Infinitesimal generator 無窮小生成元

Let $X$ be a Banach space. Let $Φ$ be a flow on $X$ . The infinitesimal generator associated with $Φ$ is defined by

G v := \partial_{t} Φ^{t} (v) |_{t = 0} = lim_{t ↘ 0} \frac{Φ^{t} (v) - v}{t}

with the domain

D (G) = {v \in X : lim_{t ↘ 0} \frac{Φ^{t} (v) - v}{t} exists in X} .

(c.f. NTHU MATH 526500 DDE Note week 2)

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