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2 We note that the maximum principle implies that, if f ≥ 0, the sequence {un } is positive and increasing. Therefore, u is positive and the whole sequence {un } converges to u. 1, corresponding, respectively, to the data f + and f − . 5). Indeed, if v is another positive solution, the maximum principle yields v(t, x) ≥ un (t, x) for any t > 0, any x ∈ B(n) and any n ∈ N, and, eventually, v ≥ u. 5) is not uniquely solvable in 1+α/2,2+α Cb ([0, +∞) × RN ) ∩ Cloc ((0, +∞) × RN ). 5) associated with the one-dimensional operator Au = u′′ +x3 u′ admits a nontrivial solution u satisfying u(0, ·) = 0.

2) holds. 2]. In particular, as far as the semigroup {T (t)} is concerned, we have the following result. 3 There exists a continuous Markov process X associated with the semigroup {T (t)}. 5) and τ (R(λ)f )(x) = E x e−λs f (Xs )ds, 0 for any f ∈ Bb (RN ). Proof. 5). 3]. The continuity of X is proved in [10]. 2). 4. The Markov process extended, first, to any simple function f and, then, to any f ∈ Bb (RN ), by approximating with simple functions. 4), applying the Fubini theorem. 6) and we denote by X U the process induced by X in U , that is Xt , ∞, XtU = t < τU , t ≥ τU , and we recall the following result (see [10]).

3]. The continuity of X is proved in [10]. 2). 4. The Markov process extended, first, to any simple function f and, then, to any f ∈ Bb (RN ), by approximating with simple functions. 4), applying the Fubini theorem. 6) and we denote by X U the process induced by X in U , that is Xt , ∞, XtU = t < τU , t ≥ τU , and we recall the following result (see [10]). 4 Let U ⊂ RN be a regular bounded domain. Then X U is the Markov process associated with the semigroup {T U (t)}. 1]. 5 Let U ∈ RN be a regular bounded domain, let τ ′ ≤ τU be a 2,p Markov time, and let λ ≥ 0.

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5-reflectionality of anisotropic orthogonal groups over valuation rings by Knuppel F.


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