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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...

In a stochastic-flow network with budget constraint, the network reliability for level (d, b), i.e., R(d,b), where d is a given demand value and b is a budget limit, is the probability of transmitt...

We consider the following stochastic partial differential equation, $$\begin{array}{@{}rcl@{}} &&dY_{t} = L^{\ast} Y_{t} dt + A^{\ast} Y_{t} \cdot dB_{t}\\ &&Y_{0} = \psi, \end{array} $$ associated with a stochastic flow {X(t,x)}, for t ≥ 0, $x \in \mathbb {R}^{d}$ , as in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162, 2008). We show that the strong solutions constructed there are ‘locally of compact support’. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space $\mathscr{S}^{\prime }$ . We show uniqueness of solutions in the case when ψ is smooth via the ‘monotonicity inequality’ for (L∗,A∗), which is a known criterion for uniqueness.

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.

In this paper, we will try to study the same result proved in \cite{10}. So, on the same model and with some assumptions, we will study the property of homeomorphism of the stochastic flow generated by the natural model in a one-dimensional case and with some modifications, based on an important theory of Hiroshi Kunita. This is the main motivation of our research.

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

Since 1954, the maximal flow problems have gained much attention in the world. They are also extended to many other fields for applications. For example, the definition of a network reliability is just to evaluate the probability of a live connection between the source node and the sink node such that the maximal flow of the network is no less than the demand d . The popular methods in the literature to evaluate the network reliability are mostly through minimal paths (MP) or minimal cuts (MC) of the network. One of them is the three-stages method: (a) searching for all MPs/MCs; (b) searching for all d -MPs (the minimal system states for d via MP)/ d -MCs (the maximal system states for d via MC); (c) calculating union probability from these d -MPs/ d -MCs. We found that a creative innovation in solving the maximal flow problems may have benefits in the evaluation of network reliability. Based on this idea, this paper proposes a new approach to tackle the problem of searching for all d -MPs by given MPs. The comparisons with the well known algorithm are made for benchmarking. More complicated examples are also examined for illustrative purposes.

The new conception of seismic data analysis is proposed. It is based on preliminary studying of seismic background. Its characteristics are a base for using mathematical models of non realizable seismic signals. The specific mathematical model of the seismic signal is proposed as well. The peculiarity of the model is that it allows you to simulate the flow of seismic waves of different classes each of them appears in the stream with specific time delay. This process takes place against the micro-seismic background noise. It is natural to model the flow of signals by the physically realizable signal. It means those signals which do not have a trace in prehistory. But this representation of the signal is unacceptable for two reasons. The first one is related to the smoothness of the signal at the time of its appearance on the seismic record. The second one is related to the fact that the fade of the signal in the noise does not allow us to determine the time of its appearance on the record accurately. The latter circumstance does not leave us the possibility to simulate the time of the signal occurrence by using the determined value. Therefore, the time of occurrence of the signal is simulated by random variable with variance depending on the level of micro-seismic background. We introduce the notion of generalized seismic signal as a function of time and of the vector of parameters, which determine its shape, the energy, the place in flow of the other signals, spectral characteristics, and in general behavior in the entire history of its existence. Any widely spread seismic signal models used in practice are a particular case of this one. Or in a more rigorous approach to the definition the different particular cases of the signals classes are transformed into the different hyper-planes into space of parameters. Запропоновано нову концепцію аналізу сейсмічних даних. Вона базується на попередньому вивченні сейсмічного фону. Для його параметризації використовується математична модель з фізично нездійсненними сигналами. Також пропонується специфічна математична модель самого сейсмічного сигналу. Особливість моделі полягає в тому, що вона дозволяє симулювати сейсмічні хвилі різними класами сигналів, кожен з яких з’являється в потоці зі своєю тимчасовою затримкою. Цей процес розглядається на мікросейсмічному фоні. Природно моделювати потік фізично здійсненними сигналами. Мається на увазі, що сигнали не мають передісторії. Але таке уявлення сигналу неприйнятно з двох причин. Перша пов'язана з гладкістю сигналу в точці його появи на сейсмічному записі. Друга пов'язана з тим, що на фоні сейсмічного шуму не можливо чітко виділити час вступу сигналу. Остання обставина не дає можливості отримати точний детермінований час вступу сигналу. Тому час вступу сигналу представляється як випадкова величина з довірчим інтервалом, що залежить від рівня мікросейсмічних шуму. Поняття узагальненого сейсмічного сигналу представлено як функцію часу і вектора параметрів, які визначають його форму, енергію, місце в потоці інших сигналів, його спектральні характеристики і в цілому його поведінку в усій історії його існування. Будь-яка широко поширена модель сейсмічного сигналу є окремим випадком запропонованої моделі. Або, більш строго, будь-який інший клас широко відомих моделей сейсмічного сигналу є гіперплощиною в просторі параметрів запропонованої моделі.

Computer and telecommunication networks are stochastic in nature, as each node and arc may have multiple capacity states besides complete failure. Various two-terminal reliability estimation algorithms for such stochastic flow networks are available in literature. These algorithms generate d-minimal cuts from minimal cut-sets of the network, where d is required demand. Different techniques are available in literature to evaluate exact reliability from such d-minimal cuts such as the recursive inclusion–exclusion method. The recursive inclusion–exclusion method has certain redundant computations while evaluating network reliability. This paper proposes a sum of disjoint products technique to minimize redundant computations in exact reliability computation from flow vectors. MATLAB simulation is performed to evaluate the performance of the proposed method and compare it with the existing methods for benchmark networks available in literature. Simulation results show that the proposed method require lesser computational efforts and memory.

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.

Reflected diffusions in convex polyhedral domains arise in a variety of applications, including interacting particle systems, queueing networks, biochemical reaction networks and mathematical finance. Under suitable conditions on the data, we establish pathwise differentiability of such a reflected diffusion with respect to its defining parameters --- namely, its initial condition, drift and diffusion coefficients, and (oblique) directions of reflection along the boundary of the domain. We characterize the right-continuous regularization of a pathwise derivative of the reflected diffusion as the pathwise unique solution to a constrained linear stochastic differential equation with jumps whose drift and diffusion coefficients, domain and directions of reflection depend on the state of the reflected diffusion. The proof of this result relies on properties of directional derivatives of the associated (extended) Skorokhod reflection map and their characterization in terms of a so-called derivative problem, and also involves establishing certain path properties of the reflected diffusion at nonsmooth parts of the boundary of the polyhedral domain, which may be of independent interest. As a corollary, we obtain a probabilistic representation for derivatives of expectations of functionals of reflected diffusions, which is useful for sensitivity analysis of reflected diffusions., Comment: 37 pages

We consider a stochastic flow $\phi_t(x,\omega)$ in $\mathbb{R}^n$ with initial point $\phi_0(x,\omega)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|\phi_t(x)\|)}{\|\phi_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin., Comment: 34 pages, 3 figures

So far, we’ve mostly used small matrices, with a clear geometrical meaning: \(2\times 2\) matrices transform the Cartesian plane, and \(3\times 3\) matrices transform the entire Cartesian space. What about yet bigger matrices? Fortunately, they may still have a geometrical meaning of their own. Indeed, in graph theory, they may help design a weighted graph, and model a stochastic flow in it. This makes a Markov chain, converging to a unique steady state. This has a practical application in modern search engines on the Internet [44].

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 chapter, we will study stochastic flows and jump-diffusions on manifolds determined by SDEs. If the manifold is not compact, SDEs may not be complete; solutions may explode in finite time. Then solutions could not generate stochastic flow of diffeomorphisms; instead they should define a stochastic flow of local diffeomorphisms. These facts will be discussed in Sect. 7.1. In Sect. 7.2, it will be shown that the stochastic flow defines a jump-diffusion on the manifold. Then, the dual process with respect to a volume element will be discussed. Further, in Sect. 7.3, the Levy process on a Lie group and its dual with respect to the Haar measure will be discussed.

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 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.

We generalize the coalescing Brownian flow, aka the Brownian web, considered as a weak flow to allow varying drift and diffusivity in the constituent diffusion processes and call these flows coalescing diffusive flows. We then identify the time-reversal of each coalescing diffusive flow and provide two distinct proofs of this identification. One of which is direct and the other proceeds by generalizing the concept of a localized disturbance flow to allow varying size and shape of disturbances, we show these new flows converge weakly under appropriate conditions to a coalescing diffusive flow and identify their time-reversals., 57 pages

The study of both sensitivity analysis and differentiability of the stochastic flow of a reflected process in a convex polyhedral domain is challenging because the dynamics are discontinuous at the boundary of the domain and the boundary of the domain is not smooth. These difficulties can be addressed by studying directional derivatives of an associated extended Skorokhod map, which is a deterministic mapping that takes an unconstrained path to a suitably reflected version. In this work we develop an axiomatic framework for the analysis of directional derivatives of a large class of Lipschitz continuous extended Skorokhod maps in convex polyhedral domains with oblique directions of reflection. We establish existence of directional derivatives at a path whose reflected version satisfies a certain boundary jitter property, and also show that the right-continuous regularization of such a directional derivative can be characterized as the unique solution to a Skorokhod-type problem, where both the domain and directions of reflection vary (discontinuously) with time. A key ingredient in the proof is establishing certain contraction properties for a family of (oblique) derivative projection operators. As an application, we establish pathwise differentiability of reflected Brownian motion in the nonnegative quadrant with respect to the initial condition, drift vector, dispersion matrix and directions of reflection. The results of this paper are also used in subsequent work to establish pathwise differentiability of a much larger class of reflected diffusions in convex polyhedral domains., 58 pages, 3 figures

The purpose of this paper is to establish a probabilistic representation formula for the Navier--Stokes equations on compact Riemannian manifolds. Such a formula has been provided by Constantin and Iyer in the flat case of $\mathbb R^n$ or of $\mathbb T^n$. On a Riemannian manifold, however, there are several different choices of Laplacian operators acting on vector fields. In this paper, we shall use the de Rham--Hodge Laplacian operator which seems more relevant to the probabilistic setting, and adopt Elworthy--Le Jan--Li's idea to decompose it as a sum of the square of Lie derivatives., 26 pages. We add Section 4 discussing the Killing vector fields on Riemannian symmetric spaces which satisfy the conditions in Section 3

Let \((M, \mathcal {F})\) be a foliated compact Riemannian manifold. We consider a family of compatible Feller semigroups in C(Mn) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond, Ann. Probab. 32:1247–1315, 2004) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.

We construct a family of SDEs with smooth coefficients whose solutions select a reflected Brownian flow as well as a corresponding stochastic damped transport process (Wt)(Wt), the limiting pair gives a probabilistic representation for solutions of the heat equations on differential 1-forms with the absolute boundary conditions. The transport process evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary where it is driven by the boundary local time, and with its normal part erased at the end of the excursions to the boundary of the reflected Brownian motion. On the half line, this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution; on the half space it agrees with the construction of N. Ikeda and S. Watanabe [29] by Poisson point processes. The construction leads also to an approximation for the boundary local time, in the topology of uniform convergence but not in the semi-martingale topology, indicating the difficulty in proving convergence of solutions of a family of random ODE’s to the solution of a stochastic equation driven by the local time and with jumps. In addition, we obtain a differentiation formula for the heat semi-group with Neumann boundary condition and prove also that (Wt)(Wt) is the weak derivative of a family of reflected Brownian motions with respect to the initial point.

We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on \({\mathbb {T}}^2\) driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. \(\alpha >\frac{1}{2}\)) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the \(L^p\)-norm and a boot-strapping argument. The upper semicontinuity of random attractors is also established. Moreover, if the viscosity constant is large enough, the system has a trivial random attractor.

In this paper we consider the Ito SDE $$d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in {\mathbb R}^d,$$ where $W_t$ is a $d$-dimensional standard Wiener process and the drift coefficient $b:[0,T]\times{\mathbb R}^d\to{\mathbb R}^d$ belongs to $L^q(0,T;L^p({\mathbb R}^d))$ with $p\geq 2, q>2$ and $\frac dp +\frac 2q

We give a new method for proving the homomorphic property of a quantum stochastic flow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic flows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of Goswami et al. (Inst H Poincare Probab Stat 41:505–522, 2005).

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.

A theory of intermittency differentiation is developed for a general class of 1D Infinitely Divisible Multiplicative Chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman-Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail., 38 pages

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.

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 study, a new method is presented to evaluate the system reliability of a flow network using spanning trees with flow. The proposed method includes two main tasks: 1. Identifying the spanning trees without flow by representing each network link as a binary string of length N (N is the number of nodes) and performing all possible combinations between N − 1 links, and 2. Using the generated spanning trees without flow to find the spanning trees with flow by calculating the total flow carried by the links. The proposed method is tested on different examples from the literature to illustrate its efficiency in generating the spanning trees with flow and calculating the system reliability.

A linear stochastic vector advection equation is considered. The equation may model a passive magnetic field in a random fluid. The driving velocity field is a integrable to a certain power and the noise is infinite dimensional. We prove that, thanks to the noise, the equation is well posed in a suitable sense, opposite to what may happen without noise., Comment: We made minor modifications

A modulated synchronous doubly stochastic flow is considered. The flow under study is considered in conditions of a fixed dead time. It means that after each registered event there is a time of the fixed duration T (dead time), during which other flow events are inaccessible for observation. When duration of the dead time period finishes, the first event to occur creates the dead time period of duration T again and etc. It is supposed that the dead time period duration is an unknown variable. Using the maximum likelihood method and a moments of observed events occurrence the problem of dead time period estimation is solved.

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.

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