On Bounded Positive Stationary Solutions for a Nonlocal Fisher-KPP Equation

We study the existence of stationary solutions for a nonlocal version of the Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) equation. The main motivation is a recent study by Berestycki et {al.} [Nonlinearity 22 (2009), {pp.}~2813--2844] where the nonlocal FKPP equation has been studied and it was shown for the spatial domain $\mathbb{R}$ andsufficiently small nonlocality that there are only two bounded non-negative stationary solutions. Here we provide a similar result for $\mathbb{R}^d$ using a completely different approach. In particular, an abstract perturbation argument is used in suitable weighted Sobolev spaces. One aim of the alternative strategy is that it can eventually be generalized to obtain persistence results for hyperbolic invariant sets for other nonlocal evolution equations on unbounded domains with small nonlocality, {i.e.}, to improve our understanding in applications when a small nonlocal influence alters the dynamics and when it does not.Comment: 24 pages, 1 figure; revised versio

Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates

We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller type such as the fractal Burgers equation. The existence of traveling wave solutions with monotone decreasing profile has been established recently (in special cases). We show the local asymptotic stability of these traveling wave solutions in a Sobolev space setting by constructing a Lyapunov functional. Most importantly, we derive the algebraic-in-time decay of the norm of such perturbations with explicit algebraic-in-time decay rates

Traveling waves for a bistable equation with nonlocal-diffusion

We consider a single component reaction-diffusion equation in one dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. Our main result shows the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. In particular, we provide an extension to classical results on traveling wave solutions involving local diffusion. This extension to evolution equations with Riesz-Feller operators requires several technical steps. These steps are based upon an integral representation for Riesz-Feller operators, a comparison principle, regularity theory for space-fractional diffusion equations, and control of the far-field behavior

Addendum to "Travelling waves for a non-local Korteweg-de Vries-Burgers equation" [J. Differential Equations 257 (2014), no. 3, 720--758]

We add a theorem to [J. Differential Equations 257 (2014), no. 3, 720--758] by F. Achleitner, C.M. Cuesta and S. Hittmeir. In that paper we studied travelling wave solutions of a Korteweg-de Vries-Burgers type equation with a non-local diffusion term. In particular, the proof of existence and uniqueness of these waves relies on the assumption that the exponentially decaying functions are the only bounded solutions of the linearised equation. In this addendum we prove this assumption and thus close the existence and uniqueness proof of travelling wave solutions

Large-time behavior in non-symmetric Fokker-Planck equations

We consider three classes of linear non-symmetric Fokker-Planck equations having a unique steady state and establish exponential convergence of solutions towards the steady state with explicit (estimates of) decay rates. First, "hypocoercive" Fokker-Planck equations are degenerate parabolic equations such that the entropy method to study large-time behavior of solutions has to be modified. We review a recent modified entropy method (for non-symmetric Fokker-Planck equations with drift terms that are linear in the position variable). Second, kinetic Fokker-Planck equations with non-quadratic potentials are another example of non-symmetric Fokker-Planck equations. Their drift term is nonlinear in the position variable. In case of potentials with bounded second-order derivatives, the modified entropy method allows to prove exponential convergence of solutions to the steady state. In this application of the modified entropy method symmetric positive definite matrices solving a matrix inequality are needed. We determine all such matrices achieving the optimal decay rate in the modified entropy method. In this way we prove the optimality of previous results. Third, we discuss the spectral properties of Fokker-Planck operators perturbed with convolution operators. For the corresponding Fokker-Planck equation we show existence and uniqueness of a stationary solution. Then, exponential convergence of all solutions towards the stationary solution is proven with an uniform rate

Private equity funds and hedge funds: a primer

Private equity funds and hedge funds are both alternative asset classes that are continuously growing in importance. Although they have different focuses, they share some characteristics. First of all, both have or allegedly have a significant impact on the economy as well as the financial system they operate in. Therefore, the question of a potential regulation of both asset classes arises. Due to the lack of sophisticated knowledge about the differences of these asset classes, market players fear that attempts to regulate hedge funds will adversely affect private equity funds. Besides the regulatory issue, there are several other links between these two asset classes that have to be looked at. The relationship between those two asset classes is therefore of general importance. Last months' developments in the hedge fund industry (e.g. rumors about turbulences as well as hedge funds forcing the dismissal of the CEO of Deutsche Börse) have now even led to a broad public debate about private equity and hedge funds. At least in Germany the debate has been partly fueled by the fact that both types of funds are highly funded by institutional investors from abroad. Due to this the debate widened and included criticism on Anglo-Saxon style capitalism as well. In the light of the last German elections, hedge funds and private equity funds have even been compared to locusts, notorious for exhausting whole countries. However, the distinction between hedge funds and private equity funds remains very vague in this discussion, so that deep mistrust is spread among the public opinion against these new, mostly unknown and misunderstood types of investors. For this reason it is important to * discuss the arguments for or against regulation, * look at the major links between the two asset classes, * look at the major differences that exist between the asset classes, and * conceive a set of criteria to clearly distinguish between both types of funds. The purpose of this paper is to comment on possible solutions to the above mentioned tasks. It outlines preliminary thoughts and findings. Further, it comments on the steps that we think should be taken to further enhance perception of private equity funds as opposed to hedge funds from a public as well as a regulatory perspective. --Private Equity Funds,Hedge Funds

On multi-dimensional hypocoercive BGK models

We study hypocoercivity for a class of linearized BGK models for continuous phase spaces. We develop methods for constructing entropy functionals that enable us to prove exponential relaxation to equilibrium with explicit and physically meaningful rates. In fact, we not only estimate the exponential rate, but also the second time scale governing the time one must wait before one begins to see the exponential relaxation in the L1 distance. This waiting time phenomenon, with a long plateau before the exponential decay "kicks in" when starting from initial data that is well-concentrated in phase space, is familiar from work of Aldous and Diaconis on Markov chains, but is new in our continuous phase space setting. Our strategies are based on the entropy and spectral methods, and we introduce a new "index of hypocoercivity" that is relevant to models of our type involving jump processes and not only diffusion. At the heart of our method is a decomposition technique that allows us to adapt Lyapunov's direct method to our continuous phase space setting in order to construct our entropy functionals. These are used to obtain precise information on linearized BGK models. Finally, we also prove local asymptotic stability of a nonlinear BGK model.Comment: 55 pages, 2 figure

Private equity entities and conglomerates: what are the differences?

We compare the characteristics of conglomerates and private equity entities. This is done by examining the differences among their business models. We analyze the relations of the two entity types to their investors on the one hand and to their investments on the other hand. The distinguishing characteristic of private equity entities is that they pursue a stand-alone-perspective with their investment policies, meaning that they treat each investment separately. Therefore, various linkages that exist in conglomerates do not occur in private equity entities. We describe these linkages in detail. We further argue that because of the lack of these linkages in private equity entities the shareholders and debtholders of a private equity entity and its portfolio companies are not faced with the following risks that are specific for a conglomerate: the asset shifting risk, the intra-group profit risk and the capital structure risk. Finally, we define crucial evaluation criteria for identifying a private equity entity and develop a way how regulators and other persons concerned with such a task could do so. --private equity,conglomerates,investment companies

Angels or demons? Evidence on the impact of private equity firms on employment

The impact of private equity firms on employment in their portfolio companies is a controversial topic widely discussed in academia and in practice in recent years. A large body of research has resulted from this debate. The studies are focused on different aspects of employment and are based on a variety of methodologies as well as samples representing e.g. different types of buyouts and geographies. The aim of this paper is to provide access to and enhance the understanding of the highly fragmented literature by way of a systematic review and to discuss areas for future research. We review evidence on employment growth, financial and non-financial indicators of employment in a total of 49 studies. The analysis of similarities and differences of the studies revealed manifold consequences of private equity on employment. Our review reveals that the impact varies across different employment indicators and between geographies. We therefore conclude that it is not possible to label private equity firms either positively or negatively as 'angels or demons' as this would not take account of their complex and heterogeneous effects on employees post-buyout. --private equity,employment,industrial relations,entrepreneurial companies,systematic review

What you get is what you need? The role of venture capitalists in managing growth of new ventures

The resource-based view is suggested as a useful concept to shed light on the particular challenges of high potential companies on their way to building a thriving and growing company. In order to be able to apply the theoretical constructs of the resource-based view in the present context, a categorization of resources is elaborated that fits particularly well to high potential companies. Existing literature provides evidence that these companies in general only dispose of a small resource base and that they are characterized by strong resource needs in all relevant resource categories. The role of venture capitalists is assumed to provide high potential companies with financial and non-financial resources that help to create core and non-core competencies. Ultimately, this bundle of core and non-core competencies would allow high potential companies to achieve a sustained competitive advantage. The empirical results of this paper are based on insights from three in-depth case studies on German start-ups, each representing a different investor-investee dyad. It comprises investees from different industries such as software, biotech, and energy as well as investors with an established track record, first fund investors, and semi-government-dependent investors. The cases show that venture capitalists provide a number of resources to their portfolio companies that allow building a competitive advantage. The role of the venture capitalists is thus to help high potential companies to complement existing resources and competences in order to develop their full economic potential. However, there is great variation between the resource provisions of the analyzed venture capital firms. --Entrepreneurship,Venture Capital,Resource-based View