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This paper is devoted to hydrodynamic limits of linear kinetic equations. We consider situations in which the thermodynamical equilibrium is described by a heavy-tail distribution function rather than a maxwellian distribution. A similar problem was addressed in [14] using Fourier transform and it was shown that the long time/small mean free path behavior of the solution of the kinetic equation is described by a fractional diffusion equation. In this paper, we propose a different method to obtain similar results. This method is somewhat reminiscent of the so-called \"moments method\" which plays an important role in kinetic theory. This new method allows us to consider space dependent collision operators (which could not be treated in [14]). We believe that it also provides the relevant tool to address nonlinear problems.

We examine the behaviour of the pseudo-marginal random walk Metropolis algorithm, where evaluations of the target density for the accept/reject probability are estimated rather than computed precisely. Under relatively general conditions on the target distribution, we obtain limiting formulae for the acceptance rate and for the expected squared jump distance, as the dimension of the target approaches infinity, under the assumption that the noise in the estimate of the log-target is additive and is independent of the position. For targets with independent and identically distributed components, we also obtain a limiting diffusion for the first component. We then consider the overall efficiency of the algorithm, in terms of both speed of mixing and computational time. Assuming the additive noise is Gaussian and is inversely proportional to the number of unbiased estimates that are used, we prove that the algorithm is optimally efficient when the variance of the noise is approximately 3.283 and the acceptance rate is approximately 7.001%. We also find that the optimal scaling is insensitive to the noise and that the optimal variance of the noise is insensitive to the scaling. The theory is illustrated with a simulation study using the particle marginal random walk Metropolis.

We describe a new MCMC method optimized for the sampling of probability measures on Hubert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned diffusions. Our algorithm is based on two key design principles: (i) algorithms which are well defined in infinite dimensions result in methods which do not suffer from the curse of dimensionality when they are applied to approximations of the infinite dimensional target measure on ℝN (ii) nonreversible algorithms can have better mixing properties compared to their reversible counterparts. The method we introduce is based on the hybrid Monte Carlo algorithm, tailored to incorporate these two design principles. The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hubert space; as a consequence the algorithm explores the approximate target measures on ℝN in a number of steps which is independent of N. We also present the underlying theory for the limiting nonreversible diffusion on Hubert space, including characterization of the invariant measure, and we describe numerical simulations demonstrating that the proposed method has favourable mixing properties as an MCMC algorithm.

In this paper, we study the qualitative behavior of the Cauchy problem of a hyperbolic model $\\left\\{ \\matrix{p_t} - \\nabla \\cdot \\left( {pq} \\right)\\, = \\,\\Delta p,\\,x \\in {{\\Cal R}^d},t > 0, \\hfill \\cr{q_t} - \\nabla \\left( {\\varepsilon |q{|^2} + p} \\right)\\, = \\,\\varepsilon \\Delta q,\\hfill \\cr\\endmatrix \\right.$ which is transformed from a singular chemotaxis system describing the effect of a reinforced random walk in [17,27]. When d = 1 and the initial data are prescribed around a constant ground state (p̄,0) with p̄≥ 0, we prove the global asymptotic stability of constant ground states, and identify the explicit decay rate of solutions under very mild conditions on initial data. Moreover, we study the diffusion (viscous) limit of solutions as ɛ ➞ 0 with convergence rates toward solutions of the non-diffusible (inviscid) problem. While the existence of global large solutions of the system in multi-dimensions remains an outstanding open question, we show that the model exhibits a strong parabolic smoothing effect: namely, solutions are spatially analytic for a short time provided that the initial data belong to Lq(ℝd) for any q > d ≥ 1. In fact, when d = 1, we obtain that the solution remains real analytic for all time.

Fully explicit closed-form expressions are developed for the fair strike prices of discrete-time variance swaps under general affine GARCH type models that have been risk-neutralized with a family of variance dependent pricing kernels. The methodology relies on solving differential recursions for the coefficients of the joint cumulant generating function of the log price and the conditional variance processes. An alternative derivation is provided in the case of Gaussian innovations. Using standard assumptions on the asymptotic behavior of the GARCH parameters as the sampling frequency increases, the diffusion limit of a Gaussian GARCH model is derived and the convergence of the variance swap prices to its continuous-time limit is further investigated. Numerical examples on the term structure of the variance swap rates and on the convergence results are also presented.

We consider the limit of a linear kinetic equation with reflectiontransmission- absorption at an interface and with a degenerate scattering kernel. The equation arises from a microscopic chain of oscillators in contact with a heat bath. In the absence of the interface, the solutions exhibit a superdiffusive behavior in the long time limit. With the interface, the long time limit is the unique solution of a version of the fractional in space heat equation with reflection-transmission-absorption at the interface. The limit problem corresponds to a certain stable process that is either absorbed, reflected or transmitted upon crossing the interface.

We investigate scaling limits of the seed bank model when migration (to and from the seed bank) is ‘slow’ compared to reproduction. This is motivated by models for bacterial dormancy, where periods of dormancy can be orders of magnitude larger than reproductive times. Speeding up time, we encounter a separation of timescales phenomenon which leads to mathematically interesting observations, in particular providing a prototypical example where the scaling limit of a continuous diffusion will be a jump diffusion. For this situation, standard convergence results typically fail. While such a situation could in principle be attacked by the sophisticated analytical scheme of Kurtz (J Funct Anal 12:55–67, 1973), this will require significant technical efforts. Instead, in our situation, we are able to identify and explicitly characterise a well-defined limit via duality in a surprisingly non-technical way. Indeed, we show that moment duality is in a suitable sense stable under passage to the limit and allows a direct and intuitive identification of the limiting semi-group while at the same time providing a probabilistic interpretation of the model. We also obtain a general convergence strategy for continuous-time Markov chains in a separation of timescales regime, which is of independent interest.

This paper investigates the fast chemical diffusion limit from a parabolic-parabolic Keller-Segel system to the corresponding parabolic-elliptic Keller-Segel system by constructing approximate solutions with an appropriate order via an asymptotic expansion. Nonlinear stability of the precise initial layer is characterized with an exact convergence rate by using basic energy method.

Multiclass queueing networks (MQN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original MQN. To validate such an approximation amounts to justifying the interchange of two limits, t → ∞ and k → ∞, with t being the time index and k, the scaling parameter. Here, we show this interchange of limits is justified under a p∗th moment condition on the primitive data, the interarrival and service times; and we provide an explicit characterization of the required order (p∗), which depends naturally on the desired order of moment of the workload process.

Consider a system of N parallel single-server queues with unit-exponential service time distribution and a single dispatcher where tasks arrive as a Poisson process of rate λ(N). When a task arrives, the dispatcher assigns it to one of the servers according to the Join-the-Shortest Queue (JSQ) policy. Eschenfeldt and Gamarnik (Math. Oper. Res. 43 (2018) 867–886) identified a novel limiting diffusion process that arises as the weak-limit of the appropriately scaled occupancy measure of the system under the JSQ policy in the Halfin–Whitt regime, where (N – λ(N))√N → β > 0 as N → ∞. The analysis of this diffusion goes beyond the state of the art techniques, and even proving its ergodicity is nontrivial, and was left as an open question. Recently, exploiting a generator expansion framework via the Stein’s method, Braverman (2018) established its exponential ergodicity, and adapting a regenerative approach, Banerjee and Mukherjee (Ann. Appl. Probab. 29 (2018) 1262–1309) analyzed the tail properties of the stationary distribution and path fluctuations of the diffusion. However, the analysis of the bulk behavior of the stationary distribution, namely, the moments, remained intractable until this work. In this paper, we perform a thorough analysis of the bulk behavior of the stationary distribution of the diffusion process, and discover that it exhibits different qualitative behavior, depending on the value of the heavy-traffic parameter β. Moreover, we obtain precise asymptotic laws of the centered and scaled steady-state distribution, as β tends to 0 and ∞. Of particular interest, we also establish a certain intermittency phenomena in the β → ∞ regime and a surprising distributional convergence result in the β → 0 regime.