Perron Method Research Articles

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampere operators. We then use the Perron method to solve Monge-Ampere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoe, Caffarelli-Viaclovsky and Hultgren-Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-[Formula: see text]-smooth and (non)-analytic periodic center manifolds.

Smoothness of invariant manifolds for stochastic evolution equations with non-dense domain

With the crucial estimation of random Stieltjes-type convolution, the Lyapunov–Perron method allows us to establish the existence of invariant dynamics for stochastic evolution equation with non-dense domain. In this paper, we develop the conditions for [Formula: see text]-smoothness of center manifold for such equation with linear multiplicative noise, unstable manifold and stable set for the one with nonlinear multiplicative noise as well. Our result may apply to a broad range of stochastic systems with non-dense domain beyond Brownian noise.

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

Viscosity solutions to the infinity Laplacian equation with lower terms

We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ where \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\). The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is the infinity Laplacian which is strongly degenerate, quasilinear and it is associated with the absolutely minimizing Lipschitz extension. When the nonhomogeneous term \(f(x,t)\) is non-decreasing in \(t\), we prove the existence of the viscosity solution via Perron's method. We also establish a uniqueness result based on the perturbation analysis of the viscosity solutions. If the function \(f(x,t)\) is nonpositive (nonnegative) and non-increasing in \(t\), we also give the existence of viscosity solutions by an iteration technique under the condition that the domain has small diameter. Furthermore, we investigate the existence and uniqueness of viscosity solutions to the boundary-value problem with singularity $$\displaylines{ \Delta_\infty^h u=-b(x)g(u), \quad \hbox{in } \Omega, \cr u>0, \quad \hbox{in } \Omega, \cr u=0, \quad \hbox{on }\partial\Omega, }$$ when the domain satisfies some regular condition. We analyze asymptotic estimates for the viscosity solution near the boundary.

Effects of COVID-19 on the relationship between inflation and REITs returns in South Africa

PurposeThis research aims to ascertain the extent to which the coronavirus disease 2019 (COVID-19) epidemic affected the relationship between inflation and real estate investment trusts (REITs) returns in South Africa.Design/methodology/approachThis research used the Johansen cointegration test and effective test in establishing if there is a long-run cointegrating equation between the variables. To ascertain if COVID-19 resulted in a different relationship regime between inflation and REITs returns, the sequential Bai–Perron method was used.FindingsBetween December 2013 and July 2022, there was no evidence of a long-run relationship between inflation and REITs returns, and a restricted vector autoregressive (VAR) model with a period lag for each variable best describing the relationship. Using the sequential Bai–Perron method, for one break, the results show February 2020 as a structural break in the relationship. A cointegrating equation is also found for the period before the structural break and another after the break. Interestingly, the relationship is negative before the break and a new positive relationship (regime) is confirmed after the noted break.Practical implicationsThis research helps REITs stakeholders to position themselves in light of any changes to macroeconomic activity within South Africa.Originality/valueThis is one of the first studies to test inflation relationship with REITs returns in South Africa and the effects of COVID-19 thereof. This research helps REITs stakeholders to position themselves in light of any changes to macroeconomic activity within South Africa.

Unstable manifolds for rough evolution equations

In this paper, we consider a class of rough nonlinear evolution equations driven by infinite-dimensional [Formula: see text]-Hölder rough paths with [Formula: see text]. First, we give a proper integral with respect to infinite-dimensional [Formula: see text]-Hölder rough paths by using rough paths theory. Second, we obtain the global in time solution and random dynamical system of rough evolution equation. Finally, we derive the existence of local unstable manifolds for rough evolution equations by a properly discretized Lyapunov–Perron method.

Convergence of Optimal Investment Problems in the Vanishing Fixed Cost Limit

We consider an optimal investment problem for an investor facing constant and proportional transaction costs and study the limit as the constant cost tends to zero. Combining the stochastic Perron's method with stability arguments for viscosity solutions, we show that the value function converges to the value function of the problem with purely proportional costs. Moreover, using a Komlos-type argument, we show that forward-convex combinations of the optimal strategies in the problem with constant costs converge to an optimal strategy without a constant cost.

Parabolic Hessian Equations Outside a Cylinder

In this article, we mainly review the parabolic Hessian equation on the exterior region. The existence and uniqueness of solutions with asymptotic properties to the exterior problem of the parabolic Hessian equation were obtained by using the Perron method.

The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group

We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron–Wiener–Brelot generalized solution by proving its existence for very general boundary data.

Optimal entry and consumption under habit formation

AbstractThis paper studies a composite problem involving decision-making about the optimal entry time and dynamic consumption afterwards. In Stage 1, the investor has access to full market information subject to some information costs and needs to choose an optimal stopping time to initiate Stage 2; in Stage 2, the investor terminates the costly full information acquisition and starts dynamic investment and consumption under partial observation of free public stock prices. Habit formation preferences are employed, in which past consumption affects the investor’s current decisions. Using the stochastic Perron method, the value function of the composite problem is proved to be the unique viscosity solution of some variational inequalities.

Pricing Bermudan Option with Variable Transaction Costs under the Information-Based Model

The Bermudan option pricing problem with variable transaction costs is considered for a risky asset whose price process is derived under the information-based model. The price is formulated as the value function of an optimal stopping problem, which is the value function of a stochastic control problem given by a non-linear second order partial differential equation. The theory of viscosity solutions is applied to solve the stochastic control problem such that the value function is also the solution of the corresponding Bellman equation. Under some regularity assumptions, the existence and uniqueness of the solution of the pricing equation are derived by the application of the Perron method and Banach Fixed Point theorem.

Pluripotential Monge-Ampère flows in big cohomology classes

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact Kähler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential Kähler-Ricci flows on compact Kähler manifolds of general type as well as on stable varieties

Periodic solutions for one-dimensional nonlinear nonlocal problem with drift including singular nonlinearities

In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$, singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

Resolutivity and invariance for the Perron method for degenerate equations of divergence type

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation divA(x,∇u)=0 in a bounded open set Ω⊂Rn. The vector-valued function A satisfies the standard ellipticity assumptions with 1<p<∞ and a p-admissible weight w. We show that arbitrary perturbations on sets of (p,w)-capacity zero of continuous (and certain quasicontinuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p,w)-capacity zero.

Existence of positive solution for elliptic equations with singular terms and combined nonlinearities

We use an approximation scheme, sub-super solution and Perron's method to show the existence and nonexistence of positive solutions for a boundary value problem with a singular nonlinearity in the equation and on the boundary condition. We also study the asymptotic behavior of the solutions with respect to the parameters.

Inertial manifolds for functional differential equations with infinite delay

In this paper, we consider a class of functional differential equations with infinite delay in a real separable Hilbert space. By using the Lyapunov–Perron method, we prove the existence of inertial manifolds for mild solutions. The obtained results can be applied to Lotka–Volterra diffusion models with infinite delay.

The first initial-boundary value problem of parabolic Monge\u2013Amp\xe8re equations outside a bowl-shaped domain

In this paper, we study the parabolic Monge–Ampère equations -u_{t}det (D^{2}u)=g outside a bowl-shaped domain with g being the perturbation of g_{0}(|x|) at infinity. Under the weaker conditions compared with the problem outside a cylinder, we obtain the existence and uniqueness of viscosity solutions with asymptotic behavior for the first initial-boundary value problem by using the Perron method.

Asymptotic phase for flows with exponentially stable partially hyperbolic invariant manifolds

We consider an autonomous system admitting an invariant manifold M. The following questions are discussed: (i) what are the conditions ensuring exponential stability of the invariant manifold? (ii) does every motion attracting by M tend to some motion on M (i.e. have an asymptotic phase)? (iii) what is the geometrical structure of the set formed by orbits approaching a given orbit? We get an answer to (i) in terms of Lyapunov functions omitting the assumption that the normal bundle of M is trivial. An affirmative answer to (ii) is obtained for invariant manifold M with partially hyperbolic structure of tangent bundle. In this case, the existence of asymptotic phase is obtained under new conditions involving contraction rates of the linearized flow in normal and tangential to M directions. To answer the question (iii), we show that a neighborhood of M has a structure of invariant foliation each leaf of which corresponds to motions with common asymptotic phase. In contrast to theory of cascades, our technique exploits the classical Lyapunov–Perron method of integral equations.

The exterior Dirichlet problems of Monge\u2013Amp\xe8re equations in dimension two

In this paper, we study the Monge–Ampère equations det D^{2}u=f in dimension two with f being a perturbation of f_{0} at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.