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From the reviews: "This huge book written in several years by one of the few mathematicians able to do it, appears as a precise and impressive study (not very easy to read) of this bothsided question that replaces, in a coherent way, without being encyclopaedic, a large library of books and papers scattered without a uniform language. Instead of summarizing the author gives his own way of exposition with original complements. This requires no preliminary knowledge. ...The purpose which the author explains in his introduction, i.e. a deep probabilistic interpretation of potential theory and a link between two great theories, appears fulfilled in a masterly manner".
M. Brelot in Metrika (1986)

Inhaltsverzeichnis

Frontmatter

Classical and Parabolic Potential Theory

Frontmatter

Chapter I. Introduction to the Mathematical Background of Classical Potential Theory

In this chapter some of the mathematical ideas of classical potential theory are introduced, under simplifying assumptions. The basic space is Euclidean N space ℝN. For a ball B(ξ, δ) in ℝN1.1$$ {l_{{N - 1}}}(\partial B(\zeta, \delta )) = {\pi_N}{\delta^{{N - 1}}}, {\pi_N} = 2{\pi^{{N/2}}}\Gamma {\left( {\frac{N}{2}} \right)^{{ - 1}}}{l_N}(B(\zeta, \delta )) = \frac{{{\pi_N}{\delta^N}}}{N} $$

Joseph L. Doob

Chapter II. Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions

Let B = B(ξ0, δ) and if ξ ∈ B denote by ξ′ the image of ξ under inversion in ∂B. That is, ξx′ is on the ray from ξ0 through ξ, and |ξ – ξ0| |ξx′ – ξ0| = δ2. To simplify the notation take ξ0 = O. Then G B , as defined by 1.1$$ {G_B}\left\{ \begin{gathered} \log \frac{{|\zeta ' - \eta ||\zeta |}}{{\delta |\zeta - \eta |}} \left( { = \log \frac{\delta }{{|\eta |}} if \zeta = 0} \right) \hfill \\ for N = 2 \hfill \\ |\zeta - \eta {|^{{2 - N}}} - {\left( {\frac{\delta }{{|\zeta |}}} \right)^{{N - 2}}} \left( { = |\eta {|^{{2 - N}}} if \zeta = 0} \right) \hfill \\ for N > 2 \hfill \\ \end{gathered} \right. $$ with the understanding that G B (ξ, ξ)= +∞, satisfies items (ix′)–(ivx′) of Section 1.8, so that harmonic measure for B is given by 1.2$$ {\mu_B}(\zeta, d\eta ) = - \frac{1}{{\pi_N^{'}}}{D_{{{n_{\eta }}}}}{G_B}(\zeta, \eta ){l_{{N - 1}}}(d\eta ) = \frac{{K(\eta, \zeta )}}{{{\pi_N}{\delta^{{N - 1}}}}}{l_{{N - 1}}}(d\eta ) $$ where l N–1 here refers to surface area on ∂B and 1.3$$ K(\eta, ) = {\delta^{{N - 2}}}\frac{{{\delta^{{N - 2}}} - |\zeta {|^2}}}{{|\eta - \zeta {|^N}}} $$

Joseph L. Doob

Chapter III. Infima of Families of Superharmonic Functions

If Γ is a class of extended real-valued functions on a set D, a function v on D will be called a majorant [minorant] of Γ if v ≥ u [v ≤ u] for every u in Γ. If D is an open subset of ℝN, the least superharmonic majorant [greatest subharmonic minorant] of Γ, if such a function exists, will be denoted by LMDΓ [GMDΓ], or by LMDu [GMDu] if Γ = {u} is a singleton.

Joseph L. Doob

Chapter IV. Potentials on Special Open Sets

Throughout this book a special open subset of ℝN is either a ball in ℝN or ℝN itself, but the latter only when N > 2. The Green function G D for D a ball was defined in Section II.1. The Green function G D for D = ℝN with N > 2 is defined as G. If μ is a measure on a special open set D, define the function G D μ on D by 1.1$$ {G_D}\mu (\zeta ) = \int_D {{G_D}(} \zeta, \eta )\mu (d\eta ) $$

Joseph L. Doob

Chapter V. Polar Sets and Their Applications

A polar subset of ℝN is a set to each point of which corresponds an open neighborhood of the point that carries a superharmonic function equal to +∞ at each point of the set in the neighborhood. An inner polar set is a set whose compact subsets are polar. It will be shown in Section VI.2 that an analytic inner polar set is polar. If a set is (inner) polar its Kelvin transforms are also.

Joseph L. Doob

Chapter VI. The Fundamental Convergence Theorem and the Reduction Operation

Theorem. Let Γ: {uα, α ∈ I} be a family of superharmonic functions defined on an open subset of ℝN, locally uniformly bounded below, and define the lower envelope u by u(ξ) = infα ∈ Iuα(ξ). Then OC (M, p)u ≤ u, 1.1$$ \mathop{u}\limits_{ + } (\zeta ) = \mathop{{\lim }}\limits_{{n \to \zeta }} \inf u(\eta ) $$and(a) OC (M, p)u is superharmonic.(b) OC (M, p)u = u on each open set on which u is superharmonic.(c) OC (M, p)u = u quasi everywhere.(d) There is a countable subset of Γ whose lower envelope has the same lower semicontinuous smoothing OC (M, p)u.

Joseph L. Doob

Chapter VII. Green Functions

Let D be a nonempty open subset of ℝN. If N > 2 or if N = 2 and D is bounded, the function G(ξ, ·) is lower bounded for each point ξ of D; so GM D G(ξ, ·) exist (Section III.1.) If N = 2, if D is unbounded, and if G(ξ, ·) has a subharmonic minorant on D for some ξ in D, then the minorant GM D G(ξ, ·) exists for every ξ in D. In fact G(ξ , ·)–G(ξ, ·) is bounded below outside each neighborhood of ξ, and G(ξ, ·) is bounded below on each compact neighborhood of ξ so that if GM D G(ξ, ·) exists, G(ξ , ·) ≥ c + GM D G(ξ, ·) GM D G(ξ , ·) ≥ c + GM D G(ξ, ·) for some constant c depending on ξ and ξ.

Joseph L. Doob

Chapter VIII. The Dirichlet Problem for Relative Harmonic Functions

The class of relative harmonic functions is suggested by the following trivial remark. Let (D, OC (M, p)D) be a measurable space, and suppose that to each point ξ of D is assigned some set (perhaps empty) {μα (ξ, ·), α ∈ Iξ} of probability measures on D. Call a function generalized harmonic if it satisfies specified smoothness conditions and if 1.1$$ v(\zeta ) = \int_D {v(\eta ({\mu_{\alpha }}(} \zeta, d\eta ) = {\mu_{\alpha }}(\zeta, v) $$ for ξ in D and α in Iξ. For example, if D is an open subset of ℝN, if for each ξ the index α represents a ball B of center ξ with closure in D, if Iξ is the class of all such balls, and if μB(ξ, v) is the unweighted average of v on ∂B, then the class of continuous functions on D satisfying (1.1) is the class of harmonic functions on D. Going back to the general case, suppose that h is a strictly positive generalized harmonic function and define μhα(ξ, ·) by 1.2$$ \mu_{\alpha }^h(\zeta, A) = \int_A {h(\eta )\frac{{{\mu_{\alpha }}(\zeta, d\eta )}}{{h(\zeta )}}} $$

Joseph L. Doob

Chapter IX. Lattices and Related Classes of Functions

In this chapter certain function classes that arise naturally in potential theory will be discussed. These classes, the corresponding identically named classes in parabolic potential theory (Section XVIII. 19) and in stochastic process theory (Chapter V of Part 2), are discussed together in Chapter I of Part 3.

Joseph L. Doob

Chapter X. The Sweeping Operation

Throughout this chapter D is a Greenian subset of ℝN, coupled with a boundary ∂D provided by a metric compactification when a boundary is relevant, and the boundary of a subset of D ∪ ∂D is that relative to the compactification. In most of the discussion the nature of the boundary is irrelevant, and ∂D can be taken, for example, as the Euclidean boundary or one-point boundary of D.

Joseph L. Doob

Chapter XI. The Fine Topology

The topology of a topological space is the class of open subsets of the space. If T1 and T2 are topologies on a space, T1 is said to be finer than T2 (and then T2 is said to be coarser than T1) if T2 ⊂ T1. For any family of extended real-valued functions on a space there is a coarsest topology making every member of the family continuous, namely, the intersection of all the topologies doing this. The fine topology of classical potential theory is defined as the coarsest topology on ℝN making continuous every super-harmonic function on ℝN. It is easy to verify that the fine and Euclidean topologies coincide when N = 1 (see Chapter XIV for classical potential theory on ℝ), and we suppose from now on in this chapter that N > 1. Concepts relative to the fine topology will be distinguished by an “f,” for example, f lim sup, ∂fA. From now on any otherwise unqualified topological concept will refer to the Euclidean topology. Since the fine topology is defined intrinsically in terms of superharmonic functions, it is not surprising that this topology plays a fundamental role in classical potential theory.

Joseph L. Doob

Chapter XII. The Martin Boundary

Let D be an open subset of ℝN. If D is a ball, its Euclidean boundary is so well adapted to it from a potential theoretic point of view that the following statements are true.

Joseph L. Doob

Chapter XIII. Classical Energy and Capacity

Consider a distribution of positive and negative electric charges on ℝ3 and the electrostatic potential induced by this charge. By definition of a conductor, if A is a connected conducting body in ℝ3, the charge on A distributes itself in such a way that the net effect is that of an all-positive or all-negative charge, and the distribution on A is in equilibrium in the sense that the restriction to A of the potential of the charge distribution in ℝ3 is a constant function.

Joseph L. Doob

Chapter XIV. One-Dimensional Potential Theory

The one-dimensional version of classical potential theory is so special that its discussion has been deferred to this chapter, and much of this theory is so elementary that it will be left to the reader to formulate and justify. A ball in ℝ with center ξ is an open interval with midpoint ξ, and the averages L(u, ξ, δ), and Aαu can play the same role when N=1 as when N>1, but more direct methods are sometimes clearer.

Joseph L. Doob

Chapter XV. Parabolic Potential Theory: Basic Facts

The potential theory based on the Laplace operator, developed in the preceding chapters, will be called classical potential theory below. The potential theory based on the heat operator Δ̇ and its adjoint $$ \mathop{\Delta }\limits^{*} $$, called parabolic potential theory, will be developed in Chapters XV to XIX. Concepts that are parabolic counterparts of classical concepts will be distinguished by dots or asterisks, depending on whether the concepts are related to Δ̇ or to $$ \mathop{\Delta }\limits^{*} $$. Just as the domains of classical potential theory are subsets of ℝN, the domains of parabolic potential theory are subsets of “space time” ℝN+1, which we denote in this context by ℝ̇N. Here N > 1, and the case N=1 is not exceptional. A point ξ̇=(ξ, s) of ℝ̇N had space coordinate ξ in ℝN and time coordinate s-ord ξ (the ordinate of ξ̇), a point of ℝ. The point η̇: (η, t) will be said to be [strictly] below ξ̇: (ξ, s) if t<-s [t <s]. If ξ̇ is a point of an open subset Ḋ of ℝ̇N, the set of points of Ḋ [strictly] below ξ ̇ relative to Ḋ is the set of points of Ḋ that are endpoints of continuous [strictly] downward-directed arcs from ξ̇. That is, η̇ is [strictly] below ξ̇ relative to Ḋ if and only if there is a continuous function f from [0, 1] into Ḋ for which f(0)=ξ̇, f(1)=η̇, and ord f is a [strictly] decreasing function. The upper [lower] half-space of ℝ̇N is the set {ord ξ̇>0} [{ord ξ̇<0}] and the abscissa hyperplane is the set {ord ξ̇=0}

Joseph L. Doob

Chapter XVI. Subparabolic, Superparabolic, and Parabolic Functions on a Slab

If Ḋ is the slab ℝN×]0, δ[, with 0<δ<+∞. the restriction to Ḋ×Ḋ of Ġ satisfies the rather vague description of the Green function Ġ given in Section XV. 7 for smooth regions. It is therefore to be expected from XV (7.3) that the upper boundary of Ḋ if δ<+∞ is a parabolic measure null set and that parabolic measure on the lower boundary is given by $$ {\mathop{u}\limits^{.}_{{\mathop{D}\limits^{.} }}}(\mathop{\zeta }\limits^{.}, d\eta ) = b(s,\zeta - \eta ){l_N}(d\eta ) = \mathop{G}\limits^{.} (\mathop{\zeta }\limits^{.}, (\eta, 0)){l_N}(d\eta ) [\mathop{\zeta }\limits^{.} = (\zeta, s)] $$ so that if u̇ is parabolic on Ḋ with boundary function f in some suitable sense on the lower boundary and if u̇ is appropriately restricted, then 1.1$$ \dot u\left( {\dot \xi } \right) = \int\limits_{^{^{\mathbb{R}^N } } } \ell \left( {s,\xi - \eta } \right)f\left( \eta \right)l_N \left( {d\eta } \right){\text{ }}\left[ {\dot \xi = \left( {\xi ,s} \right)} \right] \cdot $$

Joseph L. Doob

Chapter XVII. Parabolic Potential Theory (Continued)

If Ḋ is a nonempty open subset of ℝ̇N and if Γ is a class of functions on Ḋ, the greatest subparabolic minorant [least superparabolic majorant] of Γ, if there is one, is denoted by ĠMΓ [ĿMΓ]. For example, if Γ is a class of superparabolic functions and if Γ has a subparabolic minorant then ĠMΓ exists and is parabolic. The proof is a translation of that of Theorem III.2. The corresponding notation in the coparabolic context is $$ \mathop{G}\limits^{*} {M_{{\mathop{D}\limits^{.} }}}\Gamma $$ and $$ \mathop{L}\limits^{*} {M_{{\mathop{D}\limits^{.} }}}\Gamma $$.

Joseph L. Doob

Chapter XVIII. The Parabolic Dirichlet Problem, Sweeping, and Exceptional Sets

Let Ḋ be a nonempty open subset of ℝ̇N, and let ḣ be a strictly positive parabolic function on Ḋ. A function v̇/ḣ on Ḋ will be called ḣ-parabolic, ḣ-superparabolic, or ḣ-subparabolic if v̇ is parabolic, superparabolic, or subparabolic, respectively. The notation will be parallel to that in the classical context, with ḣ omitted when ḣ ≡ 1. Thus $$\mathop G\limits^. M_D^{\mathop h\limits^. }$$ , $$^{\dot h}\dot R_{\dot v}^{\dot A}$$ , $$\mathop \tau \limits_{{{\mathop B\limits^. }^{\mathop h\limits^. }}}^. $$ , $$\mathop {H_f^{\dot h}}\limits^* $$ … need no further identification. In the dual context in which ḣ is coparabolic we write $$\mathop G\limits^* M_{\mathop D\limits^. }^{\mathop h\limits^. }$$ , $$^{\dot h}\mathop {R_{\dot v}^{\dot A}}\limits^* $$ , $$\mathop \tau \limits_{{{\mathop B\limits^. }^{\mathop h\limits^. }}}^* $$ , $$\mathop {H_f^{\dot h}}\limits^* $$ , …

Joseph L. Doob

Chapter XIX. The Martin Boundary in the Parabolic Context

In discussing the parabolic context Martin boundary of an open subset Ḋ of ℝ̇N for N>1 we first make the obvious remark that there are necessarily two boundaries, one adapted to the operator Δ̇ and superparabolic potentials, the other adapted to the operator $$ \mathop{\Delta }\limits^{*} $$ and cosuperparabolic potentials. The first is called the exit boundary; the second is called the entrance boundary. These dual contexts are interchanged by a reflection of ℝ̇N in the abscissa hyperplane. We shall treat the exit boundary but shall omit the word “exit” unless both boundaries are involved. The following remarks are offered to orient the reader to the new features that arise in parabolic context Martin boundary theory.

Joseph L. Doob

Probabilistic Counterpart of Part 1

Frontmatter

Chapter I. Fundamental Concepts of Probability

(a) Filtrations of a Measurable Space. Let (Ω, F) be a measurable space, and let (I, <-) be a linearly ordered st. A filtration of (Ω, F) is a map t → F (t) from I into the class of sub σ algebras of F, increasing in the sense that s<-t implies that F (s) ⊂ F+(t) for t in I. The filtration F(•)) is called a filtered measurable space.

Joseph L. Doob

Chapter II. Optional Times and Associated Concepts

Let (Ω, F; F(t), t∈I) be a filtered measurable space. If I does not have a last element, extend I to I⊃{+∞}, were +∞ is a new element ordered after every element of I, and define F(+∞) as any sub σ algebra of F containing ∨ t <+∞ F(t). The choice of F(+∞) within these limits is usually irrelevant. If I has a last element, that element will be denoted by +∞ in this section. The most common choices of I are the set ℤ+ (discrete parameter case) and the set ℝ+ (continuous parameter case). The index set I is thought of as representing a set of time points, and F(t) then represents the class of events up to time t. The problems of defining what is meant by a random time T corresponding to the arrival time of an event whose arrival is determined by preceding events and of defining the class F(T) of preceding events are solved by the following definitions.

Joseph L. Doob

Chapter III. Elements of Martingale Theory

Let (Ω, F, P; F(t), t∈I be a filtered probability space, and let x(•), F(•) be a process on this space, with state space (ℝ̄, F(ℝ̄)). The process is called a supermartingale if the process random variables are integrable and if the supermartingale inequality1.1$$ x\left( s \right) \geqslant E\left\{ {x\left( t \right)|F\left( s \right)} \right\}{\text{ a}}{\text{.s}}{\text{. if }}s < t $$ is satisfied. The exceptional null set may depend on s and t. If I is a set of consecutive integers, inequality (1.1) for t=s+1 implies (1.1) for all pairs s, t with s<t. If the inequality is reversed the process is called a submartingale, and if there is equality in (1.1) the process is called a martingale. The martingale definition is sometimes also applied to complex-valued or vector-valued random variables, but in this book the state space will always be (ℝ̄, F(ℝ̄)) unless some other state space is specified. Martingale theory refers to the mathematics of supermartingales and submartingales as well as martingales.

Joseph L. Doob

Chapter IV. Basic Properties of Continuous Parameter Supermartingales

The continuity properties of continuous parameter supermartingales derived in this section are of course also valid for continuous parameter submartingales and martingales. Recall Recall that a process is said to be [almost surely] right continuous if [almost every sample function is right continuous and the process is said to be [almost surely] right continuous with left limits it [almost every sample function is right continuous and has a left limit at every point.

Joseph L. Doob

Chapter V. Lattices and Related Classes of Stochastic Processes

In this chapter certain stochastic process classes which arise naturally in martingale theory will be discussed. These classes and their relations with the identically named classes in Chapter IX of Part 1 will be studied in later chapters. See Appendix III for the lattice theory to be used.

Joseph L. Doob

Chapter VI. Markov Processes

Let {x(·) be a stochastic process with state space (XX) on a filtered probability space (Ω, ℱ, P; ℱ(t), t ∈ I). The process is called a Markov process if when s < t and A ∈ X, then 1.1$$ P\{ x(t) \in A|F(s)\} = P\{ x(t) \in A|x(s)\} a.s. $$

Joseph L. Doob

Chapter VII. Brownian Motion

Let y(0), y(1), … be mutually independent random variables with distributions q0, q1, …, respectively, on ℝN, and define $$ x(n) = \sum\nolimits_0^n {y(j)} $$, ℱ(n) = ℱ{y(0), …, y(n)} = ℱ{x(0), …, x(n)}. If the transition function p n with state space ℝN is defined by p n (ξ, A) = q n+1(A − ξ), where A is a Borel subset of ℝN and A − ξ is the translation of A by −ξ, the process {x(·), ℱ(·)} is Markovian with initial distribution q0 and successive transition functions p0, p1, …. In fact according to the application in Section I.7, 1.1$$ P\{ x(n + 1) \in A|F(n)\} = P\{ x(n) + y(n + 1) \in A|F(n)\} = {q_{{n + 1}}}(A - x(n)) = {p_n}(x(n),A) a.s. $$

Joseph L. Doob

Chapter VIII. The Itô Integral

Let {w(·), ℱ(·)} be a Brownian motion in ℝ, defined on some probability space (Ω, ℱ, P). It is supposed that ℱ(·) is right continuous and that ℱ(0) contains the null sets. The Itô integral $$ \int_0^t {\phi dw} $$ will be defined for stochastic processes ϕ(·) in the space Γ of not necessarily adapted to ℱ(·) real processes {gy(t), t ∈ ℝ+} with the following property: there is a progressively measurable process {ϕ(·), ℱ(·)}, depending on ψ(·), for which (1.1)$$ P\left\{ {\omega :\int_0^1 {\left| {\varphi \left( {s,\omega } \right)} \right|} ^2 ds < + \infty } \right\} = 1 $$ and 1.2$$ P\{ \phi (t) = \psi (t)\} = 1 $$ for l1 almost every t. In this chapter ds refers to Lebesgue measure l1. For economy in later references absolute value signs are used in (1.1) and similar contexts because the present discussion will be extended in Section 7 to cover vector processes and processes with complex state spaces. Observe that a process indistinguishable from one in Γ is itself in Γ. Let l1 × P be the completed indicated product measure on ℝ+ × Ω, defined on the completion of ℬ(ℝ+) × ℱ relative to this product measure. According to Section I.13, if {ϕ(·), ℱ(·)} is an extended real-valued adapted process and if the function ϕ(·) is l1 × P measurable, then this process is in Γ if (1.1) is satisfied.

Joseph L. Doob

Chapter IX. Brownian Motion and Martingale Theory

The applications of the Itô integral in Sections VIII. 12 to VIII. 14 exhibit aspects of the intimate relation between Brownian motion and martingale theory. In the following we shall go from simple examples of this relation to an analysis by means of martingale theory of the composition of the basic functions of the potential theory for Laplace’s equation [the heat equation] with Brownian motion [space-time Brownian motion]. This will be effected by a direct method without the use of the Itô integral, but there will be a slight repetition of some of the most elementary topics in Chapter VIII.

Joseph L. Doob

Chapter X. Conditional Brownian Motion

Let D be an open subset of ℝN, and recall that a Brownian motion in D was defined in Section VII.9 as an almost surely continuous Markov process with state space D and transition density b D . The probability space on which the process is defined may or may not be rich enough to extend the process to be a Brownian motion in ℝN.

Joseph L. Doob

Part 3

Frontmatter

Chapter I. Lattices in Classical Potential Theory and Martingale Theory

Submartingales martingales and supermatingales are analogs in the context of martingale theory of subharmonic harmonic and superharmonic functions in the context of classical potential theory. The correspondence between these two contexts has two aspects. In the first place many of the manipulations of supermartingales correspond exactly to manipulations of superharmonic functions. This has been exhibited in previous chapters by the common choice of nomenclature, for example, D, S, S m , LM, GM, τ, R•. In the second place under appropriate hypotheses the composition of a superharmonic function with Brownian motion is a supermartingale; for example, see Section 2.IX.7. In this chapter lattice aspects of classical potential theory and martingale theory will be developed simultaneously.

Joseph L. Doob

Chapter II. Brownian Motion and the PWB Method

Let D be an open nonempty subset of ℝN, coupled with a boundary ∂D provided by a metric compactification. To avoid trivial complications, we shall assume that D is connected; if D is disconnected, the results are applicable to each open connected component of D. Let h be a strictly positive harmonic function on D. The PWB method of attacking the first boundary value (Dirichlet) problem for h-harmonic functions on D was detailed in Chapter VIII of Part 1. Recall that the σ algebra of μ D h measurable boundary subsets is the σ algebra of boundary subsets A for which the boundary indicator function l A is h-resolutive and that μ D h(·, A) = H1A h. The class of Borel boundary subsets for which l A is h-resolutive is a σ algebra, and for each point ξ of D the restriction of μ D h(ξ, ·) to this σ algebra, on completion, is the measure μ D h(ξ, ·) on the σ algebra of μ D h measurable sets. The class of μ D h measurable boundary functions f which are μ D h(ξ, ·) integrable does not depend on ξ and is the class of h-resolutive boundary functions, and H f h = μ D h(· f). This class of boundary functions will also be described as the class of μ D h integrable boundary functions.

Joseph L. Doob

Chapter III. Brownian Motion on the Martin Space

Let D be a connected Greenian subset of ℝN, let K be a Martin function for D, let h be a strictly positive superbarmonic function on D, and let {wξh(·), ℱξh(·)} be an h-Brownian motion in D from ξ with lifetime Sξh. For A a subset of D let SξhA and LξhA, respectively, be the hitting and last hitting times of A by wξh(·). According to Theorem l.XII.10, if h is harmonic, the Martin boundary is h-resolutive and μ D h(ξ, dζ) = K(ζ, ξ)M h (dζ)/h(ξ), where M h is the Martin representing measure of h corresponding to K. According to Theorem II.2, the left limit wξh(Sξh−) exists almost surely and has distribution μ D h(ξ, ·) supported (Section l.XII.7) by the minimal Martin boundary ∂1mD. In particular, if ζ is a minimal Martin boundary point and if h = K(ζ, ·), then μ D h(·, {ζ}) = 1; so wξh(Sξh−) = ζ almost surely. With this choice of h we shall sometimes write wξζ(·), SξζA, LξζA, respectively, for wξh(·), SξhA, LξhA.

Joseph L. Doob

Backmatter

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