Markov SemigroupWe consider the family of convolution operators on indexed by and given for every by The Semigroup has the Feller property, that is for every :

- for every ,
- (uniformly.)

Resolvent operator The family of linear operators associated with the L\’evy process, is called resolvent operators. The resolvent operators correspond to the Laplace transform of the semigroup . They are given for every measurable function by It is often more convenient to work with the resolvent operators than with the semigroup, thanks to the smoothing effect of the Laplace transform and to the lack of memory of exponential laws. Fourier Transform Absolute Continuity

Let (*X*, *d*) be a metric space and let *I* be an interval in the real line **R**. A function *f* : *I* → *X* is **absolutely continuous** on *I* if for every positive number , there is a positive number δ so that whenever a sequence of pairwise disjoint sub-intervals [*x*_{k}, *y*_{k}] of *I*, *k* = 1, 2, …, *n* satisfies

then

The collection of all absolutely continuous functions from *I* into *X* is denoted AC(*I*; *X*).

A further generalisation is the space AC^{p}(*I*; *X*) of curves *f* : *I* → *X* such that

for some *m* in the *L*^{p} space *L*^{p}(*I*; **R**). Potential measure It corresponds to the limit case for the -resolvent kernel. Specifically, we put for every , Transience and Recurrence

- We say that a L\’evy process is transient if the potential measures are Radon measures, that is , for every compact set K

- We say that a L\’evy process is recurrent if for every open ball centred at the origin.

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