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Markov Processes and Differential Equations

Asymptotic Problems

verfasst von: Mark Freidlin

Verlag: Birkhäuser Basel

Buchreihe : Lectures in Mathematics ETH Zürich

Enthalten in: Professional Book Archive

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Über dieses Buch

Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.

Inhaltsverzeichnis

Frontmatter

1. Stochastic Processes Defined by ODE’s

Abstract

A stochastic process in the time interval [0,∞) is defined as a family of random variables X_t(ω), t ≥ 0, ω ∈ Ω, on a measurable space (Ω,F, P). To describe the probability structure of the process, one should define the family of distributions of (X_t1,…, X_tn) for any integer n and any 0 ≤ t₁ < t₂ < … < t_n. This family of finite dimensional distributions is, in general, a rather bulky subject. Therefore, as a rule, special classes of stochastic processes are considered for which such a description can be reduced to more convenient characteristics. First, we consider the basic processes which have a simple statistical structure. Then we consider relatively simple and explicitly defined transformations of these basic processes. The Wiener process, Poisson process, and continuous time Markov chains with finite number of states will be our basic processes. We assume that the main properties of these processes are known. Actually, all classes of continuous time stochastic processes, allowing deep enough theory, can be constructed from these basic processes using relatively simple transformations.

Mark Freidlin

2. Small Parameter in Higher Derivatives: Levinson’s Case

Abstract

Let

$${L^{{ \in ^2}}} = \frac{{{ \in ^2}}}{2}\sum\limits_{i,j = 1}^r {{a^{ij}}} \left( x \right)\frac{{{\partial ^2}}}{{a{x^i}\partial {x^j}}} + \sum\limits_{i = 1}^r {{b^i}} \left( x \right)\frac{\partial }{{\partial {x^i}}},x \in {R^r}{L^{{ \in ^2}}} = \frac{{{ \in ^2}}}{2}\sum\limits_{i,j = 1}^r {{a^{ij}}} \left( x \right)\frac{{{\partial^2}}}{{a{x^i}\partial {x^j}}} + \sum\limits_{i = 1}^r {{b^i}} \left( x \right)\frac{\partial }{{\partial {x^i}}},x \in {R^r}$$

(1)

where the coefficients

$${a^{ij}}\left( x \right),{b^i}\left( x \right)$$

are assumed to be bounded and Lipschitz continuous, and

$$\sum\limits_{i,j = 1}^r {{a^{ij}}} \left( x \right){\lambda _i}{\lambda _j} \geqslant a\sum\limits_1^r \lambda _i^2$$

for any real λ₁,…,λ_r and some a > 0.

Mark Freidlin

3. The Large Deviation Case

Abstract

The Levinson conditions mean, roughly speaking, that the trajectories of the degenerate process (or of the dynamical system, in the case of one equation) leave the domain G with probability 1, and that some regularity conditions on the boundary are fulfilled. In this case the degenerate problem (∈ = 0) with boundary conditions preserved on the regular (for the degenerate equation) part of the geometric boundary of the domain, has a unique solution. The solution of the perturbed problem in the Levinson case converges to the unique solution of the degenerate problem as ∈ ↓ 0.

Mark Freidlin

4. Averaging Principle for Stochastic Processes and for Partial Differential Equations

Abstract

Consider a dynamical system in R²:

$${X_t} = b\left({{X_t}}\right),{X_0} =\chi\in {R^2}$$

(1)

Assume that system (4.1) has a first integral H(x) and that this function is of C²-class:

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-0348-9191-2_4/978-3-0348-9191-2_4_IEq1_HTML.gif

. Let the function H(x) have just one critical point — a minimum at the origin 0, H(0) = 0, H(x) > 0 for x ≠ 0, and suppose that all the level sets C(y) = {x ∈ R² : H(x) = y}, y ≥ 0, are compact. Moreover, let b(y) ≠ 0 for y ≠ 0. The corresponding phase picture is given in Fig. 5.

Mark Freidlin

5. Averaging Principle: Continuation

Abstract

Consider a Hamiltonian dynamical system in the plane R²

$${\mathop X\limits^. _t} = \mathop \nabla \limits^ - H({X_t}),{X_0} = x \in {R^2},\mathop \nabla \limits^ - H(x) = \left( {\frac{{\partial H(x)}}{{\partial {x^2}}}, - \frac{{\partial H(x)}}{{\partial {x^1}}}} \right). $$

(5.1)

Mark Freidlin

6. Remarks and Generalizations

Abstract

In this and the next section, we consider some other asymptotic problems also leading to processes on graphs and corresponding problems for PDE’s.

Mark Freidlin

7. Diffusion Processes and PDE’s in Narrow Branching Tubes

Abstract

Consider a connected graph Γ in R^r with vertices O₁,…, O_m and edges I₁,…, I_n. Let G^∈ be a domain consisting of narrow tubes surrounding the edges I_i ⊂ Γ and of small neighborhoods ε_k of the vertices O_k ⊂ Γ (see Fig. 16).

Mark Freidlin

8. Wave Fronts in Reaction-Diffusion Equations

Abstract

A. N. Kolmogorov, I. G. Petrovskii and N. S. Piskunov considered in 1937 [KPP] the following problem:

$$\begin{gathered} \frac{{\partial u(t,x)}}{{\partial t}} = \frac{D}{2}\frac{{{{\partial }^{2}}u}}{{\partial {{x}^{2}}}} + f(u),t > 0,x \in {{R}^{1}} \hfill \\ u(0,x) = {{x}^{ - }}(x) = \left\{ {\begin{array}{*{20}{c}} {1,x \leqslant 0} \\ {0,x > 0.} \\ \end{array} } \right. \hfill \\ \end{gathered}$$

(8.1)

Mark Freidlin

9. Wave Fronts in Slowly Changing Media

Abstract

We consider in this section one more generalization of the KPP result. Our goal here is to study wave fronts in non-homogeneous media. Of course, one cannot expect the existence of an asymptotic speed without any assumption about the coefficients: they must be, in an asymptotic sense, homogeneous. For example, one can prove the existence of an asymptotic speed if the coefficients and f(x,u) are periodic in x or close to periodic (see [F6]). If the coefficients and f(x,u) are space-homogeneous random fields with some ergodic properties, one also can expect that an asymptotic speed will be established. The last problem is, actually, solved just in the one-dimensional case (see [F6] and references there).

Mark Freidlin

10. Large Scale Approximation for Reaction-Diffusion Equations

Abstract

We introduce in this section a new assumption concerning the nonlinear term, which allows us to calculate the asymptotics in the bistable case and for other types of local dynamics. But first we give a more general formulation of the problem of interaction between the stochastic transport and multiplication of the particles.

Mark Freidlin

11. Homogenization in PDE’s and in Stochastic Processes

Abstract

Consider a second order elliptic operator with 1-periodic coefficients

$$L = \frac{1}{2}\sum\limits_{i,j = 1}^r {{a^{ij}}} (x)\frac{{{\partial ^2}}}{{\partial {x^i}\partial {x^j}}} + \sum {{b^i}} (x)\frac{\partial }{{\partial {x^i}}},x \in {R^r}.$$

(1)

Mark Freidlin

Backmatter

Titel: Markov Processes and Differential Equations
verfasst von: Mark Freidlin
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-9191-2
Print ISBN: 978-3-7643-5392-6
DOI: https://doi.org/10.1007/978-3-0348-9191-2

Springer Professional

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Inhaltsverzeichnis

Frontmatter

1. Stochastic Processes Defined by ODE’s

2. Small Parameter in Higher Derivatives: Levinson’s Case

3. The Large Deviation Case

4. Averaging Principle for Stochastic Processes and for Partial Differential Equations

5. Averaging Principle: Continuation

6. Remarks and Generalizations

7. Diffusion Processes and PDE’s in Narrow Branching Tubes

8. Wave Fronts in Reaction-Diffusion Equations

9. Wave Fronts in Slowly Changing Media

10. Large Scale Approximation for Reaction-Diffusion Equations

11. Homogenization in PDE’s and in Stochastic Processes

Backmatter