Thesis title: Non-local Boundary Value Problems for Brownian motions
In the present thesis, we study non-local boundary behavior for the Brownian motion. In particular, we consider heat equations equipped with boundary conditions written in terms of non-local operators, covering both dynamic and spatial boundary conditions. We refer to the non-local boundary value problem (NLBVP) and we stress the fact that the literature seems to be lacking. Conversely, the non-local initial value problem also known as non-local Cauchy problem (NLCP) has been extensively investigated for decades. The results presented here give a first glance to a new scenario involving NLBVPs in bounded domains and therefore non-local operators on lower dimensional spaces with completely different (boundary) evolutions. Due to the pioneering nature of our analysis we have directed our efforts to the special case of 1-dimensional motions. Then, as a direct application of such results, we construct and study equivalent motions on graphs.
In the first part, we deal with Brownian motion on the positive half-line, where non-local operators emerge both in time and space at zero. We derive the probabilistic representation of the solution by introducing some transformation of the Brownian motion in terms of an additive part and a time change. The behavior near the boundary point zero turns out to be given by an instantaneous jump with a subsequent slowdown outside zero. We characterize the non-local operators, the stochastic dynamics and the asymptotic behavior leading to some known versions of Brownian motion.
The previous results suggest a construction for the stochastic solution to transmission problems. We consider transmission and Feller-Wentzell conditions with the probabilistic reading in terms of skew and sticky Brownian motions. We also pay special attention to the non-local operator associated with the Lévy measure of the so-called gamma subordinator. Specifically, for the gamma processes, we rewrite the governing equations and provide a representation of the real moments of its inverses through Volterra functions.
We conclude the investigation of non-local conditions by introducing metric graphs and Brownian motions on star graphs. Since each edge is isomorphic to the positive half-line, we are able to extend the previous results to such graph structures.