschel Space Observatory imagery of 11 discs selected from the most massive and extended known debris systems. All discs were found to be extended at far-infrared wavelengths, five of them being resolved for the first time. We evaluated the feasibility of the self-stirring scenario by comparing the measured disc sizes with the predictions of the model calculated for the ages of our targets. We concluded that the self-stirring explanation works for seven discs. However, in four cases, the predicted pace of outward propagation of the stirring front, assuming reasonable initial disc masses, was far too low to explain the radial extent of the cold dust. Therefore, for HD 9672, HD 16743, HD 21997, and HD 95086, another explanation is needed. We performed a similar analysis for {ss} Pic and HR 8799, reaching the same conclusion. We argue that planetary stirring is a promising possibility to explain the disk properties in these systems. In HR 8799 and HD 95086 we may already know the potential perturber, since their known outer giant planets could be responsible for the stirring process. Our study demonstrates that among the largest and most massive debris discs self-stirring may not be the only active scenario, and potentially planetary stirring is responsible for destructive collisions and debris dust production in a number of systems.
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