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Basic Theory of Hilbert \(C^*\) -Modules
This chapter covers the basic concepts and methods of Hilbert \(C^*\)...
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Quantum Markov Semigroups Based on Hilbert \(C^*\) -Modules
In this chapter, we introduce the concept of module operator semigroup and give characterizations. Then the classical solution and mild solution of...
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Kasprove’s Stabilization and Fredholm Generalized Index Theory
In this chapter, we show that every countably generated Hilbert A-module can be regarded as a complemented submodule of...
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Preliminaries
Herewe briefly reviewseveral previously obtained results – mostly when the variance is infinite – that are used or closely related to the topics...
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Introduction
At the time when Spitzer introduced 𝑎(𝑥), several works appeared which studied problems closely related to or involving the potential function,...
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Bounds of the Potential Function
In this chapter, we study asymptotic properties of the potential function 𝑎(𝑥) of a recurrent r.w. under our basic setting. We obtain a lower bound...
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Moments
We saw in Sect. 5.3 the claim that the asymptotic moments can be derived by differentiating the...
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Zuverlässigkeitstheorie und technische Systeme
Unter Zuverlässigkeit (engl. Reliability) versteht man in der Technik die Eignung eines Systems, während einer gewissen Zeitspanne die an es...
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Anhang A
Die stochastische Matrix lautet
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Pál–Bell Equation and Moment Growth
The previous chapter largely dealt with the relationship between the NTE and the NBP. The NTE is a linear equation and so there is limited...
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Some Background Markov Process Theory
Before we embark on our journey to explore the NTE in a stochastic context, we need to lay out some core theory of Markov processes that will appear...
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The Two-Sided Exit Problem for Relatively Stable Walks
This chapter is a continuation of Chapter 6. We use the same notation as therein. As in Chapter 6, we shall be primarily concerned with the event
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The Two-Sided Exit Problem – General Case
Remark 6.1.6 For general random walks (not restricted to arithmetic ones), Kesten and Maller [47] gave an analytic equivalent for the r.w. S to exit...
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Asymptotically Stable Random Walks Killed Upon Hitting a Finite Set
This section is concerned with the potential function for the r.w. S killed upon hitting a finite set. For its description, we do not need (AS). The...
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Martingale Convergence and Laws of Large Numbers
As usual, we will work in the setting that our branching Markov process,...
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Spines and Skeletons
We have seen in Chap. 6 that a natural way to study the long-term behaviour of the NBP is via spine...
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Generational Evolution
The eigenvalue problem in Theorem 4.1 is not the only one that offers insight into the evolution of...
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Survival at Criticality
We will remain in the setting of the Asmussen–Hering class of BMPs, i.e., assuming (G2), and insist throughout this chapter that we are in the...
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Martingales and Path Decompositions
In this chapter, we use the Perron–Frobenius decomposition of the NBP to show the existence of an intrinsic family of martingales. These are...
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Many-to-One, Perron–Frobenius and Criticality
Now that the precise mathematical relationship between the NTE and the NBP is clear, we now look at how we can profit from this. The first port of...