Markov SemigroupWe consider the family of convolution operators on L^{\infty}(\mathbb R^d) indexed by t \geq 0 and given for every f \in L^{\infty}(\mathbb R^d) byP_t f(x)=E_x(f(X_t))=\int_{R^d} f(x+y)\mathbb P(X_t\in dy) The Semigroup (P_t, t\geq 0) has the Feller property, that is for every f \in l_0 :

  • P_t f \in l_0 for every t\geq 0,
  • lim_{t\to 0} P_t f= f (uniformly.)

Resolvent operator The family of linear operators U^q(q>0) associated with the L\’evy process, is called resolvent operators. The resolvent operators correspond to the Laplace transform of the semigroup (P_t, t\geq 0). They are given for every measurable function f\geq 0 by U^q f(x)=\int_{o}^{\infty} e^{-qt} P_t f(x)dt=E_x(\int_{0}^{\infty}e^{-qt}f(X_t)dt)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\mathcal F g(\zeta)=\int_{R^d} e^{i\langle \zeta , x\rangle} g(x)dx, (\zeta \in \mathbb R^d). Absolute Continuity 

Let (Xd) 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 \varepsilon, there is a positive number δ so that whenever a sequence of pairwise disjoint sub-intervals [xkyk] of Ik = 1, 2, …, n satisfies

\sum_{k=1}^{n} \left| y_k - x_k \right| < \delta

then

\sum_{k=1}^{n} d \left( f(y_k), f(x_k) \right) < \varepsilon.

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

A further generalisation is the space ACp(IX) of curves f : I → X such that

d \left( f(s), f(t) \right) \leq \int_{s}^{t} m(\tau) \, \mathrm{d} \tau \mbox{ for all } [s, t] \subseteq I

for some m in the Lp space Lp(IR).  Potential measure It corresponds to the limit case q=0 for the q-resolvent kernel. Specifically, we put for every x\in \mathbb R^d, A\in \mathcal B(\mathbb R^d),U(x,A)=\int_{0}^{\infty}\mathbb P_x(X_t\in A)dt=E_x(\int_{0}^{\infty}1_{X_t\in A}dt) \in [0,\infty]. 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

U(x,K)<\infty, x\in \mathbb R^d

  •  We say that a L\’evy process is recurrent if U(0,B)=\infty for every open ball B centred at the origin. 
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