Week 17.11.2025 – 23.11.2025

The support propagation phenomenology of heat equations with multiplicative (white Gaussian) noise is well understood. If the noise coefficient is p-Hölder with p<1 at 0, the supports of non-negative solutions propagate with finite speed and hence are compact; if the noise coefficient is Lipschitz at 0, solutions are strictly positive, like those of the deterministic heat equation. After introducing this behaviour, I will first discuss its recent extension to stable, i.e. non-Gaussian, noises.

Next, we consider support propagation when the diffusion is replaced by a discontinuous motion. In this case, results from superprocess theory imply that at p = 1/2, solutions have unbounded supports. We prove that this is essentially sharp for a class of non-local stochastic evolution equations. In particular, if the noise coefficient is p-Hölder for p<1/2, the support of the solution is compact at almost all times. To prove this, we show that the process given by the solution integrated over certain half space spends most of its time at zero. This is done using an excursion decomposition of the local time at 0 of the (non-Markovian) spatially integrated process.

This talk includes joint ongoing work with Marcel Ortgiese.