By Tanizaki H.

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The uniformly elliptic case and we can split it as f + (y)Gk (t, x, y)dy u(t, x) = lim k→+∞ RN f − (y)Gk (t, x, y)dy. 6). 5), using the monotone convergence theorem, it follows that G(t + s, x, y) = G(s, x, z)G(t, z, y)dz, t, s > 0, x, y ∈ RN . 6). It remains to prove that, if c0 ≤ 0, {p(t, x; dy) : t ≥ 0, x ∈ RN } is a stochastically continuous transition function. 10), and the condition (ii) is obvious. 11) once we have proved that the function p(t, ·; B) is Borel measurable in RN , for any fixed t ≥ 0 and any B ∈ B(RN ).

5, we deduce that, for a general f ∈ Cb (RN ), T (t)f converges to f as t tends to 0, locally uniformly in RN . Actually, as next proposition shows, if f vanishes at infinity, then T (t)f converges to f in Cb (RN ), as t tends to 0. 18 Chapter 2. 7 ([116], Prop. 3) For any function f ∈ C0 (RN ), T (t)f tends to f in Cb (RN ), as t tends to 0+ . Proof. We prove the statement assuming that f ∈ Cc∞ (RN ). The general case then will follow by density. So, let us fix f ∈ Cc∞ (RN ) and x ∈ RN . Moreover, let k ∈ N be such that B(k) contains both x and supp(f ).

3). Using the classical maximum principle we prove that the sequence {Kλn } is increasing (with respect to n ∈ N). 4) with Kλ (x, y) := lim Kλn (x, y), n→+∞ x, y ∈ RN . Thus for any λ > c0 we can define the linear operator R(λ) in Cb (RN ) by setting (R(λ)f )(x) = Kλ (x, y)f (y)dy, RN x ∈ RN . 0. Introduction R(λ) is a bounded operator, with ||R(λ)||L(Cb (RN )) ≤ (λ − c0 )−1 . 3 we see that the operators {R(λ) : λ > c0 } are the resolvent operators of a linear operator A in Cb (RN ). The operator A is called weak generator.