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In recent years, the study of the theory of Brownian motion has become a powerful tool in the solution of problems in mathematical physics. This self-contained and readable exposition by leading authors, provides a rigorous account of the subject, emphasizing the "explicit" rather than the "concise" where necessary, and addressed to readers interested in probability theory as applied to analysis and mathematical physics.
A distinctive feature of the methods used is the ubiquitous appearance of stopping time. The book contains much original research by the authors (some of which published here for the first time) as well as detailed and improved versions of relevant important results by other authors, not easily accessible in existing literature.

Inhaltsverzeichnis

Frontmatter

1. Preparatory Material

Abstract
In this chapter we collect together a number of introductory topics for later use. Our aim is to give a self-contained efficient presentation rather than a full discussion of the material. A general treatise such as Chung (1982a) should be consulted for a more deliberate treatment.
Kai Lai Chung, Zhongxin Zhao

2. Killed Brownian Motion

Abstract
Let X t be the d-dimensional Brownian motion and D a domain in ℝ d (d≥ 1). Adjoin an extra point ∂ to D and set
$$ X_t^D = \left\{ {\begin{array}{*{20}c} {X_t on (t < \tau _D )} \\ {\partial on (t \geqslant \tau _D ).} \\ \end{array} } \right. $$
Kai Lai Chung, Zhongxin Zhao

3. Schrödinger Operator

Abstract
Let D be a domain in ℝ d (d≥ 1). We consider the following equation:
$$ \frac{\Delta } {2}u(x) + q(x) = 0,x \in D, $$
where \( \Delta = \sum\nolimits_{i = 1}^d {\partial ^2 /\partial x_i^2 } \) is the Laplacian and q is a Borel measurable function on D. This equation is generally taken in the weak sense as discussed in Section 2.5. Thus (1) is satisfied when uL loc 1 (D), quL loc 1 (D) and
$$ \int {_{_D } u(x)\Delta \varphi (x)dx} = - 2\int {_D q(x)u(x)\varphi (x)dx} $$
for all φ ∈ C c (D).
Kai Lai Chung, Zhongxin Zhao

4. Stopped Feynman-Kac Functional

Abstract
Let D be a domain in ℝ d , and for a function q ∈ β d , let
$$ e_q (\tau _\mathcal{D} ) = \exp (\int_0^{\tau _\mathcal{D} } {q(X_t )dt} ), $$
where {X t} is the Brownian motion in ℝd, and τD is the exit time from D defined in Section 1.5. The random variable in (1) is well defined if and only if \( \int_0^{\tau _\mathcal{D} } q (X_t )dt \) is well defined, almost surely. This is trivially the case if q ∈ 0, or if q is bounded and τD < ∞. To see that this is also the case when q ∈ J and τD < ∞, we need the Corollary to Proposition 3.8 which implies that for each t > 0,
$$ \int_0^t {\left| q \right.(X_s )\left| {ds < \infty } \right.} $$
a.s.; thus, the same is true when t is replaced by τD provided the latter is finite a.s. As before, ‘a.s.’ will be omitted in what follows when the context is obvious. Under these circumstances we have 0 < eq (τD) < ∞; and in fact for each x ∈ ℝd , if Px{τD < ∞} > 0, then
$$ 0 < E^x \left\{ {\tau _\mathcal{D} < \infty ;e_q (\tau _\mathcal{D} )} \right\} \leqslant \infty . $$
Kai Lai Chung, Zhongxin Zhao

5. Conditional Brownian Motion and Conditional Gauge

Abstract
In this section, we develop the notion of conditional Brownian motion introduced by Doob in 1957 (see Doob (1984)) for a general boundary theory. Here it will be used as a tool to sharpen our previous results to include the exit place as well as the exit time. Our treatment is essentially self-contained.
Kai Lai Chung, Zhongxin Zhao

6. Green Functions

Abstract
Let D be a bounded and regular domain in ℝ d , d ≥ 2, and q ∈ Jloc. We assume that (D, q) is gaugeable.
Kai Lai Chung, Zhongxin Zhao

7. Conditional Gauge and q-Green Function

Abstract
In Section 5.4, we proved the Gauge Theorem for a bounded domain D: if the gauge u(x) = E x {e q D )} ≢ ∞ in D, then it is bounded in \( \bar D \) . In this section, we shall prove the Conditional Gauge Theorem for a bounded Lipschitz domain D: if the conditional gauge u(x, z) = E z x {e q D )} ≢ ∞ in D × ∂D, then it is bounded in D × ∂D. For the gauge theorem, no assumption about the boundary is imposed, not even its regularity in the Dirichlet sense. By contrast, the conditional gauge theorem requires a certain smoothness of the boundary. Ad hoc assumptions on D and q may be and have been considered, but we shall settle the case in which D is a bounded Lipschitz domain in ℝ d , d≥2 and q ∈ Jloc. For the case d = 1, see Theorem 9.9 and the Appendix to Section 9.2.
Kai Lai Chung, Zhongxin Zhao

8. Various Related Developments

Abstract
In this section, D will denote a bounded domain in ℝ d , d ≥ 2, q ∈ Jloc.
Kai Lai Chung, Zhongxin Zhao

9. The Case of One Dimension

Abstract
In the case of ℝ1, the special geometry leads to new questions and concepts. At the same time, simpler analysis is often available and yields fuller results. We shall emphasize those aspects of the general theory which do not have ready extensions to higher dimension. In particular, we shall treat the gaugeability of an infinite interval.
Kai Lai Chung, Zhongxin Zhao

Backmatter

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