SPDEs on graphs: an asymptotic approach - Lecture 2

I will introduce a new class of SPDEs defined on graphs, obtained as the limit of suitable SPDEs, defined on two-dimensional domains and depending on some parameters. I will do this presenting two examples. The first example is given by some SPDEs defined on narrow channels with wings. As the width of the channel goes to zero the solutions converge to the solution of a suitable SPDE defined on the graph that can be obtained by identifying all points on the same cross section of the tubular domain. The second example is given by the analysis of the fast advection asymptotics for some stochastic reaction-diffusion-advection equations defined on the plane. To describe the asymptotics, I will consider a suitable class of SPDEs defined on a graph, corresponding to the stream function of the underlying incompressible flow.

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