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The main purpose of this work is to obtain a comparison principle forviscosity solutions of a system of elliptic Walsh's spiderHamilton-Jacobi-Bellman equations, possessing a new boundary condition callednon linear local-time Kirchhoff's transmission. The main idea is to build testfunctions at the neighborhood of the vertex solutions of ODE, withwell-designed coefficients. The key point is to impose a 'local-time'derivative at the vertex absorbing the error term induced by - what we decideto call here - the Kirchhoff's speed of the Hamiltonians.

Comparison principle for Walsh's spider HJB equations with non linear local time Kirchhoff's boundary transmission