Elliptic Diffusion Research Articles

Rigidity for Markovian maximal couplings of elliptic diffusions

Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that the reflection-coupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 633–662, 2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists a MMC under local perturbations of the starting points of the coupled diffusions. For example, in the time-homogeneous case it is shown that the only possible manifolds that may arise are Euclidean space, hyperbolic space and the hypersphere. Moreover the permissible drifts can then derive only from rotation isometries of these spaces (and dilations, in the Euclidean case). In this sense, a geometric rigidity phenomenon holds good.

Parabolic problems with general Wentzell boundary conditions and diffusion on the boundary

We show a result of maximal regularity in spaces of Holder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.

Analysis of the domain mapping method for elliptic diffusion problems on random domains

In this article, we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains. In particular, based on the decay of the Karhunen-Loeve expansion of the domain perturbation field, we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension. For the implementation of a related approximation scheme, like quasi-Monte Carlo quadrature, stochastic collocation, etc., we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field. This simplifies the implementation and yields a non-intrusive approach. Having this machinery at hand, we can easily transfer it to stochastic interface problems. The theoretical findings are complemented by numerical examples for both, stochastic interface problems and boundary value problems on random domains.

Securing Multi Key Cryptography Policy based Session Authorized Access in Wireless Broad Cast Network

A wireless Broadcast communication system is the most significant and security needs in established communication model for authorization. Securing the broadcast network from the conflict is critical issue for broad cast environment. The cryptography standard provides randomize key assumption procedure to the systematic with the recent adoption of elliptic curve cryptography and diffusion of the broadcast network prototype in distributed systems such as wireless broadcast networks, there have been rapidly increasing demands and concerns for distributed data security. To propose a novel multi key policy scheme for a data allocation system by exploiting the attribute of the system construction. The projected scheme features achieve the security difficulty could be solved by multi attribute key issuing in wireless broadcast at the session that verifies using key policy algorithm which is constructed using the secure auditing computation between the key generation in session time and the data storing center and user revocation per each characteristic could be done by alternate encryption which takes advantage of the discriminating attribute group key distribution on crypto analysis. The security analysis of key performance indicates that the RBAC method is efficient to securely supervise the data distributed in the data allotment system.

Discontinuous Galerkin Isogeometric Analysis of Elliptic Diffusion Problems on Segmentations with Gaps

We propose a new discontinuous Galerkin (dG) isogeometric analysis (IgA) technique for the numerical solution of elliptic diffusion problems in computational domains decomposed into volumetric patches with nonmatching interfaces. Due to an incorrect segmentation procedure, it may happen that the interfaces of adjacent subdomains do not coincide. In this way, gap regions, which are not present in the original physical domain, are created. In this paper, the gap region is considered as a subdomain of the decomposition of the computational domain and the gap boundary is taken as an interface between the gap and the subdomains. We apply a multipatch approach and derive a subdomain variational formulation which includes interface continuity conditions and is consistent with the original variational formulation of the problem. The last formulation is further modified by deriving interface conditions without the presence of the solution in the gap. Finally, the solution of this modified problem is approximated b...

A New Second Order Discretization with Malliavin Calculus for Elliptic and Hypoelliptic Diffusions Arising in Finance

A New Second Order Discretization with Malliavin Calculus for Elliptic and Hypoelliptic Diffusions Arising in Finance

Stochastic finite differences for elliptic diffusion equations in stratified domains

We describe Monte Carlo algorithms to solve elliptic partial differential equations with piecewise constant diffusion coefficients and general boundary conditions including Robin and transmission conditions as well as a damping term. The treatment of the boundary conditions is done via stochastic finite differences techniques which possess a higher order than the usual methods. The simulation of Brownian paths inside the domain relies on variations around the walk on spheres method with or without killing. We check numerically the efficiency of our algorithms on various examples of diffusion equations illustrating each of the new techniques introduced here.

On Error Estimates for Asymptotic Expansions with Malliavin Weights: Application to Stochastic Volatility Model

This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More generally, we also derive an error estimate for an asymptotic expansion around a general partially elliptic diffusion and a more general Wiener functional, which is applicable to various important valuation and risk management tasks in the financial business such as the ones for multidimensional diffusion and nondiffusion models. In particular, we take the Malliavin calculus approach, and estimate the error bounds for the Malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of Kusuoka-Stroock functions introduced by Kusuoka [Kusuoka S (2003) Malliavin calculus revisited. J. Math. Sci. Univ. Tokyo 10:261–277.] functions. Moreover, a numerical experiment under the Heston-type model confirms the effectiveness of our method.

Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

A generalized predictive analysis tool for multigrid methods

SummaryMultigrid and related multilevel methods are the approaches of choice for solving linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the well‐known local mode (often local Fourier) analysis approach to a wider class of problems. The semi‐algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two‐dimensional convection‐diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short‐term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright © 2015 John Wiley & Sons, Ltd.

A Regularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type

We consider a nondegenerate quasilinear parabolic stochastic partial differential equation (SPDE) with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Holder continuous in time and possesses spatial regularity. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.

Polynomial Chaos Expansion of Random Coefficients and the Solution of Stochastic Partial Differential Equations in the Tensor Train Format

We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the s...

Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

The numerical approximation of parametric partial differential equations D(u,y)=0 is a computational challenge when the dimension d of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown in [1,2] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map y↦u(y) with an algebraic convergence rate that is independent of the parametric dimension d. The analysis in [1,2] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for classes of parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z↦u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying semi-axes lengths of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case d=∞. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDEs that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector y in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on y, (iii) nonlinear, monotone parametric elliptic PDEs, and (iv) elliptic equations set on a domain that is parametrized by the vector y. We give general strategies that allow us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions.

The Fokker–Planck–Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix

The Fokker–Planck–Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix

Asymptotic Properties of Maximum Likelihood Estimation: Parameterized Diffusion in a Manifold

This paper studies maximum likelihood estimation for a parameterised elliptic diffusion in a manifold. The focus is on asymptotic properties of maximum likelihood estimates obtained from continuous time observation. These are well known when the underlying manifold is a Euclidean space. However, no systematic study exists in the case of a general manifold. The starting point is to write down the likelihood function and equation. This is achieved using the tools of stochastic differential geometry. Consistency, asymptotic normality and asymptotic optimality of maximum likelihood estimates are then proved, under regularity assumptions. Numerical computation of maximum likelihood estimates is briefly discussed.

Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

This paper presents a semi-discrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot and Sarti. We propose a new ingredient, namely working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.

On Error Estimates for Asymptotic Expansions with Malliavin Weights -- Application to Stochastic Volatility Model

This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More generally, we also derive an error estimate for an asymptotic expansion around a partially elliptic diffusion under a multi-dimensional setting, which includes various important models in finance as the special cases.In particular, we take the Malliavin calculus approach, and estimate the error bounds for the Malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of Kusuoka-Stroock functions.

Construction and Strong Feller Property of Distorted Elliptic Diffusion with Reflecting Boundary

Using Dirichlet form techniques we prove existence of a diffusion process with singular drift on an open domain Ω ⊂ ℝd, d ∈ ℕ, d ≥ 2, and generalized reflection at the boundary. The boundary is assumed to be C2-smooth except for a sufficiently small set. We prove an elliptic regularity result which gives Lp-strong Feller property for the semigroup and resolvent for p ≥ 2 with \(p > \frac{d}{2}\). This result allows us to construct the process pointwisely except for an explicitly known set. For starting points outside this known set the process solves the corresponding martingale problem. The results are applied to prove the existence of the stochastic dynamics of a finite interacting particle system and for the Ginzburg-Landau interface model with a hard wall.

Numerical Solution for a Non-Fickian Diffusion in a Periodic Potential

AbstractNumerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.

On Filtering with Observation in a Manifold: Reduction to a Classical Filtering Problem

The current paper deals with filtering problems where the observation process, conditioned on the unknown signal, is an elliptic diffusion in a differentiable manifold. Precisely, the observation model is given by a stochastic differential equation in the underlying manifold. The main new idea is to use a Le Jan--Watanabe connection, instead of a usual Levi-Civita connection, in performing the operation of antidevelopment (both connections can be constructed from a stochastic differential equation as in the observation model). The following results are obtained. First, it is shown that antidevelopment reduces the original filtering problem to a classical one, i.e., an additive white noise model. Second, a new form of the Zakai and filtering equations is derived which has a structure very similar to that of classical equations. Currently, there are few well understood general numerical solution methods for filtering problems with observation in a manifold. Reduction to classical filtering problems seems desirable, as it allows proven existing numerical methods for classical problems to be applied immediately. In view of practical implementation of this approach, the paper also gives an explicit pathwise construction of the antidevelopment.