pairwise likelihood ratios for estimation of non gaussian structural equation models we present new measures of the causal direction or direction of effect between two non gaussian random variables they are based on the likelihood ratio under the linear non gaussian acyclic model lingam we also develop simple first order approximations of the likelihood ratio and analyze them based on related cumulant based measures which can be shown to find the correct causal directions we show how to apply these measures to estimate lingam for more than two variables and even in the case of more variables than observations we further extend the method to cyclic and nonlinear models the proposed framework is statistically at least as good as existing ones in the cases of few data points or noisy data and it is computationally and conceptually very simple results on simulated fmri data indicate that the method may be useful in neuroimaging where the number of time points is typically quite small

transition probabilities for degenerate diffusions arising in population genetics we provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong markov process arising in population genetics the markov processes that we study are a generalization of the classical wright fisher process the main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward kolmogorov equation defined on a compact manifold with corners which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities

weak uniqueness for the heat equation with noise the uniqueness in law for the equation x t t 1 2 x t x t w is established for 1 2 1 the proof uses a duality technique and requires the construction of an approximating sequence of dual processes

degenerate stochastic differential equations with holder continuous coefficients and super markov chains we consider the operator formula math acting on functions in c 2 b r d we prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on ij and b i in contrast to previous work the b i need only be nonnegative on the boundary rather than strictly positive at the expense of the ij and b i being holder continuous applications to super markov chains are given the proof follows stroock and varadhan s perturbation argument but the underlying function space is now a weighted holder space and each component of the constant coefficient process being perturbed is the square of a bessel process

reactive trajectories and the transition path process we study the trajectories of a solution x t to an ito stochastic differential equation in mathbb r d as the process passes between two disjoint open sets a and b these segments of the trajectory are called transition paths or reactive trajectories and they are of interest in the study of chemical reactions and thermally activated processes in that context the sets a and b represent reactant and product states our main results describe the probability law of these transition paths in terms of a transition path process y t which is a strong solution to an auxiliary sde having a singular drift term we also show that statistics of the transition path process may be recovered by empirical sampling of the original process x t as an application of these ideas we prove various representation formulas for statistics of the transition paths we also identify the density and current of transition paths our results fit into the framework of the transition path theory by weinan and vanden eijnden

observer design for stochastic parameter systems constant gain linear observer design is utilized to reconstruct the state vector of stochastic systems with both multiplicative and additive noises it is shown that if the system model is mean square stable then it is very easy to design an effective estimation scheme

uniqueness for diffusions degenerating at the boundary of a smooth bounded set for continuous y g 0 1 0 oo consider the degenerate stochastic differential equation d x t x t 2 1 2 x t d b t g x t x t dt in the closed unit ball of r n we introduce a new idea to show pathwise uniqueness holds when y and g are lipschitz and g 1 2 1 2 1 when specialized to a case studied by swart stochastic process appl 98 2002 131 149 with y 2 and g c this gives an improvement of his result our method applies to more general contexts as well let d be a bounded open set with c 3 boundary and suppose h d r lipschitz on d as well as c 2 on a neighborhood of d with lipschitz second partials there also assume h 0 on d h 0 on d and h 0 on d an example of such a function is h x d x d we give conditions which ensure pathwise uniqueness holds for d x t h x t 1 2 x t db t b x t dt in d

particle representations for measure valued population models models of populations in which a type or location represented by a point in a metric space e is associated with each individual in the population are considered a population process is neutral if the chances of an individual replicating or dying do not depend on its type measurevalued processes are obtained as infinite population limits for a large class of neutral population models and it is shown that these measure valued processes can be represented in terms of the total mass of the population and the de finetti measures associated with an e valued particle model

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

information geometric approach to inferring causal directions while conventional approaches to causal inference are mainly based on conditional in dependences recent methods also account for the shape of conditional distributions the idea is that the causal hypothesis x causes y imposes that the marginal distribution p x and the conditional distribution p y x represent independent mechanisms of nature recently it has been postulated that the shortest description of the joint distribution p x y should therefore be given by separate descriptions of p x and p y x since description length in the sense of kolmogorov complexity is uncomputable practical implementations rely on other notions of independence here we define independence via orthogonality in information space this way we can explicitly describe the kind of dependence that occurs between p y and p x y making the causal hypothesis y causes x implausible remarkably this asymmetry between cause and effect becomes particularly simple if x and y are deterministically related we present an inference method that works in this case we also discuss some theoretical results for the non deterministic case although it is not clear how to employ them for a more general inference method

uniqueness for isotropic diffusions with a linear drift this paper proves that valued solutions to the sde are unique in distribution when d d is convex and open d c 0 is positive and locally lipschitz on d and zero on d and x d g x r is convex for r sufficiently small the proof for 0 is based on the transformation xt ectxt which removes the drift and a random time change although the set up is rather specialized the result gives uniqueness for some sde s that cannot be treated by any of the conventional techniques

distinguishing cause from effect using observational data methods and benchmarks the discovery of causal relationships from purely observational data is a fundamental problem in science the most elementary form of such a causal discovery problem is to decide whether x causes y or alternatively y causes x given joint observations of two variables x y an example is to decide whether altitude causes temperature or vice versa given only joint measurements of both variables even under the simplifying assumptions of no confounding no feedback loops and no selection bias such bivariate causal discovery problems are challenging nevertheless several approaches for addressing those problems have been proposed in recent years we review two families of such methods methods based on additive noise models anms and information geometric causal inference igci we present the benchmark causeeffectpairs that consists of data for 100 different causee ffect pairs selected from 37 data sets from various domains e g meteorology biology medicine engineering economy etc and motivate our decisions regarding the ground truth causal directions of all pairs we evaluate the performance of several bivariate causal discovery methods on these real world benchmark data and in addition on artificially simulated data our empirical results on real world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data although more benchmark data would be needed to obtain statistically significant conclusions one of the best performing methods overall is the method based on additive noise models that has originally been proposed by hoyer et al 2009 which obtains an accuracy of 63 10 and an auc of 0 74 0 05 on the real world benchmark as the main theoretical contribution of this work we prove the consistency of that method

schauder estimates for a degenerate second order elliptic operator on a cube abstract in the present article we are concerned with a class of degenerate second order differential operators l a b defined on the cube 0 1 d with d 1 under suitable assumptions on the coefficients a and b among them the assumption of their holder regularity we show that the operator l a b defined on c 2 0 1 d is closable and its closure is m dissipative in particular its closure l a b is the generator of a c 0 semigroup of contractions on c 0 1 d and c 2 0 1 d is a core for it the proof of such result is obtained by studying the solvability in holder spaces of functions of the elliptic problem u x l a b u x f x x 0 1 d for a sufficiently large class of functions f

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

kalman filtering with random coefficients and contractions the riccati transformation of linear filtering control theory is shown to be a contraction on the space of positive symmetric matrices this is used to describe the asymptotic behavior of the filter for systems with stochastic stationary parameters

on the martingale problem for degenerate parabolic partial differential operators with unbounded coefficients and a mimicking theorem for ito processes using results from our companion article 26 on a schauder approach to existence of solutions to a degenerate parabolic partial differential equation we solve three intertwined problems motivated by probability theory and mathematical finance concerning degenerate diffusion processes we show that the martingale problem associated with a degenerate elliptic differential operator with unbounded locally holder continuous coefficients on a half space is wellposed in the sense of stroock and varadhan second we prove existence uniqueness and the strong markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable holder continuity properties third for an ito process with degenerate diffusion and unbounded but appropriately regular coefficients we prove existence of a strong markov process unique in the sense of probability law whose onedimensional marginal probability distributions match those of the given ito process

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

implications of a result on observer design for stochastic parameter systems an earlier result on observer design for stochastic parameter systems is utilized to show that optimal one step predictors for such systems are mean square stable and to propose a new class of stable estimation schemes

modeling and estimation for networked systems with multiple random transmission delays and packet losses abstract in networked systems data packets are transmitted through networks from a sensor to a data processing center due to the unreliability of communication channels a packet may be delayed even lost during the transmission at each moment the data processing center may receive one or multiple data packets or nothing at all a novel model is developed to describe the possible multiple random transmission delays and data packet losses by employing a group of bernoulli distributed random variables it is transformed to a measurement model with multiple random delayed states and noises based on the model an optimal linear filter in the linear minimum variance sense is proposed by using the orthogonal projection approach which is a universal tool to find the optimal linear estimate it does not have a steady state performance since it depends on the values of random variables that depict the phenomena of delays and losses at each moment so it needs to be computed online to reduce the online computational cost a suboptimal linear filter dependent on the probabilities of random variables is also proposed however it is worth noting that it is linearly optimal among all the linear filters dependent on the probabilities it can be computed offline since it has the steady state performance a sufficient condition of existence for the steady state performance is given a simulation example shows the effectiveness

reducibility and unobservability of markov processes the linear system case a vector markov process will be called stochastically unobservable by the measurement process if there exists an initial distribution such that some marginal conditional distributions equal the corresponding unconditional ones it will be called reducible if there exists an invertible transformation such that the transformed process is stochastically unobservable necessary and sufficient conditions are derived in the context of linear diffusions it is also shown that reducibility can be regarded as a natural extension of the concept of estimability defined for linear stochastic systems

causal discovery via reproducing kernel hilbert space embeddings causal discovery via the asymmetry between the cause and the effect has proved to be a promising way to infer the causal direction from observations the basic idea is to assume that the mechanism generating the cause distribution px and that generating the conditional distribution py x correspond to two independent natural processes and thus px and py x fulfill some sort of independence condition however in many situations the independence condition does not hold for the anticausal direction if we consider px y as generated via pypx y then there are usually some contrived mutual adjustments between py and px y this kind of asymmetry can be exploited to identify the causal direction based on this postulate in this letter we define an uncorrelatedness criterion between px and py x and based on this uncorrelatedness show asymmetry between the cause and the effect in terms that a certain complexity metric on px and py x is less than the complexity metric on py and px y we propose a hilbert space embedding based method emd an abbreviation for embedding to calculate the complexity metric and show that this method preserves the relative magnitude of the complexity metric based on the complexity metric we propose an efficient kernel based algorithm for causal discovery the contribution of this letter is threefold it allows a general transformation from the cause to the effect involving the noise effect and is applicable to both one dimensional and high dimensional data furthermore it can be used to infer the causal ordering for multiple variables extensive experiments on simulated and real world data are conducted to show the effectiveness of the proposed method

mutually catalytic branching in the plane finite measure states we study a pair of populations in r 2 which undergo diffusion and branching the system is interactive in that the branching rate of each type is proportional to the local density of the other type for a diffusion rate sufficiently large compared with the branching rate the model is constructed as the unique pair of finite measure valued processes which satisfy a martingale problem involving the collision local time of the solutions the processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations in the long term limit global extinction of one type is shown the process constructed is a rescaled limit of the corresponding z 2 lattice model studied by d a dawson and e a perkins ann probab 26 1998 1088 1138 and resolves the large scale mass time space behavior of that model

stabilization of deterministic and stochastic parameter discrete systems a class of suboptimal control laws having the stabilizing property is used for treating linear discrete time systems with deterministic and stochastic parameters known stabilization results in the deterministic case are extended to systems with singular state transition matrices and optimization horizon lengths shorter than the controllability indices of these systems it is shown that in the case where the state transition and input matrices contain stochastic components the stabilization of these systems in the sense of both almost sure asymptotic and mean square exponential stability is possible using the same line of reasoning

infinite horizon optimal control of linear discrete time systems with stochastic parameters the infinite horizon optimal control problem is considered in the general case of linear discrete time systems and quadratic criteria both with stochastic parameters which are independent with respect to time a stronger stabilizability property and a weaker observability property than usual for deterministic systems are introduced it is shown that the infinite horizon problem has a solution if the system has the first property if in addition the problem has the second property the solution is unique and the control system is stable in the mean square sense a simple necessary and sufficient condition explicit in the system matrices is given for the system to have the stronger stabilizability property this condition also holds for deterministic systems to be stabilizable in the usual sense the stronger stabilizability and weaker observability properties coincide with the usual ones if the parameters are deterministic

a cyclically catalytic super brownian motion in generalization of the mutually catalytic super brownian motion in r of dawson and perkins and mytnik a function valued cyclically catalytic model x is constructed as a strong markov solution to a martingale problem starting with a finite population x 0 each pair of neighboring types will globally segregate in the long term limit noncoexistence of neighboring types also finer extinction survival properties depending on x 0 are studied in the spirit of mueller and perkins in fact x 0 can be chosen in such a way that all types survive for all finite times on the other hand sufficient conditions on x 0 are stated for the following situation given a type k and a positive time t the kth subpopulation x k dies by time t with a large probability provided that its initial value x k 0 was sufficiently small

minimal nonlinear distortion principle for nonlinear independent component analysis it is well known that solutions to the nonlinear independent component analysis ica problem are highly non unique in this paper we propose the minimal nonlinear distortion mnd principle for tackling the ill posedness of nonlinear ica problems mnd prefers the nonlinear ica solution with the estimated mixing procedure as close as possible to linear among all possible solutions it also helps to avoid local optima in the solutions to achieve mnd we exploit a regularization term to minimize the mean square error between the nonlinear mixing mapping and the best fitting linear one the effect of mnd on the inherent trivial and non trivial indeterminacies in nonlinear ica solutions is investigated moreover we show that local mnd is closely related to the smoothness regularizer penalizing large curvature which provides another useful regularization condition for nonlinear ica experiments on synthetic data show the usefulness of the mnd principle for separating various nonlinear mixtures finally as an application we use nonlinear ica with mnd to separate daily returns of a set of stocks in hong kong and the linear causal relations among them are successfully discovered the resulting causal relations give some interesting insights into the stock market such a result can not be achieved by linear ica simulation studies also verify that when doing causality discovery sometimes one should not ignore the nonlinear distortion in the data generation procedure even if it is weak

causal inference using the algorithmic markov condition inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure markovian here we argue why causal inference is also possible when the sample size is one we develop a theory how to generate causal graphs explaining similarities between single objects to this end we replace the notion of conditional stochastic independence in the causal markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules we explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities making it possible to select among markov equivalent causal graphs this insight provides a theoretical foundation of a heuristic principle proposed in earlier work we also sketch some ideas on how to replace kolmogorov complexity with decidable complexity criteria this can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution

causal inference on multidimensional data using free probability theory in this paper we deal with the problem of inferring causal relations for multidimensional data based on the postulate that the distribution of the cause and the conditional distribution of the effect given cause are generated independently we show that the covariance matrix of the mean embedding of the cause in reproducing kernel hilbert space rkhs is free independent with the covariance matrix of the conditional embedding of the effect given cause this called freeness condition induces a cause effect asymmetry that a designed measurement is 0 in the causal direction but smaller than 0 in the anticausal direction and it uncovers the causal direction one important novel aspect of this paper is that we interpret the independence as a freeness condition between covariance matrices of rkhs distribution embeddings and it has a wide applicability we show that our freeness condition based inference method succeeds in scenarios like additive noise cases where other methods fail by theoretical analysis and experimental results

c0 estimates and smoothness of solutions to the parabolic equation defined by kimura operators abstract kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics their infinitesimal generator is an elliptic differential operator whose second order coefficients matrix degenerates on the boundary of the domain in this article we consider the inhomogeneous initial value problem defined by generators of kimura diffusions and we establish c 0 estimates which allows us to prove that solutions to the inhomogeneous initial value problem are smooth up to the boundary of the domain where the operator degenerates even when the initial data is only assumed to be continuous

measure valued branching diffusions with spatial interactions a strong equation driven by a historical brownian motion is used to construct and characterize measure valued branching diffusions in which the spatial motions obey an ito equation with drift and diffusion depending on the position of an individual and the entire population

causal inference on discrete data via estimating distance correlations in this article we deal with the problem of inferring causal directions when the data are on discrete domain by considering the distribution of the cause and the conditional distribution mapping cause to effect as independent random variables we propose to infer the causal direction by comparing the distance correlation between and with the distance correlation between and we infer that x causes y if the dependence coefficient between and is smaller experiments are performed to show the performance of the proposed method

confounder detection in high dimensional linear models using first moments of spectral measures in this letter we study the confounder detection problem in the linear model where the target variable y is predicted using its n potential causes xn x1 xn t based on an assumption of a rotat

brief paper optimal estimation of linear discrete time systems with stochastic parameters this paper considers optimal linear state estimation in the general case of linear discrete time systems with stochastic parameters which are statistically independent with respect to time the estimator is derived by transforming the system to one with deterministic parameters and state dependent additive system and observation noise it is shown that mean square stability of the system is a sufficient and almost necessary condition for the existence uniqueness and stability of the time invariant estimator

a linear non gaussian acyclic model for causal discovery in recent years several methods have been proposed for the discovery of causal structure from non experimental data such methods make various assumptions on the data generating process to facilitate its identification from purely observational data continuing this line of research we show how to discover the complete causal structure of continuous valued data under the assumptions that a the data generating process is linear b there are no unobserved confounders and c disturbance variables have non gaussian distributions of non zero variances the solution relies on the use of the statistical method known as independent component analysis and does not require any pre specified time ordering of the variables we provide a complete matlab package for performing this lingam analysis short for linear non gaussian acyclic model and demonstrate the effectiveness of the method using artificially generated data and real world data

causal discovery with continuous additive noise models we consider the problem of learning causal directed acyclic graphs from an observational joint distribution one can use these graphs to predict the outcome of interventional experiments from which data are often not available we show that if the observational distribution follows a structural equation model with an additive noise structure the directed acyclic graph becomes identifiable from the distribution under mild conditions this constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the markov equivalence class of the graph thus leaving some edges undirected we provide practical algorithms for finitely many samples resit regression with subsequent independence test and two methods based on an independence score we prove that resit is correct in the population setting and provide an empirical evaluation

stability of optimum linear estimators of stochastic signals in white multiplicative noise conditions for uniform asymptotic stability in the large of the optimal minimum mean square error linear filter are developed for linear systems whose observations are corrupted by white multiplicative noise in addition to white additive noise both discrete time as well as continuous time systems are considered the multiplicative noise model may be useful in problems associated with phenomena such as fading or reflection of the transmitted signal from the ionosphere and also certain situations involving sampling gating or amplitude modulation conditions for existence uniqueness and stability of the steady state optimal filter are also considered for time invariant systems

full and reduced order observer design for discrete stochastic bilinear systems the author considers a general discrete time stochastic bilinear system model and derives the mean square optimal linear unbiased observer equations to be used in reconstructing a prespecified linear combination of state variables based on noisy output measurements

on the almost sure and mean square exponential convergence of some stochastic observers it is pointed out that linear observers used for estimating the state of the discrete time stochastic parameter systems are both almost surely and mean square ms exponentially convergent under the same conditions guaranteeing mean square convergence in addition to the mean square convergence properties of linear observers constructed for mean square stable stochastic parameter systems they also possess an almost sure exponential convergence property and the rate of ms convergence is exponential this rate depends on the parameters used in the design

the submartingale problem for a class of degenerate elliptic operators we consider the degenerate elliptic operator acting on c 2 b functions on 0 d mathcal l f x sum i 1 d a i x x i alpha i frac partial 2 f partial x i 2 x sum i 1 d b i x frac partial f partial x i x where the ai are continuous functions that are bounded above and below by positive constants the bi are bounded and measurable and the alpha i in 0 1 we impose neumann boundary conditions on the boundary of 0 d there will not be uniqueness for the submartingale problem corresponding to mathcal l if we consider however only those solutions to the submartingale problem for which the process spends 0 time on the boundary then existence and uniqueness for the submartingale problem for mathcal l holds within this class our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations rm d x t i sqrt 2a i x t x t i alpha i 2 rm d w i t b i x t rm d t rm d l t x i quad x i t geq 0 where w t i are independent brownian motions and l x i t is a local time at 0 for xi

detectability of linear discrete time systems with stochastic parameters abstract the paper introduces the concept of mean square detectability and relates this to the recently introduced concept of mean square observability it is shown that under appropriate mean square detectability and stabilizability conditions the infinite horizon optimal control problem for the general case of linear discrete time systems and quadratic criteria both with stochastic parameters which are statistically independent of time has a unique solution when the control system is mean square stable a simple necessary and sufficient condition explicit in the system matrices is given to determine if a system is mean square detectable this condition also holds for deterministic systems to be detectable in the usual sense the mean square detectability property coincides with the usual one if the parameters are deterministic

linear estimation of continuous discrete linear state space models with multiplicative noise this paper deals with the estimation of the state variable of continuous discrete linear state space models with multiplicative noise specifically the optimal minimum variance linear filter for that class of models is constructed moreover the solutions of the differential equations that describe the evolution of the two first conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the linear filter is illustrated by means of numerical experiments

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

uniqueness for a class of one dimensional stochastic pdes using moment duality we establish a duality relation for the moments of bounded solutions to a class of one dimensional parabolic stochastic partial differential equations the equations are driven by multiplicative space time white noise with a non lipschitz multiplicative functional the dual process is a system of branching brownian particles the same method can be applied to show uniqueness in law for a class of non lipschitz finite dimensional stochastic differential equations

resolvent estimates for fleming viot operators and uniqueness of solutions to related martingale problems we define a two parameter scale of banach spaces contained in the contininuous functions on the space of probability measures on a compact metric space x and show that the resolvent of the fleming viot process is a bounded operator in the scale we use this result to prove uniqueness of solutions to the martingale problem for fleming viot operators whose mutation operator a a and sampling rate x y may depend on the measure variable m1 x provided they are sufficiently close to constant in a sense made precise by the norms defining the scale we prove an existence result with no such restriction on the coefficients

existence uniqueness and the strong markov property of solutions to kimura diffusions with singular drift motivated by applications to proving regularity of solutions to degenerate parabolic equations arising in population genetics we study existence uniqueness and the strong markov property of weak solutions to a class of degenerate stochastic differential equations the stochastic differential equations considered in our article admit solutions supported in the set 0 r and they are degenerate in the sense that the diffusion matrix is not strictly elliptic as the smallest eigenvalue converges to zero at a rate proportional to the distance to the boundary of the domain and the drift coefficients are allowed to have power type singularities in a neighborhood of the boundary of the domain under suitable regularity assumptions on the coefficients we establish existence of weak solutions that satisfy the strong markov property and uniqueness in law in the class of markov processes

schauder a priori estimates and regularity of solutions to boundary degenerate elliptic linear second order partial differential equations abstract we establish schauder a priori estimates and regularity for solutions to a class of boundary degenerate elliptic linear second order partial differential equations furthermore given a c smooth source function we prove c regularity of solutions up to the portion of the boundary where the operator is degenerate boundary degenerate elliptic operators of the kind described in our article appear in a diverse range of applications including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance 10 25 generators of diffusion processes arising in mathematical biology 3 11 and the study of porous media 7 8

degenerate stochastic differential equations and super markov chains we consider diffusions corresponding to the generator for continuous with i nonnegative we show uniqueness for the corresponding martingale problem under certain non degeneracy conditions on b i i and present a counter example when these conditions are not satisfied as a special case we establish uniqueness in law for some classes of super markov chains with state dependent branching rates and spatial motions

uniqueness in law for a class of degenerate diffusions with continuous covariance we study the martingale problem associated with the operator lu s x partial su s x frac 1 2 sum i j 1 d 0 a ij s x partial ij u s x sum i j 1 d b ij x j partial iu s x where d0 d we show that the martingale problem is well posed when the function a is continuous and strictly positive definite on mathbb r d 0 and the matrix b takes a particular lower diagonal block form we then localize this result to show that the martingale problem remains well posed when b is replaced by a sufficiently smooth vector field whose jacobian matrix satisfies a nondegeneracy condition

the fundamental solution to 1d degenerate diffusion equation with one sided boundary abstract in this work we adopt a combination of probabilistic approach and analytic method to study the fundamental solution to a certain type of one dimensional degenerate diffusion equation to be specific we consider a diffusion equation on 0 whose diffusion coefficient vanishes at the boundary 0 equipped with the cauchy initial data and the dirichlet boundary condition one such diffusion equation that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0 our main goal is to extend the study to cases when the diffusion coefficient has a general order of degeneracy with a primary focus on the fundamental solution to such a degenerate diffusion equation in particular we study the regularity properties of the fundamental solution near 0 and investigate how the order of degeneracy of the diffusion operator and the dirichlet boundary condition jointly affect these properties we also provide estimates for the fundamental solution and its derivatives near 0

stochastic viability of convex sets we investigate necessary and sufficient conditions for viability of a closed convex set k under weak solutions of a stochastic differential equation these conditions are expressed in terms of the distance function to k when in addition the boundary of k is smooth then our necessary and sufficient conditions reduce to two relations that have to be verified just on the boundary of k