Random Processes for Classical Equations of Mathematical Physics - A. S. Sipin

. .

Résumé :
1. Markov Processes and Integral Equations.- 1.1. Breaking-off Markov chains and linear integral equations.- 1.2. Markov processes with continuous time and linear evolutionary equations.- 1.3. Convergent Markov chains and some boundary values problems.- 1.4. Markov chains and nonlinear integral equations.- 2. First Boundary Value Problem for the Equation of the Elliptic Type.- 2.1. Statement of the problem and notation.- 2.2. Green formula and the mean value theorem.- 2.3. Construction of a random process and an algorithm for the solution of the problem.- 2.4. Methods for simulation of a Markov chain.- 2.5. Estimation of the variance of a random variable ???.- 3. Equations with Polynomial Nonlinearity.- 3.1. Preliminary examples and notation.- 3.2. Representation of solutions of integral equations with polynomial nonlinearity.- 3.3. Definition of probability measures and the simplest estimators.- 3.4. Probabilistic solution of nonlinear equations on measures.- 4. Probabilistic Solution of Some Kinetic Equations.- 4.1. Deterministic motion of particles.- 4.2. Computational aspects of the simulation of a collision process.- 4.3. Random trajectories of particles. The construction of the basic process.- 4.4. Collision processes.- 4.5. Auxiliary results.- 4.6. Lemmas on certain integral equations.- 4.7. Uniqueness of the solution of the (X, T?, H) equation.- 4.8. Probabilistic solution of the interior boundary value problem for the regularized Boltzmann equation.- 4.9. Estimation of the computational labour requirements.- 5. Various Boundary Value Problems Related to the Laplace Operator.- 5.1. Parabolic means and a solution of the mixed problem for the heat equation.- 5.2. Exterior Dirichlet problem for the Laplace equation.- 5.3. Solution of the Neumann problem.- 5.4.Branching random walks on spheres and the Dirichlet problem for the equation ?u = u2.- 5.5. Special method for the solution of the Dirichlet problem for the Helmholtz equation.- 5.6. Probabilistic solution of the wave equation in the case of an infinitely differentiable solution.- 5.7. Another approach to the solution of hyperbolic equations.- 5.8. Probabilistic representation of the solution of boundary value problems for an inhomogeneous telegraph equation.- 5.9. Cauchy problem for the Schr?dinger equation.- 6. Generalized Principal Value Integrals and Related Random Processes.- 6.1. Random processes related to linear equations.- 6.2. Nonlinear equations.- 6.3. On the representation of a solution of nonlinear equations as a generalized principal value integral.- 6.4. Principal part of the operator and the Monte Carlo method.- 7. Interacting Diffusion Processes and Nonlinear Parabolic Equations.- 7.1. Propagation of chaos and the law of large numbers.- 7.2. Interacting Markov processes and nonlinear equations. Heuristic considerations.- 7.3. Weakly interacting diffusions.- 7.4. Moderately interacting diffusions.- 7.5. On one method of numerical solution of systems of stochastic differential equations.- Bibliographical Notes.- References.- Additional References....

Sommaire:
1. Markov Processes and Integral Equations.- 1.1. Breaking-off Markov chains and linear integral equations.- 1.2. Markov processes with continuous time and linear evolutionary equations.- 1.3. Convergent Markov chains and some boundary values problems.- 1.4. Markov chains and nonlinear integral equations.- 2. First Boundary Value Problem for the Equation of the Elliptic Type.- 2.1. Statement of the problem and notation.- 2.2. Green formula and the mean value theorem.- 2.3. Construction of a random process and an algorithm for the solution of the problem.- 2.4. Methods for simulation of a Markov chain.- 2.5. Estimation of the variance of a random variable ???.- 3. Equations with Polynomial Nonlinearity.- 3.1. Preliminary examples and notation.- 3.2. Representation of solutions of integral equations with polynomial nonlinearity.- 3.3. Definition of probability measures and the simplest estimators.- 3.4. Probabilistic solution of nonlinear equations on measures.- 4. Probabilistic Solution of Some Kinetic Equations.- 4.1. Deterministic motion of particles.- 4.2. Computational aspects of the simulation of a collision process.- 4.3. Random trajectories of particles. The construction of the basic process.- 4.4. Collision processes.- 4.5. Auxiliary results.- 4.6. Lemmas on certain integral equations.- 4.7. Uniqueness of the solution of the (X, T?, H) equation.- 4.8. Probabilistic solution of the interior boundary value problem for the regularized Boltzmann equation.- 4.9. Estimation of the computational labour requirements.- 5. Various Boundary Value Problems Related to the Laplace Operator.- 5.1. Parabolic means and a solution of the mixed problem for the heat equation.- 5.2. Exterior Dirichlet problem for the Laplace equation.- 5.3. Solution of the Neumann problem.- 5.4.Branching random walks on spheres and the Dirichlet problem for the equation ?u = u2.- 5.5. Special method for the solution of the Dirichlet problem for the Helmholtz equation.- 5.6. Probabilistic solution of the wave equation in the case of an infinitely differentiable solution.- 5.7. Another approach to the solution of hyperbolic equations.- 5.8. Probabilistic representation of the solution of boundary value problems for an inhomogeneous telegraph equation.- 5.9. Cauchy problem for the Schr?dinger equation.- 6. Generalized Principal Value Integrals and Related Random Processes.- 6.1. Random processes related to linear equations.- 6.2. Nonlinear equations.- 6.3. On the representation of a solution of nonlinear equations as a generalized principal value integral.- 6.4. Principal part of the operator and the Monte Carlo method.- 7. Interacting Diffusion Processes and Nonlinear Parabolic Equations.- 7.1. Propagation of chaos and the law of large numbers.- 7.2. Interacting Markov processes and nonlinear equations. Heuristic considerations.- 7.3. Weakly interacting diffusions.- 7.4. Moderately interacting diffusions.- 7.5. On one method of numerical solution of systems of stochastic differential equations.- Bibliographical Notes.- References.- Additional References.