  The main purpose of this work is to provide an existence and uniqueness
result for the solution of a linear parabolic system posed on a star-shaped
network, which presents a new type of Kirchhoff's boundary transmission
condition at the junction. This new type of Kirchhoff's condition-that we
decide to call here local-time Kirchhoff 's condition-induces a dynamical
behavior with respect to an external variable that may be interpreted as a
local time parameter, designed to drive the system only at the singular point
of the network. The seeds of this study point towards a forthcoming theoretical
inquiry of a particular generalization of Walsh's random spider motions, whose
spinning measures would select the available directions according to the local
time of the motion at the junction of the network.
