By Janos Englander, Brian Rider

Markov tactics and their functions to partial differential equations Kuznetsov's contributions.- Stochastic equations on projective platforms of groups.- Modeling festival among influenza strains.- Asymptotic effects for close to severe Bienaym\'e-Galton-Watson and Catalyst-Reactant Branching Processes.- a few course huge deviation effects for a branching diffusion.- Longtime habit for at the same time Catalytic Branching.- Super-Brownian movement: Lp-convergence of martingales throughout the pathwise backbone decomposition

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If n ∏ lim lim det m→∞ n→∞ =m G ρ ◦ φk (z) μ (dz) >0 for all irreducible representations ρ of Gk for all k ∈ N, then Pμ strong n lim lim m→∞ n→∞ ∏ =m G = 0. / If ρ ◦ φk (z) μ (dz)) = 0 for some irreducible representation ρ of Gk for some k ∈ N, then Pμ strong = 0. / Under a further assumption, we get a representation theoretic necessary and sufficient condition for the existence of strong solutions. 9. A Borel probability measure ν on a compact Hausdorff group Γ is conjugation invariant if Γ f (g−1 xg) ν (dx) = Γ f (x) ν (dx) for all g ∈ Γ and bounded Borel functions f : Γ → R.

44 A. Budhiraja and D. Reinhold When k = 1, the transition probabilities of a BGW process {Z p } can be written as P(Z p+1 = j|Z p = i) = μ ∗i ( j) if i ≥ 1, δ0 j if i = 0, j ≥ 0, j ≥ 0, (1) where {μ (l)}l∈N0 is the offspring distribution of a typical particle and {μ ∗i (l)}l∈N0 is the i-fold convolution of { μ (l)}l∈N0 . The process starts with Z0 particles; each of the Z p particles alive at time p lives for one unit of time and then dies, giving rise to l offspring particles with probability μ (l), l ∈ N0 .

For models with immigration, we will prove convergence of stationary distributions. We begin by describing results for models without immigration. Let S be a subset of Rk+ , for some k ∈ N. When S is endowed with a topology, we will denote by B(S) the σ -field generated by the open sets of S. Let Y ≡ {Yt }t∈R+ be an S-valued Markov process such that 0 ∈ S is an absorbing state. If Y0 = y, we write P(Yt ∈ ·) as Py (Yt ∈ ·). Similarly, when the distribution of Y0 is μ , we write P(Yt ∈ ·) as Pμ (Yt ∈ ·).

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