polynomial time 1 turing reductions from ph to p in this paper we investigate relative complexity between p and other classes of functions our particular interest is to compare p with ph and with pfh by using polynomial time reducibility and to demonstrate that a weaker notion of polynomial time reducibility is sufficiently powerful for reducing ph functions to p functions our main result is stated as follows every function in ph is polynomial time 1 turing reducible to some function in p that is ph pf p 1 some consequences of this result are as follows every function in pfh is polynomial time 1 turing reducible to some function in p if pf p 1 ph then ph collapses to a finite level furthermore if either p pfh or pfh p then ph collapses to a finite level we also give an affirmative answer to an open question posed by valiant 1979 and we show a generalized result about p rankability by hemachandra 1987

statistics of shocks in solutions of inviscid burgers equation the purpose of this paper is to analyze statistical properties of discontinuities of solutions of the inviscid burgers equation having a typical realizationb y of the brownian motion as an initial datum this case was proposed and studied numerically in the companion paper by she aurell and frisch the description of the statistics is given in terms of the behavior of the convex hull of the random process w y int limits 0 y b eta eta d eta the hausdorff dimension of the closed set of thosey where the convex hull coincides withw is also studied

statistical properties of shocks in burgers turbulence we consider the statistical properties of solutions of burgers equation in the limit of vanishing viscosity frac partial partial t u left x t right frac partial partial x left frac 1 2 u left x t right 2 right 0 with gaussian whitenoise initial data this system was originally proposed by burgers 1 as a crude model of hydrodynamic turbulence and more recently by zel dovichet al 12 to describe the evolution of gravitational matter at large spatio temporal scales with shocks playing the role of mass clusters we present here a rigorous proof of the scaling relationp s s 1 2 s 1 wherep s is the cumulative probability distribution of shock strengths we also show that the set of spatial locations of shocks is discrete i e has no accumulation points and establish an upper bound on the tails of the shock strength distribution namely 1 p s exp cs 3 fors 1 our method draws on a remarkable connection existing between the structure of burgers turbulence and classical probabilistic work on the convex envelope of brownian motion and related diffusion processes

removable lateral singularities of semilinear parabolic pdes and besov capacities suppose 1 2 lis a second order elliptic differential operator in rdanddis a bounded smooth domain in rd let q r dand let be a compact set on the lateral boundary of q we prove that is a removable lateral singularity for the equationu lu u in q if and only if cap1 2 0 where cap stands for the besov capacity on the boundary

skorokhod embeddings for two sided markov chains let x n n in mathbb z be a two sided recurrent markov chain with fixed initial state x 0 and let nu be a probability measure on its state space we give a necessary and sufficient criterion for the existence of a non randomized time t such that x t n n in mathbb z has the law of the same markov chain with initial distribution nu in the case when our criterion is satisfied we give an explicit solution which is also a stopping time and study its moment properties we show that this solution minimizes the expectation of psi t in the class of all non negative solutions simultaneously for all non negative concave functions psi

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

transforming random elements and shifting random fields consider a locally compact second countable topological transformation group acting measurably on an arbitrary space we show that the distributions of two random elements x and x in this space agree on invariant sets if and only if there is a random transformation r such that rx has the same distribution as x applying this to random fields in d dimensions under site shifts we show further that these equivalent claims are also equivalent to site average total variation convergence this convergence result extends to amenable groups

solutions of δu 4u2 with neumann s conditions using the brownian snake we consider a brownian snake ws s 0 with underlying process a reflected brownian motion in a bounded domain d we construct a continuous additive functional ls s 0 of the brownian snake which counts the time spent by the end points s of the brownian snake paths on d the random measure z sdls is supported by d then we represent the solution v of u 4u2 in d with weak neumann boundary condition 0 by using exponential moment of z under the excursion measure of the brownian snake we then derive an integral equation for v for small it is then possible to describe negative solution of u 4u2 in d with weak neumann boundary condition in contrast to the exit measure of the brownian snake out of d the measure z is more regular in particular we show it is absolutely continuous with respect to the surface measure on d for dimension 2 and 3

small values of the maximum for the integral of fractional brownian motion we consider the integral of fractional brownian motion ifbm and its functionals t on the intervals 0 t and t t of the following types the maximum mt the position of the maximum the occupation time above zero etc we show how the asymptotics of p t 1 pt t is related to the hausdorff dimension of lagrangian regular points for the inviscid burgers equation with fbm initial velocity we produce computational evidence in favor of a power asymptotics for pt the data do not reject the hypothesis that the exponent of the power law is related to the similarity parameter h of fractional brownian motion as follows 1 h for the interval t t and h 1 h for 0 t the point 0 is special in that ifbm and its derivative both vanish there

the computational complexity of maximization and integration on montre que ces deux problemes de la theorie de la complexite de calcul aussi bien que les autres resolvant la maximisation et l integration sont equivalents aux problemes de complexite de calcul discret naturel dont la plupart ont deja beaucoup ete etudies

optimal transport and skorokhod embedding the skorokhod embedding problem is to represent a given probability as the distribution of brownian motion at a chosen stopping time over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties these constructions employ a variety of techniques ranging from excursion theory to potential and pde theory and have been used in many different branches of pure and applied probability we develop a new approach to skorokhod embedding based on ideas and concepts from optimal mass transport in analogy to the celebrated article of gangbo and mccann on the geometry of optimal transport we establish a geometric characterization of skorokhod embeddings with desired optimality properties this leads to a systematic method to construct optimal embeddings it allows us for the first time to derive all known optimal skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings while previous constructions typically used particular properties of brownian motion our approach applies to all sufficiently regular markov processes

optimality functions in stochastic programming optimality functions define stationarity in nonlinear programming semi infinite optimization and optimal control in some sense in this paper we consider optimality functions for stochastic programs with nonlinear possibly nonconvex expected value objective and constraint functions we show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer we construct confidence intervals for the value of the optimality function at a candidate point and hence provide a quantitative measure of solution quality based on sample average approximations we develop an algorithm for classes of stochastic programs that include cvar problems and utilize optimality functions to select sample sizes

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

remarks on uniqueness of boundary blow up solutions abstract we show that there is at most one nonnegative boundary blow up solution for the one dimensional boundary blow up problem u p 2 u f u in 0 1 u 0 u 1 where p 1 provided f c 1 0 c 0 0 with f s 0 and f s 0 for s 0 we see that the same result still holds for some equations with special nonlinearities satisfying f s 0 and f s 0 for s 0 in higher dimensions but we conjecture that the same result should be true for equations with general nonlinearities satisfying f s 0 and f s 0 for s 0 in higher dimensions

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

intersection of independent regenerative sets we investigate the nature of the intersection of two independent regenerative sets the approach combines bochners subordination and potential theory for a pair of markov processes in duality

enumerative counting is hard abstract an n variable boolean formula may have anywhere from 0 to 2n satisfying assignments can a polynomial time machine given such a formula reduce this exponential number of possibilities to a small number of possibilities we call such a machine an enumerator and prove that if there is a good polynomial time enumerator for p i e one where for every boolean formula f the small set has at most o f 1 e numbers then p np p p and probabilistic polynomial time equals polynomial time furthermore we show that p polynomial time turing reduces to enumerating p

complexity classes of optimization functions in this paper complexity classes of functions defined via taking maxima or minima cf the work of krentel or taking middle elements cf the work of toda are examined a number of axioms for a class to be a so called p founded class of optimization functions are given it is shown that many natural function classes fulfill these axioms then these classes are examined concerning their relationship to complexity classes of sets to this end complexity preserving operators s and f for encoding function classes into set classes and vice versa are introduced it is shown how these operators translate closure properties from one class to another how they relate operators on classes of functions and classes of sets and how they encode classes of maximum minimum or median functions into well studied classes of sets the fixpoints of the compositional operator f s are examined and shown to be exactly those function classes closed under binary search let f1 and f2 be two such fixpoints and k1 and k2 be their corresponding classes of sets i e their images under the operator s then f1 f2 if and only if k1 k2 and f1 f2 if and only if k1 k2 a number of natural classes of functions are shown to be such fixpoints thus we build hierachies of function classes with the same inclusional relationships as the polynomial time hierarchy of sets and the counting hierarchy of sets and we prove many interesting structural properties of these hierarchies

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

on some natural complete operators abstract an operator is a mapping from integer functions to integer functions it is known from a result by baker gill and solovay 1975 that there exists an operator computable in polynomial time by a nondeterministic oracle turing machine called an np operator but not computable in polynomial time by any deterministic oracle turing machine we investigate several natural operators which share similar properties we use the concept of completeness to give a precise classification of the complexity of these operators for example the question of finding maximum values of polynomial time computable functions can be formulated as an operator complete for the class of np operators

lipschitz stability for stochastic programs with complete recourse this paper investigates the stability of optimal solution sets to stochastic programs with complete recourse where the underlying probability measure is understood as a parameter varying in some space of probability measures in math programming 67 1994 pp 99 108 sha piro proved lipschitz upper semicontinuity of the solution set mapping inspired by this result we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions for a subclass of recourse models we show that the solution set mapping is hausdorff lipschitz continuous with respect to the subgradient distance moreover the subgradient distance is estimated above by the kolmogorov smirnov distance of certain distribution functions related to the recourse model the lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations finally the lipschitz continuity result is used to derive some new results on the asymptotic behavior ofoptimal solutions when the probability measure underlying the recourse model is estimated via empirical measures law of iterated logarithm large deviation estimate estimate for asymptotic distribution

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 hull process of the brownian plane we study the random metric space called the brownian plane which is closely related to the brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation we obtain a number of explicit distributions for the brownian plane in particular we consider for every r 0 the hull of radius r which is obtained by filling in the holes in the ball of radius r centered at the root we introduce a quantity z r which is interpreted as the generalized length of the boundary of the hull of radius r we identify the law of the process z r r 0 as the time reversal of a continuous state branching process starting from infty at time infty and conditioned to hit 0 at time 0 and we give an explicit description of the process of hull volumes given the process z r r 0 we obtain an explicit formula for the laplace transform of the volume of the hull of radius r and we also determine the conditional distribution of this volume given the length of the boundary our proofs involve certain new formulas for super brownian motion and the brownian snake in dimension one which are of independent interest

brownian time processes the pde connection ii and the corresponding feynman kac formula we delve deeper into our study of the connection of brownian time processes btps to fourth order parabolic pdes which we introduced in a recent joint article with w zheng probabilistically btps and their cousins btps with excursions form a unifying class of interesting stochastic processes that includes the celebrated ibm of burdzy and other new intriguing processes and is also connected to the markov snake of le gall btps also offer a new connection of probability to pdes that is fundamentally different from the markovian one they solve fourth order pdes in which the initial function plays an important role in the pde itself not only as initial data we connect two such types of interesting and new pdes to btps the first is obtained by running the btp and then integrating along its path and the second type of pdes is related to what we call the feynman kac formula for btps a special case of the second type is a step towards a probabilistic solution to linearized cahn hilliard and kuramoto sivashinsky type pdes which we tackle in an upcoming paper

the hausdorff measure of the support of two dimensional super brownian motion we show that two dimensional super brownian motion is a multiple of the h hausdorff measure on its closed support where h r r 2 log 1 r log log log 1 r this complements known results in dimensions greater than 2

an exact discretization of a lax equation for shock clustering and burgers turbulence i dynamical aspects and exact solvability we consider a finite dimensional system which arises as an exact discretization of the lax equation for shock clustering which describes the evolution of the generator of a markov process with a finite number of states the spectral curve of the system is a nodal curve which is fully reducible in this work we will show that the flow is conjugate to a straight line motion and we will also show that the equation is exactly solvable en route we will establish a dictionary between an open dense set of the lower triangular infinitesimal generator matrices and an associated set of algebro geometric data

recursion theoretic characterizations of complexity classes of counting functions abstract there has been a great effort in giving machine independent algebraic characterizations of complexity classes especially of functions astonishingly no satisfactory characterization of the prominent class p is known up to now here we characterize p as the closure of a set of simple arithmetical functions under summation and weak product based on that result the hierarchy of counting functions which is the closure of p under substitution is characterized remarkably without using the operator of substitution since we can show that in the context of this hierarchy the operation of modified subtraction is as powerful as substitution this leads us to a number of consequences concerning closure of p under certain arithmetical operations analogous results are achieved for the class gap p which is the closure of p under subtraction

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

boundary singularities of solutions to elliptic viscous hamilton jacobi equations abstract we study the boundary value problem with measures for e1 u g u 0 in a bounded domain in r n satisfying e2 u on and prove that if g l 1 1 t 2 n 1 n d t is nondecreasing e1 e2 can be solved with any positive bounded measure when g r r q with q 1 we prove that any positive function satisfying e1 admits a boundary trace which is an outer regular borel measure not necessarily bounded when g r r q with 1 q q c n 1 n we prove the existence of a positive solution with a general outer regular borel measure as boundary trace and characterize the boundary isolated singularities of positive solutions when g r r q with q c q 2 we prove that a necessary condition for solvability is that must be absolutely continuous with respect to the bessel capacity c 2 q q q we also characterize boundary removable sets for moderate and sigma moderate solutions

multistep stochastic mirror descent for risk averse convex stochastic programs based on extended polyhedral risk measures we consider risk averse convex stochastic programs expressed in terms of extended polyhedral risk measures we derive computable confidence intervals on the optimal value of such stochastic programs using the robust stochastic approximation and the stochastic mirror descent smd algorithms when the objective functions are uniformly convex we also propose a multistep extension of the stochastic mirror descent algorithm and obtain confidence intervals on both the optimal values and optimal solutions numerical simulations show that our confidence intervals are much less conservative and are quicker to compute than previously obtained confidence intervals for smd and that the multistep stochastic mirror descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart

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

optimal bounds in non gaussian limit theorems for u statistics let x x 1 x be i i d random variables taking values in a measurable space x let x y and 1 x denote measurable functions of the arguments x y x assuming that the kernel is symmetric and that e x x 0 for all x and e 1 x 0 we consider u statistics of type t n 1 x j x k n 1 2 1 x j 1 j k n 1 j n it is known that the conditions e 2 x x 1 and e 2 1 x imply that the distribution function of t say f has a limit say f 0 which can be described in terms of the eigenvalues of the hilbert schmidt operator associated with the kernel x y under optimal moment conditions we prove that n sup f x f 0 x f 1 x o n 1 provided that at least nine eigenvalues of the operator do not vanish here f 1 denotes an edgeworth type correction we provide explicit bounds for n and for the concentration functions of statistics of type t

some observations on the connection between counting and recursion abstract based on valiant s class p of all functions counting the number of accepting computations of nondeterministic polynomial time turing machines the polynomial time hierarchy of counting functions is introduced the class phcf of all functions of this hierarchy and some of its subclasses are characterized by recursion theoretic means it turns out that from the recursion theoretic point of view phcf is an analogue to kalmar s class e of elementary functions to the class pspace of polynomial space computable functions as well as to the class p of polynomial time computable functions

on the equation δu eu 1 0 with measures as boundary data if is a bounded domain in mathbb r n we study conditions on a radon measure on for solving the equation u e u 1 0 in with u on the conditions are expressed in terms of orlicz capacities

asymptotic behavior of optimal solutions in stochastic programming asymptotic behavior of optimal solutions xi n of a sequence of stochastic programming problems is studied variational and generalized equations approaches are discussed an expansion of xi n in terms of a parametrized mathematical programming problem depending on a single random vector is given when optimal solutions of the parametrized program are directionally differentiable this expansion leads to a close form expression for the asymptotic distribution of xi n applicability of the involved regularity conditions to nondifferentiable cases and in particular to stochastic programming with recourse is discussed

skorokhod embeddings minimality and non centred target distributions in this paper we consider the skorokhod embedding problem for target distributions with non zero mean in the zero mean case uniform integrability provides a natural restriction on the class of embeddings but this is no longer suitable when the target distribution is not centred instead we restrict our class of stopping times to those which are minimal and we find conditions on the stopping times which are equivalent to minimality

the statistics of burgers turbulence initialized with fractional brownian noise data the statistics of the solution to the inviscid burgers equation are investigated when the initial velocity potential is fractional brownian motion using the theory of large deviations for gaussian processes we characterize the tails of the probability distribution functions pdfs of the velocity the distance between shocks and the shock strength these pdfs are shown to decay like stretched exponentials of the form our method of proof can also be used to extend these results to a much larger class of gaussian potentials this work generalizes the results of avellaneda and e 2 3 on the inviscid burgers equation with white noise initial data

unbiased shifts of brownian motion let b b t t r be a two sided standard brownian motion an unbiased shift of b is a random time t which is a measurable function of b such that b t t b t t r is a brownian motion independent of b t we characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of b for any probability distribution on r we construct a stopping time t 0 with the above properties such that b t has distribution we also study moment and minimality properties of unbiased shifts a crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on r another new result is an analogue for diffuse random measures on r of the cycle stationarity characterisation of palm versions of stationary simple point processes

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

embedding laws in diffusions by functions of time we present a constructive probabilistic proof of the fact that if b bt t 0 is standard brownian motion started at 0 and is a given probability measure on r such that 0 0 then there exists a unique left continuous increasing function b 0 r and a unique left continuous decreasing function c 0 r such that b stopped at b c inf t 0 bt b t or bt c t has the law the method of proof relies upon weak convergence arguments arising from helly s selection theorem and makes use of the levy metric which appears to be novel in the context of embedding theorems we show that b c is minimal in the sense of monroe so that the stopped process b b c bt b c t 0 satisfies natural uniform integrability conditions expressed in terms of we also show that b c has the smallest truncated expectation among all stopping times that embed into b the main results extend from standard brownian motion to all recurrent diffusion processes on the real line

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

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

positive solutions of schrodinger equations and fine regularity of boundary points given a lipschitz domain in mathbb r n and a nonnegative potential v in such that v x d x 2 is bounded we study the fine regularity of boundary points with respect to the schrodinger operator l v v in using potential theoretic methods several conditions are shown to be equivalent to the fine regularity of z in partial omega the main result is a simple explicit if is smooth necessary and sufficient condition involving the size of v for z in partial omega to be finely regular an intermediate result consists in a majorization of int a vert frac u d partial omega vert 2 dx for u positive harmonic in and a subset omega conditions for almost everywhere regularity in a subset a of are also given as well as an extension of the main results to a notion of fine mathcal l 1 vert mathcal l 0 regularity if mathcal l j mathcal l v j v 0 v 1 being two potentials with v 0 v 1 and mathcal l a second order elliptic operator

kinetic theory and lax equations for shock clustering and burgers turbulence we study shock statistics in the scalar conservation law t u x f u 0 x t 0 with a convex flux f and spatially random initial data we show that the markov property in x is preserved for a large class of random initial data markov processes with downward jumps and derivatives of levy processes with downward jumps the kinetics of shock clustering is then described completely by an evolution equation for the generator of the markov process u x t x we present four distinct derivations for this evolution equation and show that it takes the form of a lax pair the lax equation admits a spectral parameter as in manakov funct anal appl 10 328 329 1976 and has remarkable exact solutions for burgers equation f u u 2 2 this suggests the kinetic equations of shock clustering are completely integrable

limit clusters in the inviscid burgers turbulence with certain random initial velocities we study the infinite time shock limits given certain markovian initial velocities to the inviscid burgers turbulence specifically we consider the one sided case where initial velocities are zero on the negative half line and follow a time homogeneous nice markov process x on the positive half line finite shock limits occur if the markov process is transient tending to infinity they form a poisson point process if x is spectrally negative we give an explicit description when x is furthermore spatially homogeneous a levy process or a self similar process on 0 we also consider the two sided case where we suppose an independent dual process in the negative spatial direction both spatial homogeneity and an exponential levy condition lead to stationary shock limits

the inviscid burgers equation with initial data of brownian type the solutions to burgers equation in the limit of vanishing viscosity are investigated when the initial velocity is a brownian motion or fractional brownian motion function i e a gaussian process with scaling exponent 0 h 1 typea or the derivative thereof with scaling exponent 1 h 0 typeb largesize numerical experiments are performed helped by the fact that the solution is essentially obtained by performing a legendre transform the main result is obtained for typea and concerns the lagrangian functionx a which gives the location at timet 1 of the fluid particle which started at the locationa it is found to be a complete devil s staircase the cumulative probability of lagrangian shock intervals a also the distribution of shock amplitudes follows a a h law for small a the remaining regular lagrangian locations form a cantor set of dimensionh in eulerian coordinates the shock locations are everywhere dense the scaling properties of various statistical quantities are also found heuristic interpretations are provided for some of these results rigorous results for the case of brownian motion are established in a companion paper by ya sinai for typeb initial velocities e g white noise there are very few small shocks and shock locations appear to be isolated finally it is shown that there are universality classes of random but smooth non scaling initial velocities such that the long time large scale behavior is after rescaling the same as for typea orb

stochastic boundary values and boundary singularities for solutions of the equationlu uα abstract we investigate positive solutions of a nonlinear equationlu u wherelis a second order elliptic differential operator in a riemannian manifoldeand 1 2 we establish a 1 1 correspondence between the set u of positive solutions and a class z of functionals ofxwhich we call linear boundary functionals they depend only on the behavior ofxnear the martin boundarye the class z is a closed convex cone andu u is a subadditive function ofz z special roles belong to moderate solutions corresponding tozwith finite mathematical expectations and to a family of solutions determined by the range ofx a new formula is deduced connectingu zandl diffusions conditioned to hit the boundarye at a given pointy a concept of a singular boundary point foruis introduced in terms of the conditioned diffusion

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

the inviscid burgers equation with brownian initial velocity the law of the hopf cole solution of the inviscid burgers equation with brownian initial velocity is made explicit as examples of applications we investigate the smoothness of the solution the statistical distribution of the shocks we determine the exact hausdorff function of the lagrangian regular points and investigate the existence of lagrangian regular points in a fixed borel set

a decisive characterization of bpp the complexity class bpp defined by gill contains problems that can be solved in polynomial time with bounded error probability a new and simple characterization of bpp is given it is shown that a language l is in bpp iff x l y zp x y z x l y z p x y z for a polynomial time predicate p and for y z poly x the formula yp y with the random quantifier means that the probability pr y p y for a fixed this characterization allows a simple proof that bpp zpp np which strengthens the result of lautemann inform process lett 17 1983 215 217 sipser in proceedings 15th annu acm sympos theory of comput 1983 pp 330 335 that bpp 2 p 2 p several other results about probabilistic classes can be proved using similar techniques e g np r zpp np and 2 p bpp 2 p

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

reduced measures on the boundary we study the existence of solutions of the nonlinear problem delta u g u 0 in omega u mu on partial derivative omega where mu is a and g r r is a nondecreasing continuous function with g t 0 for all t 0 problem 0 1 admits a solution for every mu epsilon l 1 partial derivative omega but this need not be the case when p is a general bounded measure we introduce a concept of reduced measure mu in the spirit of brezis et al ann math stud to appear this is the closest measure to mu for which 0 1 admits a solution

assessing solution quality in stochastic programs determining whether a solution is of high quality optimal or near optimal is fundamental in optimization theory and algorithms in this paper we develop monte carlo sampling based procedures for assessing solution quality in stochastic programs quality is defined via the optimality gap and our procedures output is a confidence interval on this gap we review a multiple replications procedure that requires solution of say 30 optimization problems and then we present a result that justifies a computationally simplified single replication procedure that only requires solving one optimization problem even though the single replication procedure is computationally significantly less demanding the resulting confidence interval might have low coverage probability for small sample sizes for some problems we provide variants of this procedure that require two replications instead of one and that perform better empirically we present computational results for a newsvendor problem and for two stage stochastic linear programs from the literature we also discuss when the procedures perform well and when they fail and we propose using optimal solutions to strengthen the performance of our procedures

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

natural linear additive functionals of superprocesses we investigate natural linear additive nla functionals of a general critical xi k psi superprocess x we prove that all of them have only fixed discontinuities all homogeneous nla functionals of time homogeneous superprocesses are continuous this was known before only in the case of quadratic branching we introduce an operator mathscr e u defined in terms of xi k psi and we prove that the potential h and the log potential u of a nla functional a are connected by the equation u mathscr e u h the potential is always an exit rule for xi and the condition h mathscr e h infty a e is sufficient for an exit rule h to be a potential in an accompanying paper these results are applied to boundary value problems for partial differential equations involving nonlinear operator lu u alpha where l is a second order elliptic differential operator and 1 alpha leq 2

monte carlo approximations in bayesian decision theory abstract in decision making problems the bayesian action and its posterior expected loss usually cannot be obtained analytically in this article we study a monte carlo method for approximating the bayesian action and its posterior expected loss the monte carlo approximation to the bayesian action is obtained through a approximating the posterior expected loss function by using the monte carlo integration method and b searching the minimum of the approximated posterior expected loss function as the monte carlo sample size diverges to infinity the monte carlo approximations are shown to be convergent in general situations the rates of the convergence are also obtained under some regularity conditions on the loss function two accuracy measures of the monte carlo approximations are proposed some examples are presented for illustration

simultaneous perturbation stochastic approximation of nonsmooth functions abstract a simultaneous perturbation stochastic approximation spsa method has been developed in this paper using the operators of perturbation with the lipschitz density function this model enables us to use the approximation of the objective function by twice differentiable functions and to present their gradients by volume integrals the calculus of the stochastic gradient by means of this presentation and likelihood ratios method is proposed that can be applied to create spsa algorithms for a wide class of perturbation densities the convergence of the spsa algorithms is proved for lipschitz objective functions under quite general conditions the rate of convergence o 1 k 1 2 of the developed algorithms has been established for functions with a sharp minimum as well as the dependence of the rate of convergence is explored theoretically as well as by computer simulation the applicability of the presented algorithm is demonstrated by applying it to minimization of the mean absolute pricing error for the calibration of the heston stochastic volatility model

extra heads and invariant allocations let n be an ergodic simple point process on e d and let n be its palm version thorisson ann probab 24 1996 2057 2064 proved that there exists a shift coupling of and n that is one can select a random point y of n such that translating n by y yields a configuration whose law is that of we construct shift couplings in which y and are functions of and prove that there is no shift coupling in which n is a function of the key ingredient is a deterministic translation invariant rule to allocate sets of equal volume forming a partition of r d to the points of n the construction is based on the gale shapley stable marriage algorithm amer math monthly 69 1962 9 15 next let be an ergodic random element of 0 1 zd and let be r conditioned on r 0 1 a shift coupling x of r and is called an extra head scheme we show that there exists an extra head scheme which is a function of r if and only if the marginal e r 0 is the reciprocal of an integer when the law of is product measure and d 3 we prove that there exists an extra head scheme x satisfying e exp c x d this answers a question of holroyd and liggett ann probab 29 2001 1405 1425

nonlinear stochastic programming by monte carlo estimators abstract methods for solving stochastic programming sp problems by a finite series of monte carlo samples are considered the accuracy of solution is treated in a statistical manner testing the hypothesis of optimality according to statistical criteria the rule for adjusting the monte carlo sample size is introduced to ensure the convergence and to find the solution of the sp problem using a reasonable number of monte carlo trials issues of implementation of the developed approach in decision making and other applicable fields are considered too

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

smoothed functionals in stochastic optimization we consider the problem of finding a global extremum for a nonconvex function by constructing another smoothed function which has a better behavior than the original one then operating only with the smoothed function we find the global extremum for the original function

removable singularities for lu ψ u and orlicz capacities suppose l is a second order elliptic differential operator in rd and let 1 baras and pierre have proved in 1984 that is removable for lu u if and only if its bessel capacity cap2 0 we extend this result to a general equation lu u where u is an increasing convex function subject to 2 and 2 conditions namely we prove that is removable for lu u if and only if its orlicz capacity is zero that is the integral b dx x y 2 d dy is equal to 0 or for every measure concentrated on where b stands for any ball containing

structure of shocks in burgers turbulence with levy noise initial data we study the structure of the shocks for the inviscid burgers equation in dimension 1 when the initial velocity is given by levy noise or equivalently when the initial potential is a two sided levy process 0 when 0 is abrupt in the sense of vigon or has bounded variation with lim sup h 0h 2 0 h we prove that the set of points with zero velocity is regenerative and that in the latter case this set is equal to the set of lagrangian regular points which is non empty when 0 is abrupt we show that the shock structure is discrete when 0 is eroded we show that there are no rarefaction intervals

statistical properties of shocks in burgers turbulence ii tail probabilities for velocities shock strengths and rarefaction intervals this paper studies the structure of the random sawtooth profile corresponding to the solution of the inviscid burgers equation with white noise initial data this function consists of a countable sequence of rarefaction waves separated by shocks we are concerned here with calculating the probabilities of rare events associated with the occurrence of very large values of the normalized velocity shock strength and rarefaction intervals we find that these quantities have tail probabilities of the form exp cx3 x 1 this cubic exponential decay of probabilities was conjectured in the companion paper 1 the calculations are done using a representation of the shock strength and length of rarefaction intervals in terms of the statistics of certain conditional diffusion processes

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

uniform laws of large numbers for set valued mappings and subdifferentials of random functions we derive a uniform strong law of large numbers lln for random set valued mappings the result can be viewed as an extension of both a uniform lln for random functions and lln for random sets we apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs

inferring optimal peer assignment from experimental data this article studies the problem of optimally dividing individuals into peer groups to maximize social gains from heterogeneous peer effects the specific setting analyzed here concerns efficient ways of allocating roommates in college dormitories using confidential data on a sample of dartmouth college freshmen who were randomly assigned to be roommates the article derives efficient roommate pairing rules based on demographic and academic background which maximize different aggregate outcomes segregation by precollege academic standing and by race are seen to minimize mean enrolment into sororities and maximize mean enrolment into fraternities segregation has no effect on mean and median freshman year grade point average gpa but increases the higher and decreases the lower percentiles of the gpa distribution for both men and women efficiency loss due to legal constraints on allocations e g race blindness is shown to be significant and also larger for women for whom peer effects are more nonl

subordination of trees and the brownian map we discuss subordination of random compact mathbb r trees we focus on the case of the brownian tree where the subordination function is given by the past maximum process of brownian motion indexed by the tree in that particular case the subordinate tree is identified as a stable levy tree with index 3 2 as a more precise alternative formulation we show that the maximum process of the brownian snake is a time change of the height process coding the levy tree we then apply our results to properties of the brownian map in particular we recover in a more precise form a recent result of miller and sheffield identifying the metric net associated with the brownian map