Random Processes for Classical Equations of Mathematical Physics

Frontmatter

Chapter 1. Markov Processes and Integral Equations

Abstract

This chapter presents a survey of well-known connections between Markov processes and some equations arising from their consideration. These connections are of great interest, for they lay the foundations of the Monte Carlo method for the solution of the corresponding equations. It is natural, of course, that various types of processes lead to a variety of equations.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 2. First Boundary Value Problem for the Equation of the Elliptic Type

Abstract

As has been pointed out in Section 1.3, one of the ways to solve the first boundary problem is to reduce it to the integral equation satisfying appropriate conditions and to construct an unbiased estimator of its solution on the trajectories of the convergent Markov chain adapted to this integral equation. In Section 1.3 the scheme was realized for the simplest example of the interior Dirichlet problem for the Laplace operator. Here a more complicated and interesting case of an arbitrary elliptic operator with smooth coefficients will be considered.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 3. Equations with Polynomial Nonlinearity

Abstract

Having discussed connections between branching Markov chains and nonlinear integral equations in Section 1.4, we dealt essentially with the constructions given in [32]. Now we shall take another approach to the unbiased estimation of the solution of equations with polynomial nonlinearity. Since further considerations have a formal character and are cumbersome, it is worthwhile starting with elucidatory examples.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 4. Probabilistic Solution of Some Kinetic Equations

Abstract

Some nonlinear kinetic equations of rarefied gas dynamics are connected with the special class of inhomogeneous Markov processes. The present chapter is devoted to the construction of these processes and the investigation of their properties. We shall also consider some simulation aspects of these processes.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 5. Various Boundary Value Problems Related to the Laplace Operator

Abstract

This Chapter describes the construction of Markov chains and estimators for the solution of concrete boundary value problems connected with the Laplace operator. The main difficulty to be overcome is the obtaining of an integral equation equivalent to a given problem and then choosing a Markov chain on whose trajectories unbiased estimators are to be constructed. For various problems associated with the Laplace operator the difficulty is overcome with the help of a corresponding Green formula.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 6. Generalized Principal Value Integrals and Related Random Processes

Abstract

In previous chapters the importance of the existence of an iterative solution of the majorant equation was emphasized many times. For a lot of problems of mathematical physics, including radiation transport problems, the situation is simplified since a majorant operator coincides with an initial one. Meanwhile it is easy to list many examples where the investigation of majorant equations is very useful. The most simple one is the matrix operator

$$ A = \left( {\begin{array}{*{20}c} a & a \\ a & { - a} \\ \end{array} } \right)$$

An iterative procedure \(X_{n + 1} = AX_n + F,X_0 = F \) converges for \(|a| < 1/\sqrt 2 \) but it it is easy to see that a majorant iterative procedure converges only for \( |a| < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\).

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Chapter 7. Interacting Diffusion Processes and Nonlinear Parabolic Equations

Abstract

Let (E, ρ) be a separable metric space with the Borel σ-algebra B and μ a probabilistic measure on (E, B). We shall say that the sequence {μ _n } _n=1 ^∞ of symmetric probability measures on (E ⁿ , B ⁿ ) is μ-chaotic if for any k ≥ 1 and any collection of functions φ₁, …, φ _k such that

$$ \phi _i \in C_b \left( E \right),i = 1,2, \ldots ,k$$

,

$$ E^{\mu _n } \left( {\phi _1 \otimes \cdots \otimes \phi _k \otimes \cdots \otimes 1} \right) \to \prod\limits_{i = 1}^k {E^\mu \phi _i } $$

(7.1.1)

.

S. M. Ermakov, V. V. Nekrutkin, A. S. Sipin

Backmatter