By Grenander U.

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10). 3. 1. P-matrices and N-matrices. First, we prepare some algebraic and analytic tools. Suppose that a symmetric n × n matrix (aij ) satisfies the condition: for all real numbers t1 , . . , tn, n aij ti tj ≥ 0. i,j=1 In algebra, such matrices are called positive semidefinite. , [BJR84]) use the name positive definite. We resolve this controversy by using a short name a P-matrix. 50 3. BRANCHING EXIT MARKOV SYSTEMS We need another class of matrices which are called negative definite in [BJR84].

4). It is called the Perron solution corresponding to f. 1. If Q is a bounded open set and f is a bounded Borel function on ∂Q, then u = KQ f is the Perron solution corresponding to f. Proof. Let w ∈ U+ and let τ be the first exit time from Q. Consider a sequence Qn exhausting Q and the corresponding first exit times τn . 9) and Fatou’s lemma, w(z) ≥ lim inf Πz w(ητn ) ≥ Πz lim inf w(ητn ) ≥ Πz f(ητ ). Similarly, if v ∈ U− , then v(z) ≤ Πz f(ητ ). 1. There only the case L = ∆ is considered but the arguments can be modified to cover a general L.

3. 1. Suppose that V k are operators from B to B, V k → V and V satisfies the Lipschitz condition on each Bc. Suppose that V˜ k are operators from u Bn to B, V˜ k → V˜ and V˜ satisfies the Lipschitz condition on each Bn c . Then u V k V˜ k → V V˜ and V V˜ satisfies the Lipschitz condition on each Bn. c Proof. 19) V k (f) − V (f) ≤ εk (c) V˜ k (f˜) − V˜ (f˜) ≤ ε˜k (c) for f ∈ Bc, for f˜ ∈ Bn c with εk (c) + ε˜k (c) → 0 as k → ∞. 20) V (f) − V (g) ≤ a(c) f − g V˜ (f˜) − V˜ (˜ g) ≤ a ˜(c) f˜ − g˜ for all f, g ∈ Bc, ˜ g˜ ∈ Bn.