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Highly concentrated functions play an important role in many research fields including control system analysis and physics, and they turned out to be the key idea behind inverse Laplace transform methods as well.This paper uses the matrix-exponential family of functions to create highly concentrated functions, whose squared coefficient of variation (SCV) is very low. In the field of stochastic modeling, matrix-exponential functions have been used for decades. They have many advantages: they are easy to manipulate, always non-negative, and integrals involving matrix-exponential functions often have closed-form solutions. For the time being there is no symbolic construction available to obtain the most concentrated matrix-exponential functions, and the numerical optimization-based approach has many pitfalls, too.In this paper, we present a numerical optimization-based procedure to construct highly concentrated matrix-exponential functions. To make the objective function explicit and easy to evaluate we introduce and use a new representation called hyper-trigonometric representation. This representation makes it possible to achieve very low SCV.

The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit et al. (Phys Rev B 27:1635-1645, 1983). This is a nearest neighbor non-Markovian random walk in Z d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (Probab Theory Relat Fields 92:337-349, 1992) (independently of the physics community), is the continuous space-time counterpart: a diffusion in R d pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a path-wise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on nonrigorous renormalization group arguments (equally 'valid' for the TSAW and SRBP 123 692 I. Horváth et al. Amit et al. (1983). The proof of the CLT follows the non-reversible version of Kipnis-Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition. cases), in

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Abstract. In this paper we extend the mean-field limit of a class of stochastic models with exponential and deterministic delays to include exponential and generally-distributed delays. Our main focus is the rigorous proof of the mean-field limit.