A finite-dimensional approximation for partial differential equations on Wasserstein space

Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space

In this chapter, we will deal with zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games, via the dynamic programming principle.In Sect. 4.1, we are concerned with basic concepts and definitions and we introduce stochastic differential games, referring to (Controlled MarkovProcesses and viscosity solutions, 2nd edn. Springer, New York 2006), XI. Then, using a semi-discretization argument, we study the DPP for lower- and upper-value functions in Sect. 4.2. In Sect. 4.3, we will consider the Isaacs equations, via semigroups related to DPP. In Sect. 4.4, we consider a link between stochastic controls and differential games via risk sensitive controls.

We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

We introduce the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line, with finite second moment, endowed with the Wasserstein metric. We discuss the advantages of this approach, over a standard functional PCA of probability densities in the Hilbert space of square-integrable functions. We establish the consistency of the method by showing that the empirical GPCA converges to its population counterpart, as the sample size tends to infinity. A key property in the study of GPCA is the isometry between the Wasserstein space and a closed convex subset of the space of square-integrable functions, with respect to an appropriate measure. Therefore, we consider the general problem of PCA in a closed convex subset of a separable Hilbert space, which serves as basis for the analysis of GPCA and also has interest in its own right. We provide illustrative examples on simple statistical models, to show the benefits of this approach for data analysis. The method is also applied to a real dataset of population pyramids.

Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.

In this work, we are in the position to view a measurement of a physical observable as an experiment in the sense of probability theory. To every physical observable, a sample space called the spectrum of the observable is therefore available. We have investigated the Wasserstein spaces over the spectrums of a quantum observables, i.e. the space of all probability measures defined on the suitable σ-algebra of subsets of the spectrum equipped with certain metric. It has been shown here the existence of one-to-one correspondence between the set of all quantum states and the set of all distribution functions defined on the spectrum of an observable. Thanks to the correspondence, every quantum state can be represented by a probability measure associated with the corresponding distribution function. It has been shown that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasserstein space over the spectrum of another observable. Such homeomorphism is referred to as generalized Fourier transform. By using the work of von Renesse, it has been shown that the Newtonian dynamics of the system can be formulated in the Wasserstein space over the spectrum of each observable which is equivalent to the Schroedinger dynamics, starting from the solution of Newton equation of motion in the Wasserstein space over the spectrum of a certain observable.

Let μ∈ P2(Rd), where P2(Rd) denotes the space of square integrable probability measures, and consider a Borel-measurable function Φ:P2(Rd)→R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for Φ(μ), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws μN=1N∑ i=1NδX i, where (a) (Xi)i=1N is a sequence of i.i.d. samples from μ or (b) (Xi)i=1N is a system of interacting particles (diffusions) corresponding to a McKean–Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform strong propagation of chaos, as well as an L2 estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.

We consider the H 4 problem for a nonlinear s y stem. The corresponding dynamic programming equation (DPE) is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient (for background, see [2], [4], [13], [25] among many notable others). The solutions are typically nonsmooth, and further, there are multiple viscosity solutions – that is, one does not even have uniqueness among the class of viscosity solutions (cf. [17]). Note that the nonuniqueness is not only due to an additive constant, but is much more difficult. If one removes the additive constant issue by requiring the solution to be zero at the origin, then, for example, the linear-quadratic case typically has two classical solutions and an infinite number of viscosity solutions [17]. Since the H 4 problem is a differential game, the PDE is, in general, a Hamilton-Jacobi-Isaacs (HJI) PDE. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult, and possibly even more so in the presence of the non-uniqueness mentioned above. Some previous works in the general area of numerical methods for these problems are [3], [6], [7], [14], and the references therein.KeywordsViscosity SolutionBasis ExpansionDynamic Programming PrincipleDynamic Program EquationEigenvector MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Aiming at the heat conduction equation, the heat conduction model is a very typical example of the partial differential equation. This paper focuses on the development and preliminary knowledge of the partial differential equation, the establishment of the heat conduction model and the solution of the heat conduction model. The partial differential equation with fractional Laplace operator is a kind of typical fractional partial differential equation, which has important applications in the fields of science and engineering. Fractional Laplace operator is a kind of nonlocal quasi-differential operator, which is the infinitesimal generator of L é vy steady-state process. It is essentially different from the classical Laplace operator, which leads to the disappearance of some classical properties, which generally brings difficulties to the study of this kind of problems. It is very difficult to solve the explicit solution of partial differential equation with fractional Laplace operator. In order to improve the accuracy and speed of solving partial differential equation, a high precision method for solving partial differential equation based on deep learning network is proposed. A multilayer radial basis function neural network is constructed and its structural layers are determined. By introducing subnetworks into each layer of the multilayer radial basis function neural network, a composite multilayer radial basis function neural network is constructed to improve the real function approximation performance and operation accuracy of the multilayer radial basis function neural network. The high-precision compound multi-layer radial basis function neural network is used to solve the partial differential equation. By giving a specific example of solving the partial differential equation, the solution effect of the method is tested. The results show that the method has very high solution accuracy under the four-layer network structure, which can improve about 1.5 orders of magnitude compared with the first layer. Under different number of training samples, it has higher solution accuracy and speed, and the comprehensive performance is superior. Therefore, it is a practical but challenging work to study the existence of solutions of partial differential equations with fractional Laplace operators. This paper mainly describes the research progress and trends of the existence of solutions of several classes of partial differential equations with fractional Laplace operators, including some of the work done by the author in this field in recent years.

We consider a zero sum differential game with lack of observation on one side. The initial state of the system is drawn at random according to a probability $\mu_0$ on ${\mathbb R}^N$. Player I is informed of the initial position of state, while Player II knows only $\mu_0$. Moreover Player I observes Player II's moves, while Player II is blind and has no further information. We prove that in this game with a terminal payoff the value exists and is characterized as the unique viscosity solution of a Hamilton--Jacobi equation on a space of probability measures.

A definition of Lévy processes with values in the space of probability measures was introduced by Shiga and Tanaka (Electronic J. Prob. 11 (2006)). It is shown that the Lévy process with values in the space of probability measures in law has a modification satisfying a certain condition. The modification is a Lévy process in the sense of Shiga and Tanaka. The Lévy-Itô decomposition of sample paths of the Lévy process satisfying the condition is derived.

The aim of this paper is to address optimality of stochastic control strategies via dynamic programming subject to total variation distance ambiguity on the conditional distribution of the controlled process. We formulate the stochastic control problem using minimax theory, in which the control minimizes the payoff while the conditional distribution, from the total variation distance set, maximizes it. First, we investigate the maximization of a linear functional on the space of probability measures on abstract spaces, among those probability measures which are within a total variation distance from a nominal probability measure, and then we give the maximizing probability measure in closed form. Second, we utilize the solution of the maximization to solve minimax stochastic control with deterministic control strategies, under a Markovian and a non-Markovian assumption, on the conditional distributions of the controlled process. The results of this part include (1) minimax optimization subject to total va...

This paper proves the existence of viscosity solutions of path dependent semilinear PDEs via Perron’s method, i.e. via showing that the supremum of viscosity subsolutions is a viscosity solution. We use the notion of viscosity solutions introduced in the work of Ekren, Keller, Touzi and Zhang which considers as test functions all those smooth processes which are tangent in mean. We also provide a comparison result for semicontinuous viscosity solutions, by using a regularization technique. As an interesting byproduct, we give a new short proof for the optimal stopping problem with semicontinuous obstacles.

During the last decade we have seen a number of major developments which show that the long-time behavior of solutions of a very large class of partial differential equations (PDEs) possess a striking resemblance to the behavior of solution of finite dimensional dynamical systems, or ordinary differential equations (ODEs). The first of these advances was the discovery (by a number of researchers) that a dissipative PDE has a compact, maximal attractor X with finite Hausdorff and fractal dimensions. More recently [6, 12–15] it was shown that some of these PDEs possess a finite dimensional inertial manifold, i.e., an invariant manifold that contains the attractor X. For the later equation, the connection with ODEs is no longer a mere resemblance, instead it has become a striking reality! The reason for this is that where one restricts the PDE to the inertial manifold one obtains an ODE, which we call an inertial form for the given PDE. Since an inertial manifold contains the universal attractor, this means that the long-time behavior of solutions of a PDE with an inertial manifold is completely determined by the inertial form.KeywordsUnstable ManifoldHomoclinic OrbitHigh Frequency OscillationChaotic RegimeChaotic Time SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.