Abstract Cauchy Problems in separable Banach Spaces driven by random Measures: Asymptotic Results in the finite extinction Case

2017/10/04 by Nerlich, Alexander
#35B40 #47J35 #60F05 #60F15 #60H15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.01795

Abstract

The aim of this paper is to prove the strong law of large numbers (SLLN) as well as the central limit theorem (CLT) for a class of vector-valued stochastic processes which arise as solutions of the stochastic evolution inclusion η(t,z) N_Θ(dt ⊗ z)∈ dX(t)+A X(t)dt, where A is a multi-valued operator and N_Θ is the counting measure induced by a point process Θ. The SLLN and the CLT will be proven not only for real-valued, but also for vector-valued functionals and the applicability of these results to the (weighted) p-Laplacian evolution equation (for "small" p) will be demonstrated. The key assumption needed in this paper is that the nonlinear semigroup arising from the multi-valued operator A extincts in finite time.

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