  We study continuous processes indexed by a special family of graphs.
Processes indexed by vertices of graphs are known as probabilistic graphical
models. Burdzy and Pal in their paper proposed a continuous version of
graphical models -- processes indexed by time-like graphs. We construct a more
general family of continuous processes on a wider family of graphs, find new
properties and solve the conjecture of uniqueness of the distribution for the
process indexed by graphs with infinite number of vertices. We present new
results showing connections to the stochastic heat equation, Markov random
fields, martingales indexed by directed sets and branching Markov processes.
