stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

variational formulations of differential equations and asymmetric fractional embedding abstract variational formulations for classical dissipative equations namely friction and diffusion equations are given by means of fractional derivatives in this way the solutions of those equations are exactly the extremal of some fractional lagrangian actions the formalism used is a generalization of the fractional embedding developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 2007 033504 where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives moreover this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle

error bounds on ultrasonic scatterer size estimates precision errors that occur in estimating the average scatterer size from pulse echo ultrasound waveforms are examined in detail the method independent lower bound on estimation error is found from the cramer rao inequality for comparison with the predicted error for the measurement technique currently used to estimate scatterer sizes in soft biological tissues the probability density function for the estimate is also derived from these statistical analyses strategies for designing experiments that minimize the error are discussed it is shown that compared with biological variability measurement errors in scatterer size estimates are relatively large consequently there is reason to continue searching for more efficient estimators although the analysis and results are derived for gaussian correlation models that have been used to study the function and structure of kidneys generalization to include correlation models for other tissues is straightforward

non differentiable variational principles abstract we develop a calculus of variations for functionals which are defined on a set of non differentiable curves we first extend the classical differential calculus in a quantum calculus which allows us to define a complex operator called the scale derivative which is the non differentiable analogue of the classical derivative we then define the notion of extremals for our functionals and obtain a characterization in term of a generalized euler lagrange equation we finally prove that solutions of the schrodinger equation can be obtained as extremals of a non differentiable variational principle leading to an extended hamilton s principle of least action for quantum mechanics we compare this approach with the scale relativity theory of nottale which assumes a fractal structure of space time

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

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

time scale differential integral and variational embeddings of lagrangian systems we introduce differential integral and variational delta embeddings we prove that the integral delta embedding of the euler lagrange equations and the variational delta embedding coincide on an arbitrary time scale in particular a new coherent embedding for the discrete calculus of variations that is compatible with the least action principle is obtained

a non differentiable noether s theorem in the framework of the nondifferentiable embedding of lagrangian systems defined by cresson and greff non dierentiable embedding of lagrangian systems and partial dierential equations preprint max plank institut fur mathematik in den naturwissenschaften leipzig 16 26 2010 we prove a noether s theorem based on the lifting of one parameter groups of diffeomorphisms

selection of a stochastic landau lifshitz equation and the stochastic persistence problem in this article we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms external and internal the first case corresponds to add a noise term to a given equation using the framework of ito or stratonovich stochastic differential equations the second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations our main concerns for the preservation of properties are stability instability of equilibrium points and symplectic poisson hamiltonian structures we formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different we then apply our results to develop a stochastic version of the landau lifshitz equation we discuss in particular previous results obtained by etore et al j differ equations 257 2115 2135 2014 and we finally propose a new family of stochastic landau lifshitz equations in this article we study the persistence of properties of a given classical deterministic differential equation under a stochastic perturbation of two distinct forms external and internal the first case corresponds to add a noise term to a given equation using the framework of ito or stratonovich stochastic differential equations the second case corresponds to consider parameter dependent differential equations and to add a stochastic dynamics on the parameters using the framework of random ordinary differential equations our main concerns for the preservation of properties are stability instability of equilibrium points and symplectic poisson hamiltonian structures we formulate persistence theorem in these two cases and prove that the cases of external and internal stochastic perturbations are drastically different we then apply our results to develop a stochastic version of the landau lifshitz equation we discuss in particular previous results obtained by etore et al j differ equations 257

stability of the overdamped langevin equation in double well potential abstract in this article we discuss stability of the one dimensional overdamped langevin equation in double well potential we determine unstable and stable equilibria and discuss the rate of convergence to stable ones also we derive conditions for stability of general diffusion processes which generalize the classical and well known results of khasminskii 8

integration by parts and time reversal for diffusion processes in this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property assuming the existence of a density at every time t the proofs are based on techniques of the stochastic calculus of variations

irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise in this paper we derive several results related to the long time behavior of a class of stochastic semilinear evolution equations in a separable hilbert space h d mathfrak u t rm a mathfrak u t rm b mathfrak u t mathfrak u t dt dl t quad mathfrak u 0 x in rm h here a is a positive self adjoint operator and b is a bilinear map and the driving noise l is basically a d rm a 1 2 valued levy process satisfying several technical assumptions by using a density transformation theorem type for levy measure we first prove a support theorem and an irreducibility property of the ornstein uhlenbeck processes associated to the nonlinear stochastic problem second by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain gamma in 0 1 4 b is continuous on d rm a gamma times d rm a gamma with values in d rm a 1 2 using a coupling argument the exponential ergodicity is also proved under the stronger assumption that b is continuous on rm h times rm h while the latter condition is only satisfied by the nonlinearities of goy and sabra shell models the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2d navier stokes magnetohydrodynamics equations the 3d leray varvec alpha model the goy and sabra shell models

stochastic perturbation theory in this paper classical matrix perturbation theory is approached from a probabilistic point of view the perturbed quantity is approximated by a first order perturbation expansion in which the perturbation is assumed to be random this permits the computation of statistics estimating the variation in the perturbed quantity up to the higher order terms that are ignored in the expansion these statistics tend to be more realistic than perturbation bounds obtained in terms of norms the technique is applied to a number of problems in matrix perturbation theory including least squares and the eigenvalue problem

hamiltonian systems with levy noise symplecticity hamilton s principle and averaging principle abstract this work focuses on topics related to hamiltonian stochastic differential equations with levy noise we first show that the phase flow of the stochastic system preserves symplectic structure and propose a stochastic version of hamilton s principle by the corresponding formulation of the stochastic action integral and the euler lagrange equation based on these properties we further investigate the effective behavior of a small transversal perturbation to a completely integrable stochastic hamiltonian system with levy noise we establish an averaging principle in the sense that the action component of solution converges to the solution of a stochastic differential equation when the scale parameter goes to zero furthermore we obtain the estimation for the rate of this convergence finally we present an example to illustrate these results

almost global stochastic stability we develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point with respect to the lebesgue measure is asymptotically attracted to the origin with unit probability the method can be viewed as a dual to lyapunov s second method for stochastic differential equations and extends the deterministic result of a rantzer syst control lett 42 2001 pp 161 168 the result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization the main technical tool is the theory of stochastic flows of diffeomorphisms

statistics of ultrasonic scatterer size estimation with a reference phantom a theoretical expression for the variance of scatterer size estimates is derived for a modified least squares size estimator used in conjunction with a reference phantom method for backscatter coefficient measurement a gaussian spatial autocorrelation function is assumed simulations and phantom experiments were performed to verify the results for backscatter and size variances the dependence of size estimate errors upon free experimental parameters is explored implications of the findings for the optimization of scatterer size estimation are discussed the utility of scatterer size parametric imaging is examined through the signal to noise ratio comparison with standard ultrasonic b mode imaging

hamilton s principle in stochastic mechanics in this paper we establish three variational principles that provide new foundations for nelson s stochastic mechanics in the case of nonrelativistic particles without spin the resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature we first develop two stochastic variational principles whose hamilton jacobi like equations are precisely the two coupled partial differential equations that are obtained from the schrodinger equation madelung equations the two problems are zero sum noncooperative stochastic differential games that are familiar in the control theory literature they are solved here by means of a new absolutely elementary method based on lagrange functionals for both games the saddle point equilibrium solution is given by the nelson s process and the optimal controls for the two competing players are precisely nelson s current velocity v and osmotic velocity u respectively the first variational principle includes as special cases both the guerra morato variational principle phys rev d 27 1774 1983 and schrodinger original variational derivation of the time independent equation it also reduces to the classical least action principle when the intensity of the underlying noise tends to zero it appears as a saddle point action principle in the second variational principle the action is simply the difference between the initial and final configurational entropy it is therefore a saddle point entropy production principle from the variational principles it follows in particular that both v x t and u x t are gradients of appropriate principal functions in the variational principles the role of the background noise has the intuitive meaning of attempting to contrast the more classical mechanical features of the system by trying to maximize the action in the first principle and by trying to increase the entropy in the second combining the two variational principles we get the quantum hamilton principle i e a variational characterization of the logarithm of the wave function the lagrangian is the lagrangian of classical mechanics with the complex valued velocity v iu replacing the classical velocity the dynamics is given by a stochastic differential equation for real valued diffusions with complex valued drift and driving noise processes from the variational principle we derive a newton type law we finally define the momentum process and show that its mean and variance coincide with those of the quantum momentum operator

weak solution of a stochastic 3d cahn hilliard navier stokes model driven by jump noise abstract we investigate a stochastic 2d and 3d cahn hilliard navier stokes system with a multiplicative noise of levy type the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard system for the order phase parameter we prove that the system the existence of weak martingale solution for both 2d and 3d cases the proof of the existence is based on a classical galerkin approximation as well as some compactness methods in the 2d case we prove the pathwise uniqueness of the weak solution

barriers to transport and mixing in volume preserving maps with nonzero flux abstract we identify some geometric structures secondary tori that restrict transport and prevent mixing in perturbations of integrable volume preserving systems with nonzero net flux unlike the customary kam tori secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction we also note that secondary tori persist under the addition of a net flux which destroys all customary kam tori we introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori we then implement the algorithm and after running it present assertions regarding the distribution of the escape times of the unbounded orbits the abundance of secondary tori the size of the resonant regions and the robustness of the tori under the addition of a mean flux

an averaging principle for stochastic dynamical systems with levy noise abstract the purpose of this paper is to establish an averaging principle for stochastic differential equations with non gaussian levy noise the solutions to stochastic systems with levy noise can be approximated by solutions to averaged stochastic differential equations in the sense of both convergence in mean square and convergence in probability the convergence order is also estimated in terms of noise intensity two examples are presented to demonstrate the applications of the averaging principle and a numerical simulation is carried out to establish the good agreement

local approximation to the critical parameters of quantum wells we calculate the critical parameters for some simple quantum wells by means of the riccati pade method the original approach converges reasonably well for nonzero angular momentum quantum number l but rather too slowly for the s states we therefore propose a simple modification that yields remarkably accurate results for the latter case the rate of convergence of both methods increases with l and decreases with the radial quantum number n we compare rpm results with wkb ones for sufficiently large values of l as illustrative examples we choose the one dimensional and central field gaussian wells as well as the yukawa potential the application of perturbation theory by means of the rpm to a class of rational potentials yields interesting and baffling unphysical results

the levinson theorem the levinson theorem is a fundamental theorem in quantum scattering theory which shows the relation between the number of bound states and the phase shift at zero momentum for the schrodinger equation the levinson theorem was established and developed mainly with the jost function with the green function and with the sturm liouville theorem in this review we compare three methods of proof study the conditions of the potential for the levinson theorem and generalize it to the dirac equation the method with the sturm liouville theorem is explained in some detail references to development and application of the levinson theorem are introduced

upper and lower limits on the number of bound states in a central potential in a recent paper new upper and lower limits were given in the context of the schrodinger or klein gordon equations for the number n0 of s wave bound states possessed by a monotonically nondecreasing central potential vanishing at infinity in this paper these results are extended to the number nl of bound states for the lth partial wave and results are also obtained for potentials that are not monotonic and even somewhere positive new results are also obtained for the case treated previously including the remarkably neat lower limit nl 2l 1 1 2 with 2 max0 r r v r 1 2 valid in the schrodinger case for a class of potentials that includes the monotonically nondecreasing ones entailing the following lower limit for the total number n of bound states possessed by a monotonically nondecreasing central potential vanishing at infinity n 1 2 3 2 2 here the double braces denote the integer part

well posedness and invariant measures for a class of stochastic 3d navier stokes equations with damping driven by jump noise abstract a class of stochastic 3d navier stokes equation with damping driven by a noise of levy type is considered in this paper the existence and uniqueness of the solution for the problem 1 1 1 2 are proved for 2 with any 0 and 1 4 as 2 the main result covered various types of spde such as stochastic 3d navier stokes equations with damping stochastic tamed 3d navier stokes equations stochastic three dimensional brinkman forchheimer extended darcy model by using the exponential stability of solutions the existence of a unique invariant measures is proved

non differentiable embedding of lagrangian systems and partial differential equations abstract we develop the non differentiable embedding theory of differential operators and lagrangian systems using a new operator on non differentiable functions we then construct the corresponding calculus of variations and we derive the associated non differentiable euler lagrange equation and apply this formalism to the study of pdes first we extend the characteristics method to the non differentiable case we prove that non differentiable characteristics for the navier stokes equation correspond to extremals of an explicit non differentiable lagrangian system second we prove that the solutions of the schrodinger equation are non differentiable extremals of the newton s lagrangian

the existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and levy jumps in this paper we consider a class of stochastic pantograph differential equations with levy jumps spdewljs by using the burkholder davis gundy inequality and the kunita s inequality we prove the existence and uniqueness of solutions to spdewljs whose coefficients satisfying the lipschitz conditions and the local lipschitz conditions meantime we establish the p th exponential estimations and almost surely asymptotic estimations of solutions to spdewljs

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

uniqueness of nelsons diffusions let 2 be a generalized schrodinger operator or generator of nelsons diffusion defined on c 0 d where is a continuous and strictly positive function on an open domain d d such that lloc2 d some results are given about the two questions below i whether does generate a unique semigroup in l1 d 2dx ii whether the semigroup determined by is strong feller

large deviations and transitions between equilibria for stochastic landau lifshitz gilbert equation we study a stochastic landau lifshitz equation on a bounded interval and with finite dimensional noise we first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property next we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method an essential ingredient of the proof is the compactness or weak to strong continuity of the solution map for a deterministic landau lifschitz equation when considered as a transformation of external fields we then apply this large deviations principle to show that small noise can cause magnetisation reversal we also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise the problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

upper and lower limits for the number of s wave bound states in an attractive potential new upper and lower limits are given for the number of s wave bound states yielded by an attractive monotonic potential in the context of the schrodinger or klein gordon equation

existence of global solutions and invariant measures for stochastic differential equations driven by poisson type noise with non lipschitz coefficients the purpose of this paper is twofold firstly we investigate the problem of existence and uniqueness of solutions to stochastic differential equations with one sided dissipative drift driven by semi martingales secondly we investigate the problem of existence of an invariant measure for such equations when the coefficients are time independent

2d stochastic navier stokes equations driven by jump noise abstract in the paper we are studying the existence and uniqueness of the solution of an abstract nonlinear equation driven by a multiplicative noise of levy type our result is formulated in an abstract setting this type of equation covers the stochastic 2d navier stokes equations the 2d stochastic magneto hydrodynamic equations the 2d stochastic boussinesq model for the benard convection the 2d stochastic magnetic bernard problem the 3d stochastic leray model for the navier stokes equations and several stochastic shell models of turbulence

well posedness of stochastic second grade fluids abstract the theory of turbulent newtonian fluids shows that the choice of the boundary condition is a relevant issue because it can modify the behavior of a fluid by creating or avoiding a strong boundary layer in this study we consider stochastic second grade fluids filling a two dimensional bounded domain with the navier slip boundary condition with friction we prove the well posedness of this problem and establish a stability result our stochastic model involves a multiplicative white noise and a convective term with third order derivatives which significantly complicate the analysis

wong zakai approximation for the stochastic landau lifshitz gilbert equations abstract in this work we study stochastic landau lifshitz gilbert equations sllges in one dimension with non zero exchange energy only firstly by introducing a suitable transformation we convert the sllges to a highly nonlinear time dependent partial differential equation with random coefficients which is not fully parabolic we then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property following regular approximation of the brownian motion and using reverse transformation we show existence of strong solution of sllges taking values in a two dimensional unit sphere s 2 in r 3 the construction of the solution and its corresponding convergence results are based on wong zakai approximation

analytic continuation of taylor series and the boundary value problems of some nonlinear ordinary differential equations abstract we compare and discuss the respective efficiency of three methods with two variants for each of them based respectively on taylor maclaurin series pade approximants and conformal mappings for solving quasi analytically a two point boundary value problem of a nonlinear ordinary differential equation ode six configurations of ode and boundary conditions are successively considered according to the increasing difficulties that they present after having indicated that the taylor series method almost always requires the recourse to analytical continuation procedures to be efficient we use the complementarity of the two remaining methods pade and conformal mapping to illustrate their respective advantages and limitations we emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods particularly for the so called pade hankel method we show that this latter method is equivalent to pushing a movable pole to infinity for each configuration we determine the singularity distribution in the complex plane of the independent variable of the solution sought and show how this distribution controls the efficiency of the two methods in general the method based on pade approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

generalized fractional calculus with applications to the calculus of variations we study operators that are generalizations of the classical riemann liouville fractional integral and of the riemann liouville and caputo fractional derivatives a useful formula relating the generalized fractional derivatives is proved as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual such results are explored in the context of dynamic optimization by considering problems of the calculus of variations with general fractional operators necessary optimality conditions of euler lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of riemann liouville or caputo as an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem illustrative optimization problems are considered

scale calculus and the schrodinger equation this paper is twofold in a first part we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus the scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f t e such that f t e f t when e goes to zero this led to several new notions as representations fractal functions and e differentiability the basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative we then discuss some algebraic properties of these operators we define a natural bialgebra called quantum bialgebra associated with them finally we discuss a convenient geometric object associated with our study in a second part we define a first quantization procedure of classical mechanics following the scale relativity theory developed by nottale we obtain a nonlinear schrodinger equation v

a characterization of reciprocal processes via an integration by parts formula on the path space we characterize in this paper the class of reciprocal processes associated to a brownian diffusion therefore not necessarily gaussian as the set of probability measures under which a certain integration by parts formula holds on the path space this functional equation can be interpreted as a perturbed duality equation between malliavin derivative operator and stochastic integration an application to periodic ornstein uhlenbeck process is presented we also deduce from our integration by parts formula the existence of nelson derivatives for general reciprocal processes

second order stochastic differential equations and non gaussian reciprocal diffusions we first prove that a markov diffusion satisfies a second order stochastic differential equation involving the invariants associated to its reciprocal class as a reciprocal process some properties of the noise term are given we also prove that this equation can be viewed as an euler lagrange equation in a problem of calculus of variations in the non markovian case a bernstein bridge is shown to satisfy the same equation but in a weak sense

multiscale functions scale dynamics and applications to partial differential equations modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism randomness and differentiability depending on these choices different behaviors can be observed the natural question associated to the modeling problem is the following with a finite set of data concerning a phenomenon can we recover its underlying nature from this problem we introduce in this paper the definition of multi scale functions scale calculus and scale dynamics based on the time scale calculus see bohner m and peterson a dynamic equations on time scales an introduction with applications springer science business media 2001 which is used to introduce the notion of scale equations these definitions will be illustrated on the multi scale okamoto s functions scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime the introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant typical examples of such equations are given by the scale euler lagrange equation we illustrate our results using the scale newton s equation which gives rise to a non linear diffusion equation or a non linear schrodinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered

successive approximation of neutral stochastic evolution equations with infinite delay and poisson jumps abstract in this paper we study the existence and uniqueness of mild solutions of neutral stochastic evolution equations with infinite delay and poisson jumps in real separable hilbert spaces we study the continuous dependence of solutions on the initial value the nonlinear term in our equations are not assumed to lipschitz continuous the results of this paper generalize and improve some known results

an averaging principle for diffusions in foliated spaces consider an sde on a foliated manifold whose trajectories lay on compact leaves we investigate the effective behavior of a small transversal perturbation of order e an average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ode according to the average of the perturbing vector field with respect to invariant measures on the leaves as e goes to zero an estimate of the rate of convergence is given these results generalize the geometrical scope of previous approaches including completely integrable stochastic hamiltonian system

jump type stochastic differential equations with non lipschitz coefficients non confluence feller and strong feller properties and exponential ergodicity abstract this paper considers multidimensional jump type stochastic differential equations with super linear and non lipschitz coefficients after establishing a sufficient condition for nonexplosion this paper presents sufficient local non lipschitz conditions for pathwise uniqueness the non confluence property for solutions is investigated feller and strong feller properties under local non lipschitz conditions are investigated via the coupling method sufficient conditions for irreducibility and exponential ergodicity are derived as applications this paper also studies multidimensional stochastic differential equations driven by levy processes and presents a feynman kac formula for levy type operators

canonical stochastic dynamical systems a canonical stochastic dynamical system for time symmetric semimartingales is formulated by the stochastic least action principle in a new stochastic calculus of variations a certain class of stochastic dynamical systems gives a hamiltonian formalism of nelson s stochastic mechanics in a manner analogous to classical mechanics the notions of a stochastic poisson bracket and canonical transformation are introduced to the stochastic dynamical systems it is shown that the phase factor of the wave function plays the role of a generating function of the canonical transformation

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

the sharma parthasarathy stochastic two body problem we study the sharma parthasarathy stochastic two body problem introduced by sharma and parthasarathy in dynamics of a stochastically perturbed two body problem proc r soc a 463 979 1003 2007 in particular we focus on the preservation of some fundamental features of the classical two body problem like the hamiltonian structure and first integrals in the stochastic case numerical simulations are performed which illustrate the dynamical behaviour of the osculating elements as the semi major axis the eccentricity and the pericenter we also derive a stochastic version of gauss s equations in the planar case

generalization of the calogero cohn bound on the number of bound states it is shown that for the calogero cohn type upper bounds on the number of bound states of a negative spherically symmetric potential v r in each angular momentum state that is bounds containing only the integral 0 v r 1 2 dr the condition v r 0 is not necessary and can be replaced by the less stringent condition d dr r1 2p v 1 p 0 1 2 p 1 which allows oscillations in the potential the constants in the bounds are accordingly modified depend on p and l and tend to the standard value for p 1 2

correlation of ultrasonic scatterer size estimates for the statistical analysis and optimization of angular compounding ultrasonic scatterer size estimates generally have large variances due to the inherent noise of spectral estimates used to calculate size compounding partially correlated size estimates associated with the same tissue but produced with data acquired from different angles of incidence is an effective way to reduce the variance without making dramatic sacrifices in spatial resolution this work derives theoretical approximations for the correlation between these size estimates and the coherence between their associated spectral estimates as functions of ultrasonic system parameters a gaussian spatial autocorrelation function is assumed to adequately model scatterer shape both approximations compare favorably with simulation results which consider validation near the focus utilization of the correlation coherence expressions for statistical analysis and optimization is discussed approximations such as the invariance of phase and amplitude terms with angle are made to obtain closed form solutions to the derived spectral coherence near the focus and permit analytical optimization analysis results indicate that recommended parameter adjustments for performance improvement generally depend upon whether for the system under consideration the primary source of change in total coherence with rotation is phase term variation due to the change in the relative position of scattering sites or field amplitude term variation due to beam movement

stochastic derivatives for fractional diffusions in this paper we introduce some fundamental notions related to the so called stochastic derivatives with respect to a given fieldq in our framework we recall well known results about markov wiener diffusions we then focus mainly on the case where x is a fractional diffusion and where q is the past the future or the present of x we treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of x when x solves a stochastic differential equation driven by a fractional brownian motion with hurst index h 1 2 we give explicit formulas 1 introduction there exist various ways to generalize the notion of differentiation on deterministic functions we may think of fractional derivatives or differentiation in the sense of the theory of distributions in both cases we lose a dynamical or geometric interpretation of tangent vectors velocities e g in the present work we seek to construct derivatives on stochastic processes which conserve a dynamical meaning our goal is motivated by the stochastic embedding of dynamical systems introduced in 2 this procedure aims at comprehending the following question how can we write an equation which contains the dynamical meaning of an initial ordinary differential equation and which extends this dynamical meaning to stochastic processes we refer to 3 for more details unfortunately for most of the stochastic processes used in physical models the limit zt h zt h