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

pinning class of the wiener measure by a functional related martingales and invariance properties for a given functional y on the path space we define the pinning class of the wiener measure as the class of probabilities which admit the same conditioning given y as the wiener measure using stochastic analysis and the theory of initial enlargement of filtration we study the transformations not necessarily adapted which preserve this class we prove in this non markov setting a stochastic newton equation and a stochastic noether theorem we conclude the paper with some non canonical representations of brownian motion closely related to our study

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

the modified decomposition method for eighth order boundary value problems a modified adomian decomposition method is developed for the numerical solution of eighth order boundary value problems bvps the calculated numerical results show reliability and efficiency of the algorithm given the numerical procedure is tested on linear and nonlinear problems

malliavin calculus and euclidean quantum mechanics i functional calculus abstract we give a rigorous version of the functional calculus developed by r feynman in relation to his path integral formulation of quantum mechanics our approach is euclidean but distinct from the one founded on the feynman kac formula it uses two basic ingredients a new probabilistic interpretation of the classical heat equation introduced recently in the framework of euclidean quantum mechanics and an infinite dimensional differential calculus adapted to functionals of the diffusion processes relevant for this interpretation

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

symplectic structure for gaussian diffusions using information exclusively about classical hamiltonian boundary value problems with quadratic time dependent potentials we construct probability measures providing a symplectic structure associated with some inhomogeneous gaussian diffusion processes introduced earlier t kolsrod and j c zambrini j math phys 33 1301 1992

a complete class theorem for statistical problems with finite sample spaces this paper contains a complete class theorem theorem 3 2 which applies to most statistical estimation problems having a finite sample space this theorem also applies to many other statistical problems with finite sample spaces the description of this complete class involves a stepwise algorithm at each step of the process it is necessary to construct the bayes procedures in a suitably modified version of the original problem the complete class is a minimal complete class if the loss function is strictly convex some examples are given to illustrate the application of this complete class theorem among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function see example 4 5

quantum mechanics and stochastic control theory a time symmetric stochastic control theory is proposed as one of the representatives of quantum mechanics the main idea is based on nelson s probability theoretical approach to quantum mechanics his approach is reformulated as a time symmetric stochastic control problem several different control constraints equivalent to nelson s are obtained one of them has a close connection to the lagrangian formalism of classical mechanics this suggests to us the use of stochastic calculus of variations within the realm of this time symmetric stochastic control theory it is shown why schrodinger s original variational method of quantization was successful several advantageous points of the stochastic control theoretical approach to quantum mechanics including the analysis of the classical limit are also discussed

stochastic dynamical interpolation formula to time reversible diffusion processes a diffusion process of which the diffusion property is preserved under time reversal is called a time reversible diffusion process a typical example of such a process is found in nelson s stochastic mechanics within the extended framework of stochastic integrals proposed by ito a sort of the interpolation formula called a stochastic dynamical interpolation formula to the time reversible diffusion process is derived from a pinned process for the diffusion process it is shown that the stochastic interpolation formula is regarded as an extention of levy s interpolation formula which is obtained on the basis of a pinned brownian motion

reciprocal diffusions in flat space we describe the theory of reciprocal diffusions in flat space a reciprocal process is a markov random field on a one dimensional parameter space every markov process is reciprocal but not vice versa we descibe the first and second order mean differential characteristics of reciprocal diffusions this includes a new definition of stochastic acceleration we show that reciprocal diffusions satisfy stochastic differential equations of second order associated to a reciprocal diffusion is a sequence of conservation laws the first two of which are the familiar continuity and euler equations there are two cases where these laws can be closed after the first two they are the mutually exclusive subclasses of markov and quantum diffusions the latter corresponds to solutions of the schrodinger equation and may be part of a stochastic description of quantum mechanics

an introduction to the semiclassical limit of euclidean quantum mechanics a new class of gaussian bernstein diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new euclidean strategy for the quantization of systems of particles in a potential euclidean quantum mechanics all the measures known by other means up to now are included as particular cases of the present framework it is shown that the theory may be regarded as a well defined probabilistic version of feynman s space time approach distinct in particular from the one due to kac and with much closer relations with classical lagrangian dynamics many explicit examples are considered

feynman kac kernels in markovian representations of the schrodinger interpolating dynamics probabilistic solutions of the so called schrodinger boundary data problem provide for a unique markovian interpolation between any two strictly positive probability densities designed to form the input output statistics data for the process taking place in a finite time interval the key issue is to select the jointly continuous in all variables positive feynman kac kernel appropriate for the phenomenological physical situation we extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation and prove the existence of a continuous markovian interpolation in this case next we analyze the compatibility of this stochastic evolution with the original parabolic dynamics which is assumed to be governed by the temporally adjoint pair of parabolic partial differential equations and prove that the pertinent random motion is a diffusion process in particular in conjunction with born s statistical interpretation postulate in quantum theor

the mean field schrodinger problem ergodic behavior entropy estimates and functional inequalities we study the mean field schrodinger problem mfsp that is the problem of finding the most likely evolution of a cloud of interacting brownian particles conditionally on the observation of their initial and final configuration its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent brownian particles we undertake a fine study of the dynamics of its solutions including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger as well as a novel class of functional inequalities involving the mean field entropic cost i e the optimal value in mfsp our strategy unveils an interesting connection between forward backward stochastic differential equations and the riemannian calculus on the space of probability measures introduced by otto which is of independent interest

stochastic euler poincare reduction we prove a euler poincare reduction theorem for stochastic processes taking values on a lie group which is a generalization of the reduction argument for the deterministic case j e marsden and t s ratiu introduction to mechanics and symmetry a basic exposition of classical mechanical systems texts in applied mathematics springer 2003 we also show examples of its application to so 3 and to the group of diffeomorphisms which includes the navier stokes equation on a bounded domain and the camassa holm equation

small noise limit and convexity for generalized incompressible flows schrodinger problems and optimal transport this paper is concerned with six variational problems and their mutual connections the quadratic monge kantorovich optimal transport the schrodinger problem brenier s relaxed model for incompressible fluids the so called brodinger problem recently introduced by arnaudon et al an entropic interpolation problem for incompressible viscid fluids arxiv preprint arxiv 1704 02126 2017 the multiphase brenier model and the multiphase brodinger problem all of them involve the minimization of a kinetic action and or a relative entropy of some path measures with respect to the reversible brownian motion as the viscosity parameter nu rightarrow 0 we establish gamma convergence relations between the corresponding problems and prove the convergence of the associated pressures arising from the incompressibility constraints we also present new results on the time convexity of the entropy for some of the dynamical interpolations along the way we extend previous results by lavenant calc var partial differ equ 56 6 170 2017 and benamou et al generalized incompressible flows multi marginal transport and sinkhorn algorithm arxiv preprint arxiv 1710 08234 2017

non negative feynman kac kernels in schrodinger s interpolation problem the local formulations of the markovian interpolating dynamics which is constrained by the prescribed input output statistics data usually utilize strictly positive feynman kac kernels this implies that the related markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process we discuss an extension of the framework to encompass singular potentials and associated non negative feynman kac type kernels it allows us to deal with a class of continuous interpolations admitted by general non negative solutions of the schrodinger boundary data problem the resulting nonstationary stochastic processes are capable of both developing and destroying nodes zeros of probability densities in the course of their evolution also away from the spatial boundaries this observation conforms with the general mathematical theory due to m nagasawa and r aebi that is based on the notion of multiplicative functionals extending in turn the well know

an ancillarity paradox in the estimation of multinomial probabilities abstract let x be a multinomial n p variable where n is an ancillary statistic in section 2 it is shown that the minimax estimator of p for fixed sample size n is not minimax for squared error loss in section 3 it is shown that the minimax estimator of p for fixed sample size n is still minimax for relative squared error loss

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

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

a simplified variational characterization of schrodinger processes the variational problem e 1 2 t0 t a t xt 2 dt t0c t xt dt min where t is the drift process of a diffusion process with unit diffusion coefficient and given initial and final distributions a is a given vector field and c is a given scalar field is considered it is shown that the solution is given by a certain markovian diffusion process which in the case a 0 c 0 first was investigated by schrodinger sitzungsber preuss akad wiss phys math kl 1931 144