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We show that a single laser pulse, traveling through a dense plasma, produces a population of MeV photons of sufficient density to generate a large number of electron–positron pairs via the linear Breit–Wheeler process. While it may be expected that the photons are emitted predominantly in the forward direction, parallel to the laser propagation, we find that a longitudinal plasma electric field drives the emission of photons in the backwards direction. This enables the collision of oppositely directed, MeV-level photons necessary to overcome the mass threshold for the linear Breit–Wheeler process. Our calculations predict the production of 107 electron–positron pairs, per shot, by a laser with peak intensity of just 3 × 1022 W cm−2. By using only a single laser pulse, the scheme sidesteps the practical difficulties associated with the multiple-laser schemes previously investigated.

In this paper, we consider a generalized birth–death process (GBDP) and examine its linear versions. Using its transition probabilities, we obtain the system of differential equations that governs its state probabilities. The distribution function of its waiting time in state s given that it starts in state s is obtained. For a linear version of it, namely the generalized linear birth–death process (GLBDP), we obtain the probability generating function, mean, variance and the probability of ultimate extinction of population. Also, we obtain the maximum likelihood estimate of its parameters. The differential equations that govern the joint cumulant generating functions of the population size with cumulative births and cumulative deaths are derived. In the case of constant birth and death rates in GBDP, the explicit forms of the state probabilities, joint probability mass functions of population size with cumulative births and cumulative deaths, and their marginal probability mass functions are obtained. It is shown that the Laplace transform of an integral of GBDP satisfies its Kolmogorov backward equation with certain scaled parameters. The first two moments of the path integral of GLBDP are obtained. Also, we consider the immigration effect in GLBDP for two different cases. An application of a linear version of GBDP and its path integral to a vehicles parking management system is discussed. Later, we introduce a time-changed version of the GBDP where time is changed via an inverse stable subordinator. We show that its state probabilities are governed by a system of fractional differential equations.

A Monte Carlo method for simulating a multi-dimensional diffusion process conditioned on hitting a fixed point at a fixed future time is developed. Proposals for such diffusion bridges are obtained by superimposing an additional guiding term to the drift of the process under consideration. The guiding term is derived via approximation of the target process by a simpler diffusion processes with known transition densities. Acceptance of a proposal can be determined by computing the likelihood ratio between the proposal and the target bridge, which is derived in closed form. We show under general conditions that the likelihood ratio is well defined and show that a class of proposals with guiding term obtained from linear approximations fall under these conditions.

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

In this work, we consider parabolic equations of the form ( u ε ) t + A ε ( t ) u ε = F ε ( t , u ε ) , where ε is a parameter in [ 0 , ε 0 ) , and { A ε ( t ) , t ∈ R } is a family of uniformly sectorial operators. As ε → 0 + , we assume that the equation converges to u t + A 0 ( t ) u = F 0 ( t , u ) . The time-dependence found on the linear operators A ε ( t ) implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family A ε ( t ) and on its convergence to A 0 ( t ) when ε → 0 + , we obtain a Trotter-Kato type Approximation Theorem for the linear process U ε ( t , τ ) associated with A ε ( t ) , estimating its convergence to the linear process U 0 ( t , τ ) associated with A 0 ( t ) . Through the variation of constants formula and assuming that F ε converges to F 0 , we analyze how this linear process convergence is transferred to the solution of the semilinear equation. We illustrate the ideas in two examples. First a reaction-diffusion equation in a bounded smooth domain Ω ⊂ R 3 ( u ε ) t - d i v ( a ε ( t , x ) ∇ u ε ) + u ε = f ε ( t , u ε ) , x ∈ Ω , t > τ , where a ε converges to a function a 0 , f ε converges to f 0 . We apply the abstract theory in this example, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated with each problem. The second example is a nonautonomous strongly damped wave equation u tt + ( - a ( t ) Δ D ) u + 2 ( - a ( t ) Δ D ) 1 2 u t = f ( t , u ) , x ∈ Ω , t > τ , where Δ D is the Laplacian operator with Dirichlet boundary conditions in a domain Ω and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.

Let {Zij } be independent and identically distributed (i.i.d.) random variables with EZij = 0, E|Zij |² = 1 and E|Zij |⁴ < ∞. Define linear processes Y t j = ∑ k = 0 ∞ b k Z t − k , j with ∑ i = 0 ∞ | b i | < ∞ . Consider a p-dimensional time series model of the form xt = ∏xt−1 + ∑1/2yt, 1 ≤ t ≤ T with yt = (Y t1, … , Ytp )′ and ∑1/2 be the square root of a symmetric positive definite matrix. Let B = (1/p)XX* with X = (x₁, … , xT)′ and X* be the conjugate transpose. This paper establishes both the convergence in probability and the asymptotic joint distribution of the first k largest eigenvalues of B when xt is nonstationary. As an application, two new unit root tests for possible nonstationarity of high-dimensional time series are proposed and then studied both theoretically and numerically.

Let Xi,1≤i≤n be a sequence of linear process based on dependent random variables with random coefficients, which has a mean shift at an unknown location. The cumulative sum (CUSUM, for short) estimator of the change point is studied. The strong convergence, Lr convergence, complete convergence and the rate of strong convergence are established for the CUSUM estimator under some mild conditions. These results improve and extend the corresponding ones in the literature. Simulation studies and two real data examples are also provided to support the theoretical results.

In this paper, we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes place in the space of càdlàg functions on [0 , 1] with the Skorokhod M 2 topology.

The purpose of this paper is to study the convergence of the quasi-maximum likelihood (QML) estimator for long memory linear processes. We first establish a correspondence between the long-memory linear process representation and the long-memory AR ( ∞ ) process representation. We then establish the almost sure consistency and asymptotic normality of the QML estimator. Numerical simulations illustrate the theoretical results and confirm the good performance of the estimator.

In this paper, we study the complete convergence and complete moment convergence of linear processes generated by negatively dependent random variables under sub-linear expectations. The obtained results complement the ones of Meng, Wang, and Wu (Commun. Stat., Theory Methods 52(9):2931–2945, 2023) in the case of negatively dependent random variables under sub-linear expectations.