By Lucia M.

We examine the equationIf Ω is of sophistication , we exhibit that this challenge has a non-trivial answer u λ for every λ ∊ (8π, λ*). the price λ* relies on the area and is bounded from lower than via 2 j zero 2 π, the place j zero is the 1st 0 of the Bessel functionality of the 1st form of order 0 (λ*≥ 2 j zero 2 π > eight π). additionally, the family members of resolution u λ blows-up as λ → eight π.

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We will to show in this exercise that there are significant differences between the cases when γ < 1 and γ = 1. (a) Show that E[Xn | X0 = i] = iγ n . Hence, when γ < 1, the expected size of the population goes to 0 at an exponential rate. On the other hand, when γ = 1, the expected size remains constant, this in spite of the fact that as n → ∞ P(Xn = 0|X0 = i) −→ 1. 11. In the present case, the explanation is simple: as n → ∞, with large probability Xn = 0 but, nonetheless, with positive probability Xn is enormous.

Now suppose that i→j but P(ρj < ρi |X0 = i) = 0. 1 Classification of States 51 (m) P(ρi < ρj |X0 = i) = 1. 3), this means that P(ρi < ρj |X0 = i) = 1 for all m ≥ 1, which, since ρ (m) ≥ m, leads to P(ρj = ∞|X0 = i) = 1 and therefore rules out i→j . Hence, we have already shown that i→j =⇒ P(ρj < ρi |X0 = i) > 0, and the opposite implication needs no comment. To prove that i→j =⇒ P(ρi < ∞|X0 = j ) = 1, first observe that P(ρj < ρi < ∞ | X0 = i) = lim n→∞ = lim n→∞ = lim n→∞ ∞ m=1 ∞ m=1 ∞ m=1 P(ρj = m < ρi ≤ m + n | X0 = i) E 1 − Fn,i (Xm , .

2 Return Times As the contents of Sects. 2 already indicate, return times ought to play an important role in the analysis of the long time behavior of Markov chains. In particu(0) (m) lar, if ρj ≡ 0 and, for m ≥ 1, the time of mth return to j is defined so that ρj = ∞ (m−1) if ρj (m) = ∞ or Xn ̸= j for every n > ρ (m−1) and ρj Xn = j } otherwise, then we say that j is recurrent if (1) P(ρj P(ρj(1) (m−1) = inf{n > ρj : < ∞|X0 = j ) = 1 < ∞|X0 = j ) < 1; and we can hope that when j and that it is transient if is recurrent, then the history of the chain breaks into epochs which are punctuated by the successive returns to j .

### A blowing-up branch of solutions for a mean field equation by Lucia M.

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