Explosion In Finite Time Research Articles

Entrance and exit at infinity for stable jump diffusions

In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty \leq a<b\leq \infty $ in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior from the boundary. Feller’s technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille–Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \begin{equation*}dZ_{t}=\sigma (Z_{t-})\,dX_{t},\end{equation*} driven by stable Lévy processes for $\alpha \in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller’s work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha \in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha \in [1,2)$. Accordingly, we derive necessary and sufficient conditions on $\sigma $. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti–Kiu representation and new Wiener–Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz–Bogdan–Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt–Nagasawa duality and Getoor’s characterisation of transience and recurrence.

Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of n weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions.

On the slit motion obeying chordal Komatu–Loewner equation with finite explosion time

This paper studies the behavior of solutions near the explosion time to the chordal Komatu-Loewner equation for slits, motivated by the preceding studies by Bauer and Friedrich (2008) and by Chen and Fukushima (2018). The solution to this equation represents moving slits in the upper half-plane. We show that the distance between the slits and driving function converges to zero at its explosion time. We also prove a probabilistic version of this asymptotic behavior for stochastic Komatu-Loewner evolutions under some natural assumptions.

Global well‐posedness and blow‐up criterion for the periodic quasi‐geostrophic equations in Lei‐Lin‐Gevrey spaces

In this paper we consider a periodic 2‐dimensional quasi‐geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution for small initial data in the Lei‐Lin‐Gevrey spaces . Moreover, we establish an exponential type explosion in finite time of this solution.

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

AbstractThe explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

Deterministic versus stochastic aspects of superexponential population growth models

Deterministic population growth models with power-law rates can exhibit a large variety of growth behaviors, ranging from algebraic, exponential to hyperexponential (finite time explosion). In this setup, selfsimilarity considerations play a key role, together with two time substitutions. Two stochastic versions of such models are investigated, showing a much richer variety of behaviors. One is the Lamperti construction of selfsimilar positive stochastic processes based on the exponentiation of spectrally positive processes, followed by an appropriate time change. The other one is based on stable continuous-state branching processes, given by another Lamperti time substitution applied to stable spectrally positive processes.

Blow-up of a Stable Stochastic Differential Equation

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete—in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.

Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

To treat networked systems involvinguncertainty due torandomness with both continuous dynamics and discrete events,this paper focuses ondiffusions modulated by acontinuous-time Markov chain.In our paper [19], we considered ordinary differential equationswith Markovian switching. This paper further treats more complexcases, namely,stochastic differential equations with Markovian switching.Our goal is to stabilize the systemsunder consideration.One of the difficulties is that the systems grow much faster thanthe allowable rates in the literature of stochastic differentialequations. As a result,the underlying systems have finite explosion time. To overcome the difficulties,we developfeedback controls to extend the local solutionsto global solutions and to stabilize the resulting systems.The feedback controls are Brownian type of perturbations.We establish the existence of global solution, prove the stability of theresulting systems, obtain boundedness in probability as $t\to\infty$, and provide sufficientconditions for almost sure stability. Then we present numerical examples to illustrate themain results.

Mathematical issues in eternal inflation

In this paper, we consider the problem of the existence and uniqueness of solutions to the Einstein field equations for a spatially flat Friedmann–Lemaître–Robertson–Walker universe in the context of stochastic eternal inflation, where the stochastic mechanism is modelled by adding a stochastic forcing term representing Gaussian white noise to the Klein–Gordon equation. We show that under these considerations, the Klein–Gordon equation actually becomes a stochastic differential equation. Therefore, the existence and uniqueness of solutions to Einstein’s equations depend on whether the coefficients of this stochastic differential equation obey Lipschitz continuity conditions. We show that for any choice of , the Einstein field equations are not globally well-posed, hence, any solution found to these equations is not guaranteed to be unique. Instead, the coefficients are at best locally Lipschitz continuous in the physical state space of the dynamical variables, which only exist up to a finite explosion time. We further perform Feller’s explosion test for an arbitrary power-law inflaton potential and prove that all solutions to the Einstein field equations explode in a finite time with probability one. This implies that the mechanism of stochastic inflation thus considered cannot be described to be eternal, since the very concept of eternal inflation implies that the process continues indefinitely. We therefore argue that stochastic inflation based on a stochastic forcing term would not produce an infinite number of universes in some multiverse ensemble. In general, since the Einstein field equations in both situations are not well-posed, we further conclude that the existence of a multiverse via the stochastic eternal inflation mechanism considered in this paper is still very much an open question that will require much deeper investigation.

Modeling Event Clustering Using the m-Memory Cox-Type Self-Exciting Intensity Model

In the analysis of point processes or recurrent events, the self-exciting component can be an important factor in understanding and predicting event occurrence. A Cox-type self-exciting intensity point process is generally not a proper model because of its explosion in finite time. However, the model with $m$-memory is appropriate to analyze sequences of recurrent events. It assumes the most recent $m$ events multiplicatively affect the conditional intensity of event occurrence. Aside from the interpretability, one advantage is the simplicity of the estimation and inference--the Cox partial likelihood can be applied and the resulting estimator is consistent and asymptotically normal. Another advantage is that the model can be applied to the analysis of case-cohort data via the pseudo-likelihood approach. The simulation studies support the asymptotic theory. Application is illustrated with analysis of a bladder cancer dataset and of an Australian stock index dataset, which shows evidence of self-excitation.

Lyapunov exponent and almost sure asymptotic stability of a stochastic SIRS model

Epidemiological models with bilinear incidence rate usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable nontrivial equilibrium (i. e. interior equilibrium) corresponding to the endemic state. In this paper, we consider an epidemiological model, which is a SIRS (susceptible-infected-removed-susceptible) model in uenced by random perturbations. We prove that the solutions of the system are positive for all positive initial conditions and that the solutions are global, that is, there is no finite explosion time. We present necessary and suficient condition for the almost sure asymptotic stability of the steady state of the stochastic system.

Stochastic Viscoelastic Wave Equations with Nonlinear Damping and Source Terms

The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.

Almost Sure and $p$th-Moment Stability and Stabilization of Regime-Switching Jump Diffusion Systems

This work focuses on regime-switching jump diffusions, which include three classes of random processes, Brownian motions, Poisson processes, and Markov chains. First, a scalar linear system is treated as a benchmark model. Then stabilization of systems with one-sided linear growth is considered. Next, nonlinear systems that have a finite explosion time are treated, in which regularization (explosion suppression) and stabilization are achieved by introducing appropriate diffusions together with Poisson and Markov chain perturbations. This work reveals the impact of various random effects on the underlying systems for almost sure and $p$th-moment stability and provides insight on stability and stabilization of switching jump diffusion systems.

Lyapunov exponent of a stochastic SIRS model

We consider a SIRS (susceptible–infected–removed–susceptible) model influenced by random perturbations. We prove that the solutions are positive for positive initial conditions and are global, that is, there is no finite explosion time. We present necessary and sufficient conditions for the almost sure asymptotic stability of the steady state of the stochastic system.

Study of the non-viscous equatorial shallow water system

The aim of this article is to prove the convergence of solutions of penalized systems describing given by equatorial shallow water equations without viscosity towards the solution of the limit system, generalizing the results of A. Dutrifoy, A. Majda and S. Schochet (2009) in [3] to all initial data even ill-prepared. A similar result is proved by I. Gallagher and L. Saint-Raymond (2006) in [4] but with a viscous fluid. We will also give a necessary and sufficient condition for a global existence of the solution of the limit system proving that the Kelvin waves lead to an explosion in finite time as a Burger equation.

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.

Regularization and Stabilization of Randomly Switching Dynamic Systems

This work concerns regularization (or explosion suppression) and stabilization of randomly switching dynamic systems by selecting suitable input functions. Starting with a randomly switching system modulated by a continuous-time Markov chain, our aim is to find a feedback control (input function) so that the system becomes stable. But before addressing the stabilization issue, another problem needs to be resolved first. Because of the fast growth in the continuous state variable, the solution of the system has a finite explosion time with probability one. To ensure the existence of global solutions, a feedback control (a noise perturbation) is added to make the system regular. Then another feedback control (a second perturbing noise) is added to ensure the stability of the resulting systems. Owing to the nonlinearity, closed-form solutions are often virtually impossible to obtain. Thus, a discrete-time approximation algorithm is constructed. Under simple conditions, a suitably interpolated sequence of the discrete-time algorithm is shown to converge to the limit switching diffusion. Finally, regularization and stabilization of the approximating sequence are dealt with.

Large time behavior of reaction–diffusion equations with Bessel generators

We investigate explosion in finite time of one-dimensional semilinear equations of the form ∂ u t ∂ t ( x ) = 1 2 ∂ 2 u t ∂ x 2 ( x ) + φ ′ ( x ) φ ( x ) ∂ u t ∂ x ( x ) − a x 2 u t ( x ) + u t 1 + β ( x ) with initial value ϕ ⩾ 0 , where φ ∈ C 2 ( R ) is positive and a ⩾ 0 , β > 0 are constants. In the free case a = 0 we provide conditions on φ under which any positive nontrivial solution is non-global. In the case a > 0 and φ ( x ) = x μ + 1 / 2 , μ ∈ R , which includes in the special case μ = − 1 / 2 the equation ∂ u t ∂ t ( x ) = 1 2 ∂ 2 u t ∂ x 2 ( x ) − a x 2 u t ( x ) + u t 1 + β ( x ) , we use the Feynman–Kac formula for Bessel processes to give conditions on the equation parameters ensuring finite-time blowup and existence of nontrivial positive global solutions.

Stability of Regime-Switching Jump Diffusions

This work is concerned with the stability of a class of switching jump-diffusion processes. The processes under consideration can be thought of as a number of jump-diffusion processes modulated by a random switching device. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. A distinct feature of the two-component process $(X(t),\alpha(t))$ considered in this paper is that the switching process $\alpha(t)$ depends on the $X(t)$ process. This paper focuses on the long-time behavior, namely, stability of the switching jump diffusions. First, the definitions of regularity and stability are recalled. Next it is shown that under suitable conditions, the underlying systems are regular or have no finite explosion time. To study stability of the trivial solution (or the equilibrium point 0), systems that are linearizable (in the $x$ variable) in a neighborhood of 0 are considered. Sufficient conditions for stability and instability are obtained. Then, almost sure stability is examined by treating a Lyapunov exponent. The stability conditions present a gap for stability and instability owing to the maximum and minimal eigenvalues associated with the drift and diffusion coefficients. To close the gap, a transformation technique is used to obtain a necessary and sufficient condition for stability.

Lack of strong completeness for stochastic flows

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If -- in addition -- the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition $x$, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently $x$, then the maximal flow is called {\em strongly complete}. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\em not} strongly complete thus answering the question in the negative.