convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay this paper can be regarded as the continuation of the work contained in papers milosevic 2011 2013 at the same time it represents the extension of the paper wu et al 2010 in this paper the one sided lipschitz condition is employed in the context of the backward euler method for a class of neutral stochastic differential equations with constant delay sufficient conditions for this method to be well defined are revealed under certain nonlinear growth conditions the convergence in probability is established for the continuous forward backward euler method as well as for the discrete backward euler method additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without requiring for the drift coefficient to satisfy the linear growth condition this manuscript represents the continuation of the previous work of the author related to a class of highly nonlinear neutral stochastic differential equations with time dependent delay main results of this manuscript are convergence in probability and almost sure exponential stability of the backward euler approximate solution for a class of stochastic differential equations with constant delay conditions under which both the exact and approximate solutions share the property of the almost sure exponential stability are revealed this paper also illustrates that in some cases stochastic differential equations with constant delay should be studied separately from those with time dependent delay in that sense it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time dependent delay by the constant delay

random dynamical systems generated by stochastic navier stokes equations on a rotating sphere abstract in this paper we first prove the existence and uniqueness of the solution to the stochastic navier stokes equations on the rotating 2 dimensional unit sphere then we show the existence of an asymptotically compact random dynamical system associated with the equations

analysis of time fractional and space nonlocal stochastic incompressible navier stokes equation driven by white noise abstract in this paper we consider the stochastic time space fractional incompressible navier stokes equation driven by white noise we firstly derive the regularity of the nonlocal stochastic convolution then we prove the global existence and uniqueness of the mild solution finally the holder continuity for the stochastic time space fractional incompressible navier stokes equation driven by white noise is established the optimal holder continuity accords with the result of integer order time incompressible navier stokes equation

strong convergence rates of modified truncated em methods for neutral stochastic differential delay equations abstract the aim of this paper is to investigate strong convergence of modified truncated euler maruyama method for neutral stochastic differential delay equations introduced in lan 2018 strong convergence rates of the given numerical scheme to the exact solutions at fixed time t are obtained under local lipschitz and khasminskii type conditions moreover convergence rates over a time interval 0 t are also obtained under additional polynomial growth condition on g without the weak monotonicity condition which is usually the standard assumption to obtain the convergence rate two examples are presented to interpret our conclusions

strong convergence rates of modified truncated em method for stochastic differential equations abstract motivated by truncated euler maruyama em method introduced by mao 2015 a new explicit numerical method named modified truncated euler maruyama method is developed in this paper strong convergence rates of the given numerical scheme to the exact solutions to stochastic differential equations are investigated under given conditions in this paper compared with truncated em method the given numerical simulation strongly converges to the exact solution at fixed time t and over a time interval 0 t under weaker sufficient conditions meanwhile the convergence rates are also obtained for both cases two examples are provided to support our conclusions

local and global strong solutions to the stochastic incompressible navier stokes equations in critical besov space abstract considering the stochastic navier stokes system in r d forced by a multiplicative white noise we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space b p r d p 1 r d the proof is based on the contraction mapping principle stopping time and stochastic estimates then we prove the global existence of strong solutions in probability if the initial data are sufficiently small which contain a class of highly oscillating large data

global strong solution to the three dimensional stochastic incompressible magnetohydrodynamic equations the three dimensional incompressible magnetohydrodynamic equations with stochastic external forces are considered first the existence and uniqueness of local strong solution to the stochastic magnetohydrodynamic equations are proved when the external forces satisfy some conditions the proof is based on the contraction mapping principle stopping time and stochastic estimates the strong solution is a weak solution for the fluid variables with a given complete probability space and a given brownian motion then the global existence of strong solutions in probability is established if the initial data are sufficiently small and the noise is multiplicative and non degenerate

the approximation of a crank nicolson scheme for the stochastic navier stokes equations in this paper we prove the convergence of stochastic navier stokes equations driven by white noise a linearized version of the implicit crank nicolson scheme is considered for the approximation of the solutions to the n s equations the noise is defined as the distributional derivative of a wiener process and approximated by using the generalized l 2 projection operator optimal strong convergence error estimates in the l 2 norm are obtained

the existence of energy solutions to 2 dimensional non lipschitz stochastic navier stokes equations in unbounded domains in this paper we consider the existence and uniqueness of weak energy solutions to a stochastic 2 dimensional non lipschitz navier stokes equation perturbed by the cylindrical wiener process w t in a bounded or unbounded domain d with the smooth boundary d or d r2 dx t ax t b x t dt f t x t dt g t x t dw t where a is the stokes operator and f g satisfy the non lipschitz condition

stochastic three dimensional rotating navier stokes equations averaging convergence and regularity we consider stochastic three dimensional rotating navier stokes equations and prove averaging theorems for stochastic problems in the case of strong rotation regularity results are established by bootstrapping from global regularity of the limit stochastic equations and convergence theorems

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

optimal transportation with traffic congestion and wardrop equilibria in the classical monge kantorovich problem the transportation cost depends only on the amount of mass sent from sources to destinations and not on the paths followed by this mass thus it does not allow for congestion effects using the notion of traffic intensity we propose a variant taking into account congestion this variant is a continuous version of a well known traffic problem on networks that is studied both in economics and in operational research the interest of this problem is in its relations with traffic equilibria of wardrop type what we prove in the paper is exactly the existence and the variational characterization of equilibria in a continuous space setting

the exponential behaviour and stabilizability of stochastic 2d navier stokes equations abstract some results on the pathwise exponential stability of the weak solutions to a stochastic 2d navier stokes equation are established the first ones are proved as a consequence of the exponential mean square stability of the solutions however some of them are improved by avoiding the previous mean square stability in some more particular and restrictive situations also some results and comments concerning the stabilizability and stabilization of these equations are stated

martingale solutions for the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes abstract in this paper the three dimensional stochastic nonhomogeneous incompressible navier stokes equations driven by levy processes consisting of the brownian motion the compensated poisson random measure and the poisson random measure are considered in a bounded domain we obtain the existence of martingale solutions the construction of the solution is based on the classical galerkin approximation method the stopping times the stochastic compactness method and the jakubowski skorokhod theorem

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

first order mean field games with density constraints pressure equals price in this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality a weak solution of the system contains an extra term an additional price imposed on the saturated zones we show that this price corresponds to the pressure field from the models of incompressible euler equations a la brenier by this observation we manage to obtain a minimal regularity which allows us to write optimality conditions at the level of single agent trajectories and to define a weak notion of nash equilibrium for our model

action functional for diffusions in discontinuous media the action functional i e the rate function governing the large deviations is obtained for a family of stochastic processes with discontinuous drift and small diffusion a well known method of continuous mapping is developed which proves to be efficient in a so called stable case

irreducibility of the 3 d stochastic navier stokes equation abstract a 3 dimensional navier stokes equation with random force is investigated a form of irreducibility of interest in ergodic theory is proved under a full noise assumption the basic tool is the fact that even if the equation is a priori non well posed the solutions depend continuously on the noise around regular solutions

strong convergence of split step backward euler method for stochastic age dependent capital system with markovian switching abstract in this paper we present and analyze the split step backward euler method for the stochastic capital system with markovian switching under the one sided local lipschitz condition on the drift and local lipschitz condition on the diffusion we prove the split step backward euler method converges with strong order of one half to the true solution a numerical example is provided to illustrate the theoretical results

first order stationary mean field games with congestion abstract mean field games mfgs are models for large populations of competing rational agents that seek to optimize a suitable functional in the case of congestion this functional takes into account the difficulty of moving in high density areas here we study stationary mfgs with congestion with quadratic or power like hamiltonians first using explicit examples we illustrate two main difficulties the lack of classical solutions and the existence of areas with vanishing densities our main contribution is a new variational formulation for mfgs with congestion with this formulation we prove the existence and uniqueness of solutions finally we consider applications to numerical methods

convergence and stability of two classes of theta milstein schemes for stochastic differential equations abstract this paper examines convergence and stability of the two classes of theta milstein schemes for stochastic differential equations sdes with non global lipschitz continuous coefficients the split step theta milstein sstm scheme and the stochastic theta milstein stm scheme for 1 2 1 this paper concludes that the two classes of theta milstein schemes converge strongly to the exact solution with the order 1 for 0 1 2 under the additional linear growth condition for the drift coefficient these two classes of the theta milstein schemes are also strongly convergent with the standard order this paper also investigates exponential mean square stability of these two classes of the theta milstein schemes for 1 2 1 these two theta milstein schemes can share the exponential mean square stability of the exact solution for 0 1 2 similar to the convergence under the additional linear growth condition these two theta milstein schemes can also reproduce the exponential mean square stability of the exact solution

strong convergence of split step backward euler method for stochastic differential equations with non smooth drift in this paper we are concerned with the numerical approximation of stochastic differential equations with discontinuous nondifferentiable drifts we show that under one sided lipschitz and general growth conditions on the drift and global lipschitz condition on the diffusion a variant of the implicit euler method known as the split step backward euler ssbe method converges with strong order of one half to the true solution our analysis relies on the framework developed in d j higham x mao and a m stuart strong convergence of euler type methods for nonlinear stochastic differential equations siam journal on numerical analysis 40 2002 1041 1063 and exploits the relationship which exists between explicit and implicit euler methods to establish the convergence rate results

martingale and weak solutions for a stochastic nonlocal burgers equation on finite intervals abstract this work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal burgers equation on a one dimensional bounded domain the existence of a martingale solution is shown by using a galerkin approximation prokhorov s theorem and skorokhod s embedding theorem the same galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal burgers equation on a bounded domain

regularity via duality in calculus of variations and degenerate elliptic pdes abstract a technique based on duality to obtain h 1 or other sobolev regularity results for solutions of convex variational problems is presented this technique first developed in order to study the regularity of the pressure in the variational formulation of the incompressible euler equation has been recently re employed in mean field games here it is shown how to apply it to classical problems in relation with degenerate elliptic pdes of p laplace type this allows to recover many classical results via a different point of view and to have inspiration for new ones the applications include among others variational models for traffic congestion and more general minimization problems under divergence constraints but the most interesting results are obtained in dynamical problems such as mean field games with density constraints or density penalizations

global l2 solutions of stochastic navier stokes equations this paper concerns the cauchy problem in r d for the stochastic navier stokes equation 1 u u u u p f u u p g u o w u 0 u 0 div u 0 driven by white noise w under minimal assumptions on regularity of the coefficients and random forces the existence of a global weak martingale solution of the stochastic navier stokes equation is proved in the two dimensional case the existence and pathwise uniqueness of a global strong solution is shown a wiener chaos based criterion for the existence and uniqueness of a strong global solution of the navier stokes equations is established

local martingale and pathwise solutions for an abstract fluids model we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system the primary application of this abstract framework is to infer the local existence of strong pathwise solutions to the 3d primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise instead of developing our results specifically for the 3d primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations atmosphere oceans coupled atmosphere ocean on the sphere on the beta plane approximation etc and the incompressible navier stokes equations in applications all of the details are given for the beta plane approximation of the oceans equations

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

asymptotic behaviors of stochastic two dimensional navier stokes equations with finite memory the stochastic 2d navier stokes equations with finite memory are studied for the differentiable memory function the almost sure exponential stability of the weak solution is shown by employing a non negative semimartingale convergence argument for the nondifferentiable memory function the exponential stability in mean square for the weak solution is proved by using the differentiability property of the moment function

geodesics in the space of measure preserving maps and plans we study brenier s variational models for incompressible euler equations these models give rise to a relaxation of the arnold distance in the space of measure preserving maps and more generally measure preserving plans we analyze the properties of the relaxed distance we show a close link between the lagrangian and the eulerian model and we derive necessary and sufficient optimality conditions for minimizers these conditions take into account a modified lagrangian induced by the pressure field moreover adapting some ideas of shnirelman we show that even for non deterministic final conditions generalized flows can be approximated in energy by flows associated to measure preserving maps

on inviscid limits for the stochastic navier stokes equations and related models we study inviscid limits of invariant measures for the 2d stochastic navier stokes equations as shown by kuksin j stat phys 115 1 2 469 492 2004 the noise scaling sqrt nu is the only one which leads to non trivial limiting measures which are invariant for the 2d euler equations using a moser type iteration for stochastic drift diffusion equations we show that any limiting measure mu 0 is in fact supported on bounded vorticities relationships of mu 0 to the long term dynamics of 2d euler in l infty with the weak topology are discussed we also obtain a drift independent modulus of continuity for a stationary deterministic model problem which leads us to conjecture that in fact mu 0 is supported on c 0 moreover in view of the batchelor krainchnan 2d turbulence theory we consider inviscid limits for a weakly damped stochastic navier stokes equation in this setting we show that only an order zero noise scaling with respect to leads to a nontrivial limiting measure in the inviscid limit

stochastic non newtonian fluid motion equations of a nonlinear bipolar viscous fluid an analysis based on the galerkin method is developed to examine the behaviour of a nonlinear bipolar viscous fluid mathematically modelled by stochastic non newtonian fluid motion equations existence and uniqueness of solutions to the stochastic equations are derived

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

qualitative properties of stationary measures for three dimensional navier stokes equations abstract the paper is devoted to studying the distribution of stationary solutions for 3d navier stokes equations perturbed by a random force under a non degeneracy assumption we show that the support of such a distribution coincides with the entire phase space and its finite dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the lebesgue measure similar assertions are true for weak solutions of the cauchy problem with a regular initial function the results of this paper were announced in the short note a shirikyan controllability of three dimensional navier stokes equations and applications in semin equ deriv partielles 2005 2006 ecole polytech palaiseau 2006

large deviations for the stochastic quasigeostrophic equation with multiplicative noise this paper proves a large deviation principle for the stochastic quasigeostrophic equation with multiplicative noise the proof is based on the laplace principle and on weak convergence methods we also give a rigorous mathematical result on the existence of a strong solution and uniqueness of mild solutions to the stochastic quasigeostrophic equation in the dimension d 2 3 finally large deviations for short time are considered

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation