Solutions of a class of nonlinear master equations

Balancing Queues by Mean Field Interaction

“…First of all, similar to the proof of Lemma 3.1 in [10] we have that Lemma 3. For any u ∈ P(E), every q-solution of (13) is a solution of the martingale problem defined by Definition 2.…”

“…Although the infinitesimal generator defined by (12) is different from that in [10], the proof is similar.…”

“…If "h(i, g(t))" in (12) is replaced by "−g(t)", then Theorem 2 has been proved in [10] (see also [15] and [12]), where Q = (q i j ) is not necessarily triagular. Although the infinitesimal generator defined by (12) is different from that in [10], the proof is similar.…”

“…The third step is to prove that the limit of P n u is exactly as we wanted it. The detailed procedure of the proof is similar to that of Theorem 1.1 and its Corollary in [10]. The key point is that for every T > 0,…”

“…Based on the estimates in Lemma 4, we can prove (21) by a method similar to that used to prove Lemma 2.3 in [10].…”