Random Processes for Classical Equations of Mathematical Physics

Inhaltsverzeichnis

• 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 Schrödinger 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.