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In Section 3 we use this set up to prove Theorem 3.1 , which characterizes the boundaries of graphs of harmonic functions using the moment conditions arising from conservation laws..
The main objective in this section is to prove the following theorem concerning the conditional distribution of the number of edges given the colouring..
In section 4, we prove a converse to the comparison theorem for BSDEs and then, with the help of this result we study the case when the generators do not depend on the variable y...
Section 7 deals with an ap- plication to normal martingales, and in the appendix (Section 8 ) we prove the forward-backward Itˆ o type change of variable formula which is used in
In Section 3, we show that the birth-and-assassination process is the scaling limit, as n goes to infin- ity, of the rumor scotching process when G is the complete graph over n
Remark 24 Recall that to prove Proposition 8, we used Proposition 1 to establish the fact that if there is a coalescent process satisfying conditions B1 and B2 of Theorem 2, then
In Section 5.3 we make a cut-off for small times, showing that these times are negligible in the limit as κ → ∞ , perform a space-time scaling and compactification of the
Furthermore, Kesten and Zhang (1997) showed a central limit theorem for the passage time in the undirected critical model.. Here, we partially verify (1.14) for the directed
