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This study focuses on characterizing the fatigue damage accumulated in nonlinear aeroelastic systems subjected to stochastic inflows through both numerical simulations and wind tunnel experiments. In the mathematical model, nonlinearities are assumed to exist either in the structure (via a cubic hardening nonlinearity in the pitch stiffness), or in the flow (via dynamic stall condition), or simultaneously in both the structural and aerodynamic counterparts. The aerodynamic loads in the attached flow and dynamic stall conditions are estimated using Wagner's formulation and semi-empirical Leishman–Beddoes model, respectively. To augment the findings to in-field flow conditions, the oncoming wind flow is considered to be randomly time-varying in nature. The stochastic input flow fluctuations are modeled using a Karhunen–Loeve Expansion formulation. The response dynamics and the associated fatigue damage of the aeroelastic system, possessing different sources of nonlinearities, are systematically investigated under isolated cases of deterministic and stochastic input flows. Specifically, the pertinent role of stochasticity in the input flow is brought out by presenting the response dynamics and the associated fatigue damage accumulation for different values of noise intensity and time scale of the input flow fluctuation. It is demonstrated that under fluctuating flow conditions, the dynamics intermittently switch between attached flow and the dynamic stall regimes even at low mean flow speeds. The intermittent nature of the response varies as the time scale and intensity of the oncoming flow are varied. The role of torsional stresses as the predominant component dictating the fatigue damage accumulation irrespective of the source of nonlinearity is illustrated. Using the rainflow counting method and Miner's linear damage accumulation theory, it is shown that the accumulated fatigue damage is substantially higher under stochastic flow conditions as compared to deterministic input flows. Importantly, it is observed that different time scales and intensities of the oncoming flow fluctuation play a pivotal role in dictating the fatigue damage in aeroelastic systems. Finally, fatigue damage is observed to be significantly higher for torsionally dominant oscillations in the dynamical stall regime compared to the oscillations at the attached flow regime. The numerical findings are strengthened by drawing comparisons with the preliminary results obtained from wind tunnel experiments performed on a NACA 0012 airfoil undergoing dynamic stall. To the best of our knowledge, this is the first study that systematically bridges the dichotomy between the stall-induced dynamical signatures in stochastic aeroelastic systems and maps the same to the corresponding structural damage.

Many real-world networks such as freight, power and long distance transportation networks are represented as multi-source multi-sink stochastic flow network. The objective is to transmit flow successfully between the source and the sink nodes. The reliability of the capacity vector of the assigned components is used an indicator to find the best flow strategy on the network. The Components Assignment Problem (CAP) deals with searching the optimal components to a given network subject to one or more constraints. The CAP in multi-source multi-sink stochastic flow networks with multiple commodities has not yet been discussed, so our paper investigates this scenario to maximize the reliability of the capacity vector subject to an assignment budget. The mathematical formulation of the problem is defined, and a proposed solution based on genetic algorithms is developed consisting of two steps. The first searches the set of components with the minimum cost and the second searches the flow vector of this set of components with maximum reliability. We apply the solution approach to three commonly used examples from the literature with two sets of available components to demonstrate its strong performance.

The goal of this paper is to establish the Lipschitz and W2,∞ estimates for a second-order parabolic PDE ∂tu(t,x)=12Δu(t,x)+f(t,x) on Rd with zero initial data and f satisfying a Ladyzhenskaya–Prod...

We have studied the efficiency of organization of transfer trains at developed railroad junctions based on the criterion of total hours of railroad wagon downtime under accumulation. We have constructed discrete-event simulation models (Java SE, AnyLogic) of the organization of transfer trains in line with a "rigid schedule" and under accumulation to the standard for trains. The models take into consideration the stochastic nature of wagon arrivals to the accumulation points and meeting the established norm for the mass of trains. It is considered at modeling that track development of railroad stations and intranodal passages is rational and does not significantly affect delay in the motion of transfer trains. The transfer trains schedule is rational and rhythmic. The throughput capacity of open tracks at a node and the processing capability of subsystems at railroad stations are sufficient. We have experimentally compared the efficiency of four models of the organization of transfer trains, including a "flexible model", which combines dispatch upon accumulation to the standard of a train and a "rigid schedule" at the same time. It was established that the minimal hours of wagon accumulation are achieved when transfer trains are organized via a combined variant: a simultaneous dispatch in line with a "rigid schedule" and at accumulation to the standard. Organization of transfer trains in line with a "rigid schedule" requires the organization of additional trains, otherwise there is an increase in the queue of wagons at accumulation points. When increasing the number of daily trains by 10‒11 %, the process of forming and dispatching becomes sufficiently reliable and efficient. The specified recommendations would make it possible to improve the typical process of operation of railroad junctions in terms of dispatcher control over the formation, dispatch, and passing of transfer trains during intranodal traffic

In this paper, we give a probabilistic interpretation of Sobolev solutions to parabolic semilinear McKean–Vlasov partial differential equations (PDEs for short) in terms of mean-field backward stochastic differential equations (BSDEs for short). This probabilistic interpretation can be viewed as a generalization of the Feynman–Kac formula. The method is based on the stochastic flow technique which is different from classical stochastic differential equations (SDEs for short) due to the influence of mean-field term in McKean–Vlasov SDEs.

Sufficient conditions as well as necessary and sufficient ones are found for the completeness of the stochastic flow generated by equations with the so-called current velocities (Nelson's symmetric...

This paper contains a list of corrections to the paper ‘ Solutions of SPDEs associated with a stochastic flow’.

The paper is devoted to the modeling of a single-phase flow through saturated porous media. A statistical approach where permeability is considered as a lognormal random field is applied. The impact of permeability, random boundary conditions and wells pressure on the flow in a production well is studied. A numerical procedure to generate an ensemble of realizations of the numerical solution of the problem is developed. A global sensitivity analysis is performed using Sobol indices. The impact of different model parameters on the total model uncertainty is studied.

For stochastic flow network (SFN), given all the lower (or upper) boundary points, the classic problem is to calculate the probability that the capacity vectors are greater than or equal to the lower boundary points (less than or equal to the upper boundary points). However, in some practical cases, SFN reliability would be evaluated between the lower and upper boundary points at the same time. The evaluation of SFN reliability with upper and lower boundary points at the same time is the focus of this paper. Because of intricate relationships among upper and lower boundary points, a decomposition approach is developed to obtain several simplified subsets. SFN reliability is calculated according to these subsets by means of the inclusion-exclusion principle. Two heuristic options are then established in order to calculate SFN reliability in an efficient direction based on the lower and upper boundary points.

We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear multiplicative noise acting on a finite set of modes of the dominant linear flow is introduced. Existence of a stochastic flow and a local stochastic invariant manifold for this stochastic form of SHE are both addressed in this work. We show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation, and obtain numerical evidence suggesting that this picture is a good approximation for the full system as well.

In this paper we investigate the regularity properties of strong solutions to SDEs driven by L\'evy processes with irregular drift coefficients. Under some mild conditions, we show that the singular SDE has a unique strong solution for each starting point and the family of all the solutions forms a stochastic flow. Moreover, the Malliavin differentiability of the strong solutions is also obtained. As an application, we also prove a path-by-path uniqueness result for some related random ODEs., Comment: 28 pages

Consider a manifold $M$ endowed locally with a pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let $\text{Diff}(\Delta^H, M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroups generated by vector fields in the corresponding distributions. We decompose a stochastic flow with jumps, up to a stopping time, as $\varphi_t = \xi_t \circ \psi_t$, where $\xi_t \in \text{Diff}(\Delta^H, M)$ and $\psi_t \in \text{Diff}(\Delta^V, M)$. Our main result provides Stratonovich stochastic differential equations with jumps for each of these two components in the corresponding infinite dimensional Lie groups. We present an extension of the Ito-Ventzel-Kunita formula for stochastic flows with jumps generated by classical Marcus equation (as in Kurtz, Pardoux and Protter [11]). The results here correspond to an extension of Catuogno, da Silva and Ruffino [4], where this decomposition was studied for the continuous case.

In this chapter, we show that solutions of a continuous symmetric stochastic differential equation (SDE) on a Euclidean space define a continuous stochastic flow of diffeomorphisms and that solutions of an SDE with diffeomorphic jumps define a right continuous stochastic flow of diffeomorphisms. Sections 3.1 and 3.2 are introductions. Definitions of these SDEs and stochastic flows will be given and the geometric property of solutions will be explained as well as basic facts. Rigorous proof of these facts will be given in Sects. 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9. In Sect. 3.3, we study another Ito SDE with parameter, called the master equation. Applying results of Sect. 3.3, we show in Sect. 3.4 that solutions of the original SDE define a stochastic flow of C∞-maps. For the proof of the diffeomorphic property, we need further arguments. In Sect. 3.5 we consider backward SDE and backward stochastic flow of C∞-maps. Further, the forward–backward calculus for stochastic flow will be discussed in Sects. 3.5, 3.6 and 3.8. These facts will be applied in Sects. 3.7 and 3.9 for proving the diffeomorphic property of solutions.

In this article, we are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time tau. The main motivations are giving a probabilistic representation of the Sobolev's solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when tau is the first exit time of a forward SDE from a cylindrical domain. Euler schemes and bounds for the discrete-time approximation error are provided., 38, Monte Carlo Methods and Applications (MCMA) 2016

In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

We obtain an explicit form of supports of strongly mixing evolution system of measures naturally connected with non-homogenous Markov process induced by time-dependent SPDEs. We show that considered supports one can get as a net of limit sets determined by a two-parameters semigroup of set-valued maps induced by transition probabilities.

A Richardson triple is an ideal fluid flow map $g_{t/\ep,t,\ep t} = h_{t/\ep}k_t l_{\ep t}$ composed of three smooth maps with separated time scales: slow, intermediate and fast; corresponding to the big, little, and lesser whorls in Richardson's well-known metaphor for turbulence. Under homogenisation, as $\lim \ep\to0$, the composition $h_{t/\ep}k_t $ of the fast flow and the intermediate flow is known to be describable as a single stochastic flow $\dd g$. The interaction of the homogenised stochastic flow $\dd g$ with the slow flow of the big whorl is obtained by going into its non-inertial moving reference frame, via the composition of maps $(\dd g)l_{\ep t}$. This procedure parameterises the interactions of the three flow components of the Richardson triple as a single stochastic fluid flow in a moving reference frame. The Kelvin circulation theorem for the stochastic dynamics of the Richardson triple reveals the interactions among its three components. Namely, (i) the velocity in the circulation integrand acquires is kinematically swept by the large scales; and (ii) the velocity of the material circulation loop acquires additional stochastic Lie transport by the small scales. The stochastic dynamics of the composite homogenised flow is derived from a stochastic Hamilton's principle, and then recast into Lie-Poisson bracket form with a stochastic Hamiltonian. Several examples are given, including fluid flow with stochastically advected quantities, and rigid body motion under gravity, i.e., the stochastic heavy top in a rotating frame., Comment: Printed in J Nonlinear Science

In this paper we study the reversibility of a stochastic flow based on its integral representation when the diffusion vector field commutes with drift vector fields. This allows us to write the complete solution of stochastic characteristic system as a composition of the deterministic flows generated by the drift and diffusion vector fields combined with the Wiener process acting in the equation. This explicit integral representation of the solution implied reversibility of the complete solution taking into account the reversibility of each deterministic flow in the finite composition. The unique solution satisfies a nonlinear SPDE.

In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that in fact under the similar conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_k M$, $k\geq1$, vanish., Comment: 9 pages. This version 2 contains a general result about stochastic deformations

We prove the existence of a stochastic flow of Hölder homeomorphisms for solutions of SDEs with singular time dependent drift having only certain integrability properties. We also show that the solution map x → X x is differentiable in a weak sense.

We study invariance and monotonicity properties of Kunita-type stochastic differential equations in $\RR^d$ with delay. Our first result provides sufficient conditions for the invariance of closed subsets of $\RR^d$. Then we present a comparison principle and show that under appropriate conditions the stochastic delay system considered generates a monotone (order-preserving) random dynamical system. Several applications are considered., 27 pages

In this article we prove that stochastic differential equation (SDE) with Sobolev drift on a compact Riemannian manifold admits a unique ν -almost everywhere stochastic invertible flow, where ν is the Riemannian measure, which is quasi-invariant with respect to ν . In particular, we extend the well-known DiPerna-Lions flows of ODEs to SDEs on a Riemannian manifold.

A time-based stochastic flow network (TBSFN), in which each arc has several possible capacities/states and a lead time, is addressed to discuss the system reliability of spare routing for a computer network. The minimum transmission time to send a given amount of data through a single minimal path is uncertain. Although the transmission time will be shortened even if the data are sent through p ( p > 1) disjoint minimal paths simultaneously, it is still variable in a TBSFN. This paper is concerned with evaluating the probability that a specified amount of data can be sent through p minimal paths simultaneously within a time threshold. Such a probability is named the system reliability, which can be treated as a performance index for measuring the transmission ability. We present an efficient methodology to assess the system reliability. Furthermore, a spare routing for boosting the system reliability is established in advance to indicate the main and spare p minimal paths. Subsequently, the system reliability of the spare routing can be computed easily, which shows the contribution of the spare design. From the viewpoint of decision support, we may conduct the sensitive analysis to find out the most important arc which will increase/decrease the system reliability most significantly.

We study transport properties of isotropic Brownian flows. Under a transience condition for the two-point motion, we show asymptotic normality of the image of a finite measure under the flow and -- under slightly stronger assumptions -- asymptotic normality of the distribution of the volume of the image of a set under the flow. Finally, we show that for a class of isotropic flows, the volume of the image of a nonempty open set (which is a martingale) converges to a random variable which is almost surely strictly positive., Comment: To appear in "Stochastic processes and their applications"

In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.

In this paper we study how σ-finite measures on ℝdevolve under a class of "stochastic flows" associated to stochastic differential equations with (resp. without) jumps in ℝd. First we show the related measure evolution processes are càdlàg (resp. continuous), strongly Markovian and weakly Fellerian. Then we extend the existing results on incompressibility in Harris [8] and Kunita [14], and prove strong Markov property of the process describing how compact subsets evolve under incompressible "stochastic flows" under a certain condition.

In this paper we prove Kunita-type stochastic differential equation (SDE) [Formula: see text], t > s, driven by a spatial continuous semimartingale F (x, t) =(F1 (x, t), …, Fm (x, t)), x ∈ ℜm, with local characteristic (a, b), in the sense of Kunita (Chap. 3 of Ref. 10), can produce a stochastic flow of homeomorphisms of ℜm into itself almost surely under the (a, b) satisfies non-Lipschitz conditions. This result bases on recent works12 by the author.

The equivalent probabilistic formulation of Dupire’s PDE is the put-call duality equality. In local volatility models including exponential Levy jumps, we give a direct probabilistic proof for this result based on stochastic flow arguments. This approach also enables us to check the equivalent probabilistic formulation of various generalizations of Dupire’s PDE recently obtained by Pironneau [C. R. Acad. Sci. Paris Ser. I 344(2) 127–133 (2007)] by the adjoint equation technique in the case of complex options.

Our aim is to study under what conditions the exact and numerical solution (based on equidistant nonrandom partitions of integration time-intervals) to a stochastic differential delay equation (SDDE) share the property of mean-square exponential stability. Our approach is trying to avoid the use of Lyapunov functions or functionals. We show that under a global Lipschitz assumption an SDDE is exponentially stable in mean square if and only if for some sufficiently small stepsize Δ the Euler–Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same “if and only if” result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. Replacing the exact and EM numerical solution with stochastic processes, we also discuss whether a family of stochastic processes share the stability property. This new approach allows us to discuss (i) whether a family of SDDEs share the stability property, and (ii) whether an SDDE with variable time lag shares stability property with the modified EM method. As another application of this general approach we consider a linear SDDE with variable time lag and establish an “if and only if” result. It should also be mentioned that the questions addressed, results proved, as well as style of analysis borrow heavily from [14] but the computations involved to cope with time delay are nontrivial.

In this paper we consider a stochastic flow in Rn which leaves a closed convex set K invariant. By using a recent characterization of the invariance, involving the distance function, we study the corresponding transition semigroup Pt and its infinitesimal generator N. Due to the invariance property, N is a degenerate elliptic operator. We study existence of an invariant measure ν of Pt and the realization of N in L2 (H, ν).

Estimation of the states of a synchronous double stochastic flow of events with an arbitrary (finite) number of states was considered. The flow is a mathematical model of the information flows of the customers circulating in the queuing systems and networks. An explicit form of the a posteriori probabilities representing the total characteristics of the flow of event was determined. Decision about the state of flow is made using the criterion for maximum of the a posteriori probability. Numerical calculations of the a posteriori probabilities and statistical experiments with a simulation model are illustrated.

A.A. Kirillov has given a parametrization of the space U ∞ of univalent functions on the closed unit disk, which are C ∞ up to the boundary, by Diff(S 1 )/S 1 where Diff(S 1 ) denotes the group of orientation preserving diffeomorphisms of the circle S 1 . In the same spirit, the space J ∞ of C ∞ Jordan curves in the complex plane can be parametrized by the double quotient SU(1, 1)\ Diff(S 1 )/ SU(1, 1). As a consequence, J ∞ carries a canonical Riemannian metric. We construct a canonical Brownian motion on U ∞ . Classical technologies of the theory of univalent functions, like Beurling–Ahlfors extension, Loewner equation, Beltrami equation, developed in the context of Kunita’s stochastic flows, are the tools for obtaining this result which should be seen as a first step to the construction of a canonical Brownian motion on J ∞ .

This paper concerns the fluid dynamics modelled by the stochastic flow \left\{ \begin{array}{l} \boldsymbol{\dot{\eta}}\left( t,x\right) =\boldsymbol{u}\left( t,\boldsymbol{\eta} \left( t,x\right) \right) +\boldsymbol{\sigma}\left( t,\boldsymbol{\eta}\left( t,x\right) \right) \circ\dot{W}, \\ \\ \boldsymbol{\eta}(0,x)=x, \end{array} \right. where the turbulent term is driven by the white noise $\dot{W}$. The motivation for this setting is to understand the motion of fluid parcels in turbulent and randomly forced fluid flows. Stochastic Euler equations for the undetermined components $\boldsymbol{u}(t,x)$ and $\boldsymbol{\sigma}(t,x)$ of the spatial velocity field are derived from the first principles. The resulting equations include as particular cases the deterministic and randomly forced counterparts of these equations.In the second part of the paper, we prove the existence and uniqueness of a strong local solution to the stochastic Navier--Stokes equation in $W_{p}^{1}(\boldsymbol{R}^{d}),d >1,p > d. ...

The mean busy period of an infinite multilinear queue with a doubly stochastic flow of incoming queries having two intensities and exponential server is calculated.

A stochastic model is proposed for multiparticle Lagrangian motion in the upper ocean. The model is based on hydrodynamics equations with random forcing, includes a few well interpreted and well estimated parameters, and implies a common description of the one-particle motion via a Langevin equation for the particle velocity. The dependence of the top Lyapunov exponent on the model parameters is studied as a part of a Lagrangian predictability problem. In particular, it is found that the Coriolis effect can radically improve the prediction of a Lagrangian particle position based on observations of other particles.

We consider a $d$-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data., 15 pages

Large deviations for stochastic flow solutions to SDEs containing a small parameter are studied. The obtained results are applied to establish a C p,r -large deviation principle for stochastic flows and for solutions to anticipating SDEs. The recent results of Millet-Nualart-Sans and Yoshida are improved and refined.

We consider the Lyapunov exponents of flows generated by a class of Markovian velocity fields. The existence of the exponents is obtained for flows on a compact set, but with the most general form of the velocity field. As a particular class, we study the homogeneous and incompressible flows. In this case, the exponents are nonrandom, free of the initial position of the particle path, and their sum is zero. We numerically compute the top Lyapunov exponent on R 2 for a range of parameters to conjecture that it is strictly positive.

We study a stochastic flow of C 1 -homeomorphisms of R . At certain stopping times, the spatial derivative of the flow is a diffusion in the space variable and its generator is given. This answers several questions posed in a previous study by Bass and Burdzy (1999, Ann. Probab. 27, 50–108) .

The problem of estimating the drift of a stochastic flow given Lagrangian observations is an estimation problem for a multidimensional diffusion with a degenerate diffusion matrix. The maximum-likelihood estimator of the constant drift is considered. A long-time asymptotic of its mean-square error (MSE) is computed. It is shown that the time-space average of the observed Lagrangian velocities has the same asymptotic. These estimators are compared to the least-squares estimator based on Eulerian data. In the most important, for applications, two-dimensional case the Lagrangian estimator is typically preferable for incompressible flows, while for flows close to potential the Eulerian estimator is better.

This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows., Comment: Published in at http://dx.doi.org/10.1214/11-AOP653 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on global averages over a stochastic flow. We show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the Fokker-Planck operator and its adjoint. We first analyze the interplay of the deterministic dynamics with the noise in the neighborhood of a periodic orbit of a map, by using a discretized version of Fokker-Planck formalism. Then we propose a method to determine the 'optimal resolution' of the state space, based on solving Fokker-Planck's equation locally, on sets of unstable periodic orbits of the deterministic system. We test our hypothesis on unimodal maps., 45 pages, 11 figures

Birth and Death on a flow refers to a particle system on a stochastic flow. Particles are born in a point process and move on the flow subject to position-dependent killing. They die eventually and leave the flow. The particle process is a measure-valued, Markov process tracking these motions. Its law depends on the distribution of births, the coefficients of the flow, and the rate of killing. We parametrize the system and derive a quasi-likelihood for chronicles of integral data on the particle process.

It is known that the two-point generator of an SDE determines the law of its flow uniquely. We describe an explicit way to find the coefficients of the SDE from its two-point generator. Indeed, this method of construction produces a sequence of SDEs in a way that the associated sequence of stochastic flows converges to the stochastic flow of the desired SDE.

Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Imagine an infinite population with immigration labelled at each generation by N:={1,2,...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focussing on simple distinguished coalescents, i.e such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0,1] denoted by M=(\Lambda_{0},\Lambda_{1}). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the \Lambda-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures \Lambda_{0} and \Lambda_{1} specify respectively the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time, all individuals are immigrant children., Comment: 30 pages

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming--Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincar\'{e} Probab. Statist. 41 (2005) 307--333]. Two scaling limit theorems for the generalized Fleming--Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147--181]., Comment: Published in at http://dx.doi.org/10.1214/10-AOP629 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Sensitivity analysis represents an important aspect of network flow design problems. For example, what is the incremental increase in system flow of increasing the diameters of specified pipes in a water distribution network? Although methods exist for solving this problem in the deterministic case, no comparable methodology has been available when the network's arc capacities are subject to random variation. This paper provides this methodology by describing a Monte Carlo sampling plan that allows one to conduct a sensitivity analysis for a variable upper bound on the flow capacity of a specified arc. The proposed plan has two notable features. It permits estimation of the probabilities of a feasible flow for many values of the upper bound on the arc capacity from a single data set generated by the Monte Carlo method at a single value of this upper bound. Also, the resulting estimators have considerably smaller variancesthan crude Monte Carlo sampling would produce in the same setting. The success of the technique follows from the use of lower and upper bounds on each probability of interest where the bounds are generated from an established method of decomposing the capacity state space.

For a superprocess under a stochastic flow, we prove that it has a density with respect to the Lebesgue measure for d=1 and is singular for d>1. For d=1, a stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov's L_p-theory for linear SPDE. A snake representation for this superprocess is established. As applications of this representation, we prove the compact support property for general d and singularity of the process when d>1.