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When dealing with markov chains, one often needs to look at the semigroup associated to it:

$$ T(t)f (x) = \Bbb{E}^{\Bbb{P_x}}\big[f(X_t)\big]$$

More precisely, these semigroups are probability semigroups, That is, they satisfy (see Liggett - Continuous time Markov processes pg 93):

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There is a one to one correspondence between Feller processes and probability Semigroups.

The Hille Yosida theorem gives a one to one correspondence between probability semigroups and probability generators.

The generator is the time derivative of the semigroup.

Often it is a second order operator.

according to liggett (pg 98)

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it need not be a second order, it can be $L f = f'$ the generator associated with the process $X_t = x_0 + t$.

Is there an example of a probability generator that is of higher order? say, a 4th order differential operator?

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The generator $\mathcal L$ satisfies a maximum principle: If $f$ (in the domain of $\mathcal L$) attains its global maximum at $x_0$, then $\mathcal L f(x_0)\le 0$. This rules out higher-than-second-order differential operators. For example, $f(x)=\cos x$ attains its maximum value of $1$ at $x_0=0$, but $f^{(4)}(0) =\cos(0)=1>0$.

The maximum principle follows from the formula $\mathcal Lf(x)=\lim_{t\downarrow 0}[T(t)f(x)-f(x)]/t$, because if $f$ attains its maximum value at $x_0$ then $T(t)f(x_0)=\Bbb E^{x_0}[f(X_t)]\le\Bbb E^{x_0}[f(x_0)]=f(x_0)$.

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