invariant palm and related disintegrations via skew factorization let g be a measurable group with haar measure acting properly on a space s and measurably on a space t then any finite jointly invariant measure m on s t admits a disintegration nu otimes mu into an invariant measure on s and an invariant kernel from s to t here we construct and by a general skew factorization which extends an approach by rother and zahle for homogeneous spaces s over g this leads to easy extensions of some classical propositions for invariant disintegration previously known in the homogeneous case the results are applied to the palm measures of jointly stationary pairs where is a random measure on s and is a random element in t

convex invariant means and a pathwise central limit theorem on etudie le probleme de choisir un point aleatoirement a partir de la ligne des reels on definit une notion de distribution d espace uniforme on considere trois sources supplementaires d uniformite l invariance d echelle convexe la consistance avec des methodes de moyennes et la linearite mesure de mokobodski

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

random recursive construction of self similar fractal measures the noncompact case the self similarity of sets measures is often defined in a constructive way in the present paper it will be shown that the random recursive construction model of falconer graf and mauldin williams for statistically self similar sets may be generalized to the noncompact case we define a sequence of random finite measures which converges almost surely to a self similar random limit measure under certain conditions on the generating lipschitz maps we determine the carrying dimension of the limit measure

an implicit algorithm within the arbitrary lagrangian eulerian formulation for solving incompressible fluid flow with large boundary motions abstract the objective of the paper is to present an implicit algorithm for incompressible fluid flow solution using the arbitrary lagrangian eulerian ale formulation and to investigate solution accuracy and stability of the algorithm the governing equations of the implicit procedure are derived using isoparametric interpolations for the fluid velocities and pressure the details suitable for general use are presented in our derivations of the fundamental equations and of the basic finite element balance equations the penalty method is utilized to eliminate the pressure on the element level accuracy and stability of the solutions are demonstrated in three examples for which the analytical solutions are known in the first example the burger s equation analogue to 1 d fluid flows is solved without and with fe mesh motion to show that the mesh motion practically does not affect the solutions in the second example the solitary wave motion with large displacements of the free boundary is solved as a benchmark problem in the third example the fluid flow in an infinite contractible and expandable pipe with prescribed large radial wall motion is solved this example is specifically attractive for biological flows as in blood vessels and lung airways all solutions presented show that the proposed algorithm is sufficiently accurate and stable since the algorithm is implicit high accuracy of results can be achieved with a relatively large time step

sample path properties of self similar processes with stationary increments a real valued process x x t u is self similar with exponent h h ss if x a d ahx for all a 0 sample path properties of h ss processes with stationary increments are investigated the main result is that the sample paths have nowhere bounded variation if 0 1 in which case they are singular however nowhere bounded variation may occur also for h 1 examples exhibiting this combination of properties are constructed as well as many others most are obtained by subordination of random measures to point processes in ir2 that are poincar6 i e invariant in distribution for the transformations t x at b ax of r in a final section it is shown that the self similarity and stationary increment properties are preserved under composition of independent processes x1 a x2 x1 x2 t l some interesting examples are obtained this way

symplectic integration of hamiltonian systems with additive noise hamiltonian systems with additive noise possess the property of preserving symplectic structure numerical methods with the same property are constructed for such systems special attention is paid to systems with separable hamiltonians and to second order differential equations with additive noise some numerical tests are presented

equidistribution from fractal measures we give a fractal geometric condition for a measure on 0 1 to be supported on points x that are normal in base n i e such that n kx k in mathbb n equidistributes modulo 1 this condition is robust under c 1 coordinate changes and it applies also when n is a pisot number rather than an integer as applications we obtain new results and strengthen old ones about the prevalence of normal numbers in fractal sets and new results on measure rigidity specifically completing host s theorem to multiplicatively independent integers and proving a rudolph johnson type theorem for certain pairs of beta transformations

convergence under q of p invariant measures on the circle abstract consider the semigroup of maps tn n n where tn x nx mod 1 suppose p and q are multiplicatively independent integers in that they are not both powers of the same integer further suppose that is a borel probability measure invariant ergodic and of positive entropy for tp we show that the sequence of measures tiq converges weak to lebesgue measure on a subsequence of values i of uniform full density

an energy conserving method for stochastic maxwell equations with multiplicative noise abstract in this paper it is shown that three dimensional stochastic maxwell equations with multiplicative noise are stochastic hamiltonian partial differential equations possessing a geometric structure i e stochastic multi symplectic conservation law and the energy of system is a conservative quantity almost surely we propose a stochastic multi symplectic energy conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy the mean square convergence result of the method in temporal direction is tested numerically and numerical comparisons with finite difference method are also investigated

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

a local version of the projection theorem we prove the following theorem suppose that 1 m n are integers and is a borel measure on r such that for a e x 1 the upper and lower m densities of at x are positive and finite 2 if is a tangent measure of at x then for all v g n m the orthogonal projection of the support of onto v is a convex set then is m rectifiable

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

tangent measure distributions of fractal measures tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in euclidean spaces the idea is loosely speaking to attach to every point of the set a family of random measures called the alpha dimensional tangent measure distributions at the point which describe asymptotically the alpha dimensional scenery seen by an observer zooming down towards this point this tool has been used by bandt ba and graf g to study the regularity of the local geometry of self similar sets but in this paper we show that its scope goes much beyond this situation and in fact it may be used to describe a strong regularity property possessed by every measure we show that for every measure mu on a euclidean space and any dimension alpha at mu almost every point all alpha dimensional tangent measure distributions are palm measures this means that the local geometry of every dimension of general measures can be described like the local geometry of self similar sets by means of a family of statistically self similar random measures we believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

invariant transports of stationary random measures and mass stationarity we introduce and discuss balancing invariant transports of stationary random measures on a polish abelian group the first main result is an associated fundamental invariance property of palm measures derived from a generalization of neveu s exchange formula the second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transports we then introduce in a non stationary setting the concept of mass stationarity with respect to a random measure formalizing the intuitive idea that the origin is a typical location in the mass the third main result of the paper is that a measure is a palm measure if and only if it is mass stationary

a completely explicit scheme of cauchy problem in bslm for solving the navier stokes equations abstract this paper presents a backward semi lagrangian method bslm with third order convergence in both time and space for solving incompressible navier stokes equations the third order backward differentiation formula for the total time derivative and the projection method for the steady state stokes equation are used a fourth order difference scheme together with a local bi cubic interpolation is used to solve the resulted two governing equations for the velocity and pressure this paper mainly focuses on the development of an efficient scheme for solving the nonlinear cauchy problem of the characteristic curve we employ a modified linear multi step method of the implicit type based on the error correction strategy a novel contribution of this paper is the design of a completely explicit formula for the three foot points the proposed method is superior to existing methods in terms of computational costs and accuracy allowing the use of a large time step size the fully explicit formula for the foot points significantly improves the performance of a common bslm for a wide range of practical applications

on multifractal formalism for self similar measures with overlaps let mu be a self similar measure generated by an ifs varphi phi i i 1 ell of similarities on mathbb r d d ge 1 when varphi is dimensional regular see definition 1 1 we give an explicit formula for the l q spectrum tau mu q of mu over 0 1 and show that tau mu is differentiable over 0 1 and the multifractal formalism holds for mu at any alpha in tau mu 1 tau mu 0 we also verify the validity of the multifractal formalism of mu over tau mu infty tau mu 0 for two new classes of overlapping algebraic ifss by showing that the asymptotically weak separation condition holds for one of them the proof appeals to the recent result of shmerkin ann math 2 189 2 319 391 2019 on the l q spectrum of self similar measures

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

dimension growth for iterated sumsets we study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set f subseteq mathbb r satisfies overline dim mathrm b f f overline dim mathrm b f or even dim mathrm h n f rightarrow 1 our results apply to for example all uniformly perfect sets which include ahlfors david regular sets our proofs rely on hochman s inverse theorem for entropy and the assouad and lower dimensions play a critical role we give several applications of our results including an erd s volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

efficient numerical integrators for stochastic models abstract the efficient simulation of models defined in terms of stochastic differential equations sdes depends critically on an efficient integration scheme in this article we investigate under which conditions the integration schemes for general sdes can be derived using the trotter expansion it follows that in the stochastic case some care is required in splitting the stochastic generator we test the trotter integrators on an energy conserving brownian model and derive a new numerical scheme for dissipative particle dynamics we find that the stochastic trotter scheme provides a mathematically correct and easy to use method which should find wide applicability

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

galton watson iterated function systems iterated function systems ifs are interesting parametric models for generating fractal sets and functions the general idea is to compress deform and translate a given set or function with a collection of operators and to iterate the procedure under weak assumptions ifs possess a unique fixed point which is in general fractal ifs were introduced in a deterministic context then were generalized to the random setting on abstract spaces in the early 1980 s their adaptation to random signals was carried out by hutchinson and r schendorff 9 by considering random operators this study extends their model with not only random operators but also a random underlying construction tree we show that the corresponding ifs converges under certain hypothesis to a unique fractal fixed point properties of the fixed point are also described

on the complexity of the set of codings for self similar sets and a variation on the construction of champernowne abstract let f p 0 p n be a collection of points in r d the set f naturally gives rise to a family of iterated function systems consisting of contractions of the form s i x x 1 p i i 0 n where 0 1 and x r d given f and it is well known that there exists a unique non empty compact set x satisfying x i 0 n s i x for each x x there exists a sequence a j j 1 0 n n satisfying x lim j s a 1 s a j 0 we call such a sequence a coding of x in this paper we prove that for any f and k n there exists k f 0 such that if 1 k f 1 then every point in the interior of x has a coding which is k simply normal similarly we prove that there exists u n i f 0 such that if 1 u n i f 1 then every point in the interior of x has a coding containing all finite words for some specific choices of f we obtain lower bounds for k f and u n i f we also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction our proofs rely on a variation of a well known construction of a normal number due to champernowne and an approach introduced by erd s and komornik

entropy of convolutions on the circle given ergodic p invariant measures f ig on the 1 torus w e give a sharp condition on their entropies guaranteeing that the entropy of the convolution 1 n converges to logp we also prove a variant of this result for joinings of full entropy on in conjunction with a method of host this yields the following denote aeq x qx mod 1 then for every p invariant ergodic with positive entropy 1 n p ni1 n 0 aecn converges weak to lebesgue measure as n i 1 under a certain mild combinatorial condition onfckg for instance the condition is satisfled if p 10 and ck 2 k 6 k or ck 2 2 k this extends a result of johnson and rudolph who considered the sequence ck q k when p and q are multiplicatively independent we also obtain the following corollary concerning hausdorfi dimension of sum sets for any sequence fsig of p invariant closed subsets of i f p dimh si j log dimh si j 1 then dimh s1 sn i 1

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

characterization of palm measures via bijective point shifts the paper considers a stationary point process n in r d a point map picks a point of n in a measurable way it is called bijective thorisson h 2000 coupling stationarity and regeneration springer new york if it is generating by suitable shifts a bijective mapping on n mecke math nachr 65 1975 335 344 proved that the palm measure of n is point stationary in the sense that it is invariant under bijective point shifts our main result identifies this property as being characteristic for palm measures this generalizes a fundamental classical result for point processes on the line see e g theorem 11 4 in kallenberg o 2002 foundations of modern probability 2nd ed springer new york and solves a problem posed in thorisson h 2000 coupling stationarity and regeneration springer new york and ferrari p a landim c and thorisson h 2004 ann inst h poincare probab statist 40 141 152 our second result guarantees the existence of bijective point maps that have almost surely with respect to the palm measure of n no fixed points this answers another question asked by thorisson our final result shows that there is a directed graph with vertex set n that is defined in a translation invariant way and whose components are almost surely doubly infinite paths this generalizes and complements one of the main results in holroyd a e and peres y 2003 electron comm prohab 8 17 27 no additional assumptions as ergodicity nonlattice type conditions or a finite intensity are made in this paper

critical branching in a highly fluctuating random medium a model of one dimensional critical branching superprocess is constructed in a random medium fluctuating both in time and space the medium describes a moving system of point catalysts and branching occurs only in the presence of these catalysts although the medium has an infinite overall density the clumping features of the branching model can be exhibited by rescaling time space and mass by an exactly calculated scaling power which is stronger than in the constant medium case the main technique used is the asymptotic analysis of a generalized diffusion reaction equation in the space time random medium which given the medium prescribes the evolution of the laplace transition functional of the markov branching process

some issues in discrete approximate solution for stochastic differential equations abstract an evaluation method for numerical schemes of stochastic differential equations is treated discussing the source of errors in the discrete numerical solution we highlight the effect of pseudo random numbers upon the numerical solution and point out the significance of the independencies of the series of them required in the numerical schemes to discriminate the stochastic and deterministic parts in the errors more clearly we propose a new two dimensional linear test equation of multiplicative type whose analytical solution can be obtained readily our results are illustrated through some numerical examples

a second order time accurate ale lagrange galerkin method applied to wind engineering and control of bridge profiles abstract in this paper we propose a second order time accurate ale formulation of lagrange galerkin methods for the transient solution of navier stokes equations for an incompressible fluid in a moving domain following a previous work rev europeenne elem finis 8 1999 659 we show that this time discretization based on a backward differentiation formula and the construction of characteristic paths in a grid relative velocity field leads to second order time accuracy numerical results are compared with those given by a first order approach the numerical diffusion is dramatically reduced which makes the use of turbulence model quasi mandatory for realistic wind engineering applications the software developed was also used to solve elementary control problems the solution method is based on the linearization of the discretized problem including the method of characteristics the formulation and implementation were successfully completed and will be detailed elsewhere

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

some local approximations of dawson watanabe superprocesses let be a dawson watanabe superprocess in rd such that t is a s locally finite for every t 0 then for d 2 and fixed t 0 the singular ran dom measure t can be a s approximated by suitably normalized restrictions of lebesgue measure to the neighborhoods of suppff when d 3 the local distributions of near a hitting point can be approximated in total vari ation by those of a stationary and self similar pseudo random measure f by contrast the corresponding distributions for d 2 are locally invariant fur ther results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities our main proofs are based on a detailed analysis of the historical structure of f

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

multistep characteristic method for incompressible flow in porous media in this paper we present a multistep difference scheme for the problem of miscible displacement of incompressible fluid flow in porous media the discretization involves a three level time scheme based on the characteristic method and a five point finite difference scheme for space discretization we prove that the convergence is of order o h 2 dt 2 which is in contrast to the convergence of order o h dt proved for a singlestep characteristic with the same space discretization numerical experiments demonstrate the stability and second order convergence of the scheme

self similar random measures are locally scale invariant in an earlier paper patzschke and u zahle 11 have proved the existence of a fractional tangent measure at the typical point of a self similar random measure under rather special technical assumptions in the present paper we remove the most restrictive one here we suppose the open set condition for the similarities a constant positive lower bound for the random contraction ratios and vanishing on the boundary of the open set with probability 1 the tangent measure isd scale invariant whered is the similarity dimension of moreover we approximate the tangential distribution by means of and use this in order to prove that the hausdorff dimension of the tangent measure equalsd since the former coincides with the hausdorff dimension of we obtain an earlier result of mauldin and williams 9 as a corollary

weak second order s rock methods for stratonovich stochastic differential equations it is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used this has led to a plethora of implicit or semi implicit methods with a wide variety of stability properties however for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis such as those arising from stochastic partial differential equations explicit methods with extended stability regions can be very effective in the present paper our aim is to derive explicit runge kutta schemes for non commutative stratonovich stochastic differential equations which are of weak order two and which have large stability regions this will be achieved by the use of a technique in chebyshev methods for ordinary differential equations

thick points of super brownian motion we determine for a super brownian motion xt t 0 in d d 3 the precise gauge function such that almost surely on survival up to time t open image in new window improving a result of barlow evans and perkins about the most visited sites of super brownian motion we also determine upper and lower bounds for the hausdorff dimension spectrum of thick points refining the multifractal analysis of super brownian motion by taylor and perkins the upper bound conjectured to be sharp involves a constant which can be characterized in terms of the upper tails of the associated equilibrium palm distribution

discrete gradient approach to stochastic differential equations with a conserved quantity general autonomous stochastic differential equations sdes driven by one dimensional brownian motion in the stratonovich sense with a conserved quantity i y are considered relying on this conserved quantity an equivalent skew gradient sg form of original sdes is constructed with the aim of constructing conserved numerical methods direct discrete gradient approaches which approximate directly the sg system lead to two conserved methods in indirect discrete gradient approaches we first split the sg system into subsystems which preserve the original conserved quantity then apply direct discrete gradient approaches to subsystems to produce two composition schemes the mean square convergence of order 1 for these four methods depends on the assumptions that the coefficients of sdes are lipschitz continuous with bounded second moments along the solution of sdes are twice continuous differentiable and have derivatives that are uniformly bounded three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods