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We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no "really nice" set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.

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We present a satisfactory definition of the important class of Lévy processes indexed by a general collection of sets. We use a new definition for increment stationarity of set-indexed processes to obtain different characterizations of this class. As an example, the set-indexed compound Poisson process is introduced. The set-indexed Lévy process is characterized by infinitely divisible laws and a Lévy-Khintchine representation. Moreover, the following concepts are discussed: projections on flows, Markov properties, and pointwise continuity. Finally the study of sample paths leads to a Lévy-Itô decomposition. As a corollary, the semimartingale property is proved.

We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.There are several different approaches to the fundamental concept of Fractional Poisson process (FPP) on the real line. The "renewal" definition extends the characterization of the Poisson process as a sum of independent non-negative exponential random variables. If one changes the law of interarrival times to the Mittag-Leffler distribution (see [32,33,44]), the FPP arises. A second approach is given in [6], where the renewal approach to the Fractional Poisson process is developed and it is proved that its one-dimensional distributions coincide with the solution to fractionalized state probabilities. In [34] it is shown that a kind of Fractional Poisson process can be constructed by using an "inverse subordinator", which leads to a further approach.In [26], following this last method, the FPP is generalized and defined afresh, obtaining a Fractional Poisson random field (FPRF) parametrized by points of the Euclidean space R 2 + , in the same spirit it has been done before for Fractional Brownian fields, see, e.g., [17,20,22,30].The starting point of our extension will be the set-indexed Poisson process which is a well-known concept, see, e.g., [17,22,37,38,47].In this paper, we first present a martingale characterization of the Fractional Poisson process. We extend this characterization to FPRF using the concept of increasing path and strong martingales. This characterization permits us to give a definition of a set-indexed Fractional Poisson process. We study the fractional differential equation for FPRF. Finally, we study Mixed-Fractional Poisson processes.The paper is organized as follows. In the next section, we collect some known results from the theory of subordinators and inverse subordinators, see [8,36,49,50] among others. In Section 2, we prove a martingale characterization of the FPP, which is a generalization of the Watanabe Theorem. In Section 3, another generalization called "Mixed-Fractional Poisson process" is introduced and some distributional properties are studied as well as Watanabe characterization is given. Section 4 is devoted to FPRF. We begin by computing covariance for this process, then we give some characterizations using increasing paths and intensities. We present a Gergely-Yeshow characterization § N.