Lőrinczi József: A probabilistic analysis of the location of global maxima of solutions to some non-local equations in bounded domains

Using a family of singular integral operators, which are infinitesimal generators of Lévy processes, I will consider related non-local (integro-differential) equations in convex bounded domains under Dirichlet exterior condition and derive a lower bound on the location of global extrema of their solutions from the boundary of the domain. As it will be seen, this bound has an impact on various spectral geometric and variational quantities in terms of mean survival times of the process. The main tool of this analysis will be a stochastic representation of the solutions by running suitable jump processes and averaging over their paths.