Celebratio Mathematica — Varadhan

Diffusion theory

Works connected to George C. Papanicolaou

Filter the Bibliography List

year	title	people

			clear

G. C. Papanicolaou and S. R. S. Varadhan: “A limit theorem with strong mixing in Banach space and two applications to stochastic differential equations,” Comm. Pure Appl. Math. 26 : 4 (July 1973), pp. 497–524. MR 0383530 Zbl 0253.60065 article

In [Papanicolaou and Hersh 1972] the asymptotic behavior of the expected value of the solution of an abstract stochastic equation was investigated. This work was motivated by the results of Khasminskii [1966] and Stratonovich [1968] for stochastic ordinary differential equations and other works (cf. [Kubo 1963; Lax 1966; Papanicolaou and Keller 1971]) concerned with operator equations. The results obtained in [Papanicolaou and Hersh 1972] were limited by severe restrictions on the allowed from of the stochastic perturbation. Recently, Cogburn and Hersh [Cogburn and Hersh 1973] have generalized the results of [Papanicolaou and Hersh 1972] considerably by allowing a much broader class of stochastic perturbations and requiring only a strong mixing condition. Our aim here is to improve the results of [Cogburn and Hersh 1973] by giving an estimate for the error committed in the asymptotic approximation. If \( \varepsilon \) denotes the small parameter of the problem we show that the error is \( O(\varepsilon) \) as \( \varepsilon\to 0 \). The results in [Papanicolaou and Hersh 1972; Khasminskii 1966; Stratonovich 1968; Cogburn and Hersh 1973] show only that the error is \( o(1) \). Our estimate is best possible since it is achieved for the classical central limit theorem which is a special case of our Theorem 2.

G. C. Papanicolaou and S. R. S. Varadhan: “Diffusion in regions with many small holes,” pp. 190–206 in Stochastic differential systems (Vilnius, Lithuania, 28 August–2 September, 1978). Edited by B. Grigelionis. Lecture Notes in Control and Information Sciences 25. Springer (Berlin), 1980. MR 609184 Zbl 0485.60076 incollection

Let \( D \) be a bounded open set containing the origin, having \( C^2 \) boundary and with diameter less than or equal to one. For each \( N = 1,2,\dots \), let \( y_1^{(N)}, y_2^{(N)},\dots,y_N^{(N)} \) be points in \( \mathbb{R}^3 \) and define sets \( D_i^{(N)} \) by \[ D_i^{(N)} = \bigl\{x\in \mathbb{R}^3\mid N(x-y_i^{(N)})\in D\bigr\}, \] \( i = 1,2,\ldots,N \). We shall call the set \( D_i^{(N)} \) the hole centered at \( y_i^{(N)} \) with diameter less than or equal to \( N^{-1} \). Let \( G^{(N)} \) denote the region \[ G^{(N)} = \mathbb{R}^3-\bigcup_{i=1}^N D_i^{(N)} \] which is \( \mathbb{R}^3 \) with holes of diameter \( \leq N^{-1} \) centered at \( y_1^{(N)},\dots,y_N^{(N)} \). We shall analyze the asymptotic behavior of \( u^{(N)}(x,t) \) as \( N\to\infty \) which is the solution of \begin{align*} \frac{\partial}{\partial t} u^{(N)}(x,t) &= \frac{1}{2}\Delta u^{(N)}(x,t), && t > 0,\ x\in G^{(N)},\\ u^{(N)}(x,t) &=0, && t > 0,\ x\in\partial G^{(N)} = \bigcup_{i=1}^N\partial D_i^{(N)},\\ u^{(N)}(x,0) &=f(x), && x\in G^{(N)}, \end{align*} with \( f(x) \) a given bounded continuous function with compact support in \( \mathbb{R}^3 \).

George C. Papanicolaou	Related
Bronius Grigelionis	Related