3-Characterizations of finite groups by Makhnev A. A.

By Makhnev A. A.

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3. 5) is not uniquely solvable in Cb (RN ). 4 The Markov process In this section we briefly consider the Markov process associated with the semigroup {T (t)} and we show the Dynkin formula. In the whole section we assume that c(x) ≤ 0, x ∈ RN . 1) We introduce a few notations. Let E be a topological space and let B be the σ-algebra of Borel subsets of E. Moreover, let Ω be an arbitrary set, F be a σ-algebra on it and τ : Ω → [0, +∞] be a F -measurable function. For any t ≥ 0, we denote by Ft a σ-algebra on the set Ωt = {ω : t < τ (ω)}, such that (Fs ) ⊂ Ft ⊂ F for any 0 < s < t.

4) i=1 where KM > 0 is such that N N ||qij Dij ϑ||L∞ (B(M)) + i,j=1 ||bi Di ϑ||L∞ (B(M)) ≤ KM , i=1 N ||qij Dj ϑ||L∞ (B(M)) ≤ KM , 2 i = 1, . . , N. 4. 16), the function u˜n satisfies the estimate √ un (t, x)| ≤ C||˜ un ||L∞ (D(M+1)) ≤ C(exp(c0 ) ∨ 1)||f ||∞ , | tD˜ for any x ∈ B(M ), any t < 1 = dist (B(M ), ∂B(M + 1)) and some positive constant C, independent of n. This yields ||Di u ˜n (t, ·)||L∞ (B(M)) ≤ t−1/2 C ′ ||f ||∞ , t ≤ 1, for any i = 1, . . , N , where C ′ = C(exp(c0 ) ∨ 1). 5) ′ for any n > M , where KM > 0 is a constant independent of n.

M) (M+1) Thus, the functions u∞ and u∞ coincide in the domain D′ (M ) and, 1+α/2,2+α therefore, we can define the function u ∈ Cloc ((0, +∞)×RN ) by setting ′ u = u(M) ∞ in D (M ). Moreover, the diagonal subsequence defined by u ˜n = u(n) n , n ∈ N, converges to u in C 1+β/2,2+β ([T1 , T2 ] × K) for any compact set K ⊂ RN and any 0 < T1 < T2 . Hence, letting n go to +∞ in the differential equation satisfied by u ˜n , it follows that u satisfies the equation Dt u(t, x) − Au(t, x) = 0, t > 0, x ∈ RN .

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