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The following is the definition of infinitesimal generator from Oksendal.

Let $\{X_t,t\in[0,T]\}$ be a time-homogeneous It\^o diffusion in $\mathbb{R}^d$. The $\textit{infinitesimal generator}$ $\mathscr{A}$ of $X_t$ is defined by $\mathscr{A}f(x)=\lim_{t\downarrow 0}\frac{\mathbb{E}_x[f(X_t)]-f(x)}{t},\,\,\,\,x\in\mathbb{R}^d$ The set of functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that the limit exists at $x$ is denoted by $\mathscr{D}_A(x)$, while $\mathscr{D}_A$ denotes the set of functions for which the limit exists for all $x\in\mathbb{R}^d$.

But there is another definition of infinitesimal generator defined by semi-group, which works on the both time-inhomogeneous and time-homogeneous process.

So are those two really equivalent to each other?

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I think I figure it out. Those definitions are equivalent because when we define it by semi-group, we also require that it should be independent of time t. Therefore it is implicitly assumed that the infinitesimal generator should only work on time-homogeneous Cauchy. But it is raised a question that do we have a "similar" generator which could help us deal with time-inhomogenous Cauchy? So does anyone know the second question? – MathGarfield Oct 11 '12 at 11:40
Maybe Hille-Yosida theorem can help. – Davide Giraudo Oct 11 '12 at 12:05
Could you be explicit about the definition in terms of a semi-group? A time-inhomogeneous process doesn't have a transition semigroup in quite the same sense as a time-homogeneous one so I want to be sure I know what you mean here. Also, what do you mean by "time-homogeneous Cauchy" in your comment? – Nate Eldredge Oct 11 '12 at 12:55
For a time-homogenous Markov process X_t, we define the infinitesimal generator as following: \mathscr{A}f(x)=\lim_{t\rightarrow 0}\frac{T_t f(x)-f(x)}{t}, where T_t is the semi-group associated with the transition probability function of X_t. For a time-inhomogenous Markov process, we define the infinitesimal generator by: \mathscr{A}_s f(x)=\lim_{t\rightarrow 0}\frac{T_{s,s+t}f(x)-f(x)}{t}, where T is the semi-group associated to the nonstationary transition probiblity function of X_t. – MathGarfield Oct 21 '12 at 13:01
the time-homogenous Cauchy in the comment means that the elliptic differential operater is independent of time. – MathGarfield Oct 21 '12 at 13:03

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