In this talk, we start by giving a short Introduction on Walsh’s spider diffusions living on a star-shaped network. We give the main properties of this type of processes, focusing especially on their local time at the vertex. Thereafter, we will give the main ideas, steps and mathematical directions that I have recently undertaken in order to finally be able to prove existence and weak uniqueness of a spider's motion having a spinning measure depending on its own local time.

We will also talk on how the uniqueness issue was solved with the aid of the well-posedness of the corresponding linear parabolic operator. This operator describes general heat equations posed on graphs, having a new class of boundary transmission at the vertices, called: local-time Kirchhoff's boundary condition.

Finally, we will show how these latest results were the seeds of obtaining a comparison theorem for fully nonlinear HJB equations posed on graphs with discontinuities, answering at the same time to a Lions-Souganidis problem that dates back to a few years ago.

In this talk, I will present the transition probability of TASEP in half-space starting with general deterministic initial condition. I will show how properties of half-space TASEP are manifested on the formula. Taking the 1:2:3 KPZ scaling, I will present the transition probability of the half-space KPZ fixed point.