transition functions paths and path integrals the main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which kolmogorov achieved via pde theory the theory of partial differential equations and his criterion for path continuity section 2 describes how ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process x by solving an sde stochastic differential equation driven by brownian motion of course sdes have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems though 2 begins heuristically the mathematics is then tightened up this paper is after all a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation once kolmogorov s precise language is available 2 then takes a quick sight seeing trip through some of the great developments by doob ito and their successors in an age in which so many do simulations of ito equations i have explained precisely in the briefest possible fashion what the exact theory is it is easy and usable you will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems and of what the malliavin calculus is about you will however be advised on guide books on these and other areas including physicists favourites large deviations measure valued diffusions etc so that you can later explore at your leisure confession the paper consists very largely of selected tracks remixed from the album rogers williams 1987 diffusions markov processes and martingales chichester wiley tributes to kolmogorov s work in probability and statistics have appeared in every book ever written on probability and in ann probability 17 1989 815 964 ann statist 18 1990 987 1031 bull lond math soc 22 1990 31 100 teor veroyatnost primenen 34 1989 no 1 usp mat nauk 43 1988 no 6 the official biography by shiryaev will be a wonderful volume for me this paper is a further expression of my thanks to kolmogorov and as he would have wished to levy doob and ito too

markov snakes and superprocesses we suggest the name markov snakes for a class of path valued markov processes introduced recently by j f le gall in connection with the theory of branching measure valued processes le gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v v2 we establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of markov snakes and vice versa by using this theorem we get limit theorems for discrete markov snakes

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

singularity of super brownian local time at a point catalyst in a one dimensional single point catalytic continuous super brownian motion studied by dawson and fleischmann the occupation density measure c at the catalyst s position c is shown to be a singular diffuse random measure the source of this qualitative new effect is the irregularity of the varying medium c describing the point catalyst the proof is based on a probabilistic characterization of the law of the palm canonical clusters x appearing in the levy khintchine representation of c in a historical process setting and the fact that these x have infinite left upper density with respect to lebesgue measure at the palm time point

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

sample path large deviations for super brownian motion we derive two large deviation principles of freidlin wentzell type for rescaled super brownian motion for one of the appearing rate functions an integral representation is given and interpreted as kakutani hellinger energy as a tool we develop estimates for the laplace functionals of historical super brownian motion and certain maximal inequalities also it is shown that the holder norm of index 1 2 of the processt f x t possesses some finite exponential moments provided the functionf is smooth

asymptotic expansions in limit theorems for stochastic processes iii for families of processes with independent increments t 0 t t with frequent small jumps limit theorems for expectations of the functionals f 0 t are proved of the form where did are positive numbers a di are linear integro differential or differential operators acting on functionals and some differentiability conditions are imposed on the functional f the case of power tails of the jump distribution is considered

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

the brownian cactus ii upcrossings and local times of super brownian motion we study properties of the random metric space called the brownian map for every r 0 we consider the connected components of the complement of the open ball of radius r centered at the root and we let mathbf n r varepsilon be the number of those connected components that intersect the complement of the ball of radius r varepsilon we then prove that varepsilon 3 mathbf n r varepsilon converges as varepsilon rightarrow 0 to a constant times the density at r of the profile of distances from the root in terms of the brownian cactus this gives asymptotics for the number of points at height r that have descendants at height r varepsilon our proofs are based on a similar approximation result for local times of super brownian motion by upcrossing numbers our arguments make a heavy use of the brownian snake and its special markov property

regular transition functions and regular superprocesses the class of regular markov processes is very close to the class of right processes studied by meyer getoor and others we say that a transition function p is regular if it is the transition function of a well defined regular markov process a characterization of regular transition functions is given which implies that if p is regular then the dawson watanabe and the flemingviot supertransition functions over p belong to the same class

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

a path integration formulation of stochastic lagrangian models of turbulent flow abstract in analogy with the feynman kac path integration formulation of quantum mechanics a general path integration formulation of stochastic lagrangian models of turbulent flow is developed with respect to the equivalence with the usual eulerian and stochastic lagrangian descriptions we then discuss efficient numerical methods for the realisation of the lagrangian solutions via path integration which provide an effective alternative to the simpler monte carlo approach

on tractability of path integration many applications require approximate values of path integrals a typical approach is to approximate the path integral by a high dimensional integral and apply a monte carlo randomized algorithm however monte carlo algorithm requires roughly e 2 integrand evaluations to provide an e approximation moreover the error bound of e is guaranteed only in a stochastic sense do we really need to use randomized algorithms for path integrals perhaps we can find a deterministic algorithm that is more effective even in the worst case setting to answer this question we study the worst case complexity of path integration which roughly speaking is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e we consider path integration with respect to a gaussian measure and for various classes of integrands tractability of path integration means that the complexity depends polynomially on 1 e we show that for the class of r times frechet differentiab

superdiffusions and parabolic nonlinear differential equations we establish connections between superdiffusion processes and one class of nonlinear parabolic differential equations analytic results due to brezis and friedman baras and pierre and others are used to investigate the graphs of superdiffusions a survey of the literature and general comments are presented in section 4

sur la mesure de sortie du super mouvement brownien we study some properties of the exit measure of super brownian motion from a smooth domaind inr d in particular we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary as an application we compute the hausdorff dimension of the support of the exit measure in dimension 2 we prove that the exit measure is absolutely continuous with respect to the lebesgue measure on the boundary in connection with dynkin s work our results give some information on the behavior of solutions of u u 2 ind and are related to the characterization of removable singularities at the boundary as a consequence of our estimates we give a sufficient condition for the uniqueness of the positive solution of u u 2 ind that tends to on an open subseto of d and to 0 on the complement in d of the closure ofo our proofs use the path valued process studied in l2 l3

mixed interpolating smoothing splines and the v spline abstract in their monograph bezhaev and vasilenko have characterized the mixed interpolating smoothing spline in the abstract setting of a hilbert space in this paper we derive a similar characterization under slightly more general conditions this is specialized to the finite dimensional case and applied to a few well known problems including the spline a piecewise polynomial spline in tension and near interpolation as well as interpolation and smoothing in particular one of the main objectives in this paper is to show that the spline is actually a mixed spline an observation that we believe was not known prior to this work we also show that the spline is a limiting case of smoothing splines as certain weights increase to infinity and a limiting case of near interpolants as certain tolerances decrease to zero we conclude with an iteration used to construct curvature bounded spline curves

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

nonlinear parabolic p d e and additive functionals of superdiffusions suppose that e is an arbitrary domain in r d l is a second order elliptic differential operator in s r x e and s e is the extremal part of the martin boundary for the corresponding diffusion let 1 2 we investigate a boundary value problem u r lu u in s u v on s e u 0 on x e involving two measures and for the existence of a solution we give sufficient conditions in terms of a martin capacity and necessary conditions in terms of hitting probabilities for an l superdiffusion x if a solution exists then it can be expressed by an explicit formula through an additive functional a of x an l superdiffusion is a branching measure valued process a natural linear additive nla functional a of x is determined uniquely by its potential h defined by the formula p a 0 h r x dr dx for all m the determining set of a every potential h is an exit rule for and it has a unique decomposition into extremal exit rules if and are measures which appear in this decomposition then can be replaced by an integral equation u r x p r x t dy u t y ds h r x where p r x t dy is the transition function of we prove that h is the potential of a nla functional if and only if has a solution u moreover u r x log p r x e a 0 by applying these results to homogeneous functionals of time homogeneous superdiffusions we get a stronger version of theorems proved in an earlier publication the foundation for our present investigation is laid by a general theory developed in the accompanying paper

a new algorithm and worst case complexity for feynman kac path integration abstract we study algorithms for approximation of feynman kac path integrals in the worst case setting the algorithms use a finite number of samples of the initial condition and potential functions we present a new algorithm and an explicit bound on its cost to compute an e approximation to the feynman kac path integral we also establish bounds on the worst case complexity of feynman kac path integration the upper bound is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem the lower bound is given in terms of the complexity of a certain weighted integration problem for some classes of functions these two bounds coincide modulo a multiplicative factor in this case the new algorithm is almost optimal the new algorithm requires precomputation of some real coefficients that are combinations of multivariate integrals with special weights this precomputation is difficult and limits the application of the new algorithm we report the results of the precomputation for specific cases