Seminar on Stochastic Processes, 1988

Frontmatter

The Riesz transforms associated with second order differential operators

Abstract

This is an expository paper about some recent works on the Riesz transforms associated with a general symmetric, elliptic second order differential operator. Except in the third part, there are no new results and the proofs are often ommited or just sketched.

Dominique Bakry

The Optional Stochastic Integral

Abstract

In this paper we shall study the optional (or compensated) stochastic integral _CH·X. The two main problems connected with this integral will be considered. First, we wish to express H_C·X in terms of an ordinary predictable stochastic integral H’·X, where H’ is a suitable predictable process associated with the optional process H. An attempt in this direction was first undertaken by Yor [8]; however, even for bounded, scalar H, the problem remained open. We shall show in this case that H_C·X — H’·X exists as a certain limit in M², the space of cadlag (Hilbert-valued) square integrable martingales, (cf. §3). Secondly, we shall develop H_C·X for processes H and X which take their values in a separable Hilbert space. These integrals, in turn, will allow us in a later paper to develop H_C·X for certain nuclear-valued processes. Full details of the proofs of the theorems presented here will appear elsewhere.

James K. Brooks, David Neal

On Brownian Excursions in Lipschitz Domains Part II. Local Asymptotic Distributions

Abstract

In this paper, we continue the study initiated in Burdzy and Williams (1986) of the local properties of Brownian excursions in Lipschitz domains. The focus in part I was on local path properties of such excursions. In particular, a necessary and sufficient condition was given for Brownian excursions in a Lipschitz domain to share the local path properties with Brownian excursions in a half-space. This condition holds for C ^1,α -domains (α > 0), but there is a C ¹-domain for which it fails. Here we consider the distributions of a selection of local events for excursions. In particular, we focus on the asymptotics of these distributions as the region of locality shrinks to a point. We show that when a Lipschitz domain is locally approximated by a half-space, the asymptotics for excursions in the two domains are comparable.

Krzysztof Burdzy, Ellen H. Toby, Ruth J. Williams

Gauge Theorem for Unbounded Domains

Abstract

Let {x_t, t⩾0} be the Brownian motion process in R^d, d⩾1; D a domain (nonempty, open and connected set) in R^d; q a Borel function on D. Put

$${\tau_D} = \inf \left\{ {t > 0:{X_t} \notin D} \right\}, $$

and (1)

$$u(x) = {E^X}\left\{ {{\tau_D} < \infty; \;\exp \left[ {\int\limits_0^\tau {{}^Dq\left( {{X_t}} \right)dt} } \right]} \right\} $$

(1)

where E^x (P^x) denotes the expectation (probability) under X₀ = x. The function u is called the gauge for (D,q), provided it is well-defined, namely when the integral involved exists. A result of the following form is called gauge theorem: (2) either u ≡ +∞ in D, or u is bounded in D. Let D̄ denote the closure of D in R^đ (no point at infinity). It is easy to show that if it is bounded in D, then the same upper bound serves for u in D̄, so that u is in fact bounded in Rd since it is equal to one in R^đ - D̄. In this case we say that (D,q) is gaugeable.

Kai Lai Chung

Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains

Abstract

We begin with a resume. Let {P(t), t ≥ 0} be a semigroup of stochastic matrices with elements p _ij (t), (i,j) ∈ I ×I, where I is a countable set, satisfying the condition

$$\mathop {\lim }\limits_{t \downarrow 0} p_{ii} (t) = 1 $$

(1)

. It is known that p’ _ij (0) = q _ij exists and

$$ 0 \leqslant {q_i} = - {q_{{ii}}} \leqslant + \infty, \;0 \leqslant {q_{{ij}}} < \infty, i \ne j; $$

(2)

$$\sum\limits_{{j \ne i}} {{q_{{ij}}} \leqslant {q_i}.} $$

(3)

The state i is called stable if q _i < +∞, and instantaneous if q _i = +∞ (Lévy’s terminology). The matrix Q = (q _ij ) is called conservative when equality holds in (3) for all i.

Kai Lai Chung

Conditional Brownian Motion, Whitney Squares and the Conditional Gauge Theorem

Abstract

Let (X, P ^x ) be Brownian motion killed at τ _D = inf {t > 0: X _t ∉ D}, D a domain in ℝ² and (X, P _z ^x ) this motion conditioned on X _τD = z. For Kato class potentials q we show \( E_x^x\left[ {\exp \left\{ { - \int\limits_0^{{\tau D}} {q\left( {{X_s}} \right)ds} } \right\}} \right] \)is bounded from zero and infinity with little or no assumption on the smoothness of the boundary.

Michael Cranston

Local Field Gaussian Measures

Abstract

A pervasive undercurrent in the study of Gaussian measures is that they are the class of probability measures which it is natural to study if one requires that we see probabilistic properties which are consonant with the linearity and orthogonality properties of the spaces on which the measures are defined. For instance, one entry point into the theory of Gaussian random variables on an arbitrary real vector space with suitable measurable structure is to define a random variable X as being Gaussian if whenever X₁, X₂ are two independent copies of X, then the pair (α₁₁X₁ + α₁₂X₂, α₂₁X₁+ α₂₂ X₂) has the same law as (X_1, X₂) for each pair of orthonormal vectors (α₁₁, α₁₂), (α₂₁,α₂₂) ∈ ℝ². It can be shown that, in the appropriate special cases, this abstract definition is equivalent to the usual concrete definitions for ℝⁿ-valued Gaussian random variables and Gaussian stochastic processes.

Steven N. Evans

Some Formulas for the Energy Functional of a Markov Process

Abstract

In this paper we shall establish two formulas relating the energy functional of a Markov process to that of a subprocess. Let X be a right Markov process and M an exact multiplicative functional of X. Writing (X, M) for the corresponding subprocess, let L and L ^M denote the energy functional of X and (X, M) respectively. Suppose that M doesn’t vanish on [0,ζ[, and define an additive functional A by dA _t = -dM _t /M _t- . Then given an X-excessive measure ξ and an X-excessive function u we have

$$ {L^M}\left( {\xi, u} \right) = L\left( {\xi, u} \right) + {v^{\xi }}(u) $$

(1.1)

where υ^ξ is the Revuz measure of A relative to X and ξ. Formula (1.1) appears as (3.27) in [GSt] in the special case M _t = e ^-qt .

P. J. Fitzsimmons, R. K. Getoor

Note on the 3G Theorem (d = 2)

Abstract

In this note, Theorem 2 in [1] is improved as follows.

Ira W. Herbst, Zhongxin Zhao

The Independence of Hitting Times and Hitting Positions to Spheres for Drifted Brownian Motions

Abstract

A drifted Brownian motion X_t is a diffusion process on R ⁿ whose infinitesimal generator has the form

$$ {\text{L}} = \frac{1}{2}\Delta + {\text{b,}} $$

where ... is the usual Laplace operator and

$$ {\text{b}} = \sum\limits_{{{\text{i}} = 1}}^{\text{n}} {{{\text{b}}^{\text{i}}}({\text{x}})\frac{\partial }{{\partial {{\text{x}}^{\text{i}}}}}} $$

is a smooth vector field on R ⁿ. When b ≡ 0, X_t becomes the usual n-dimensional Brownian motion.

Harry Randolph Hughes, Ming Liao

The Exact Hausdorff Measure of Brownian Multiple Points, II

Abstract

The purpose of this note is to sharpen a result established in [5] concerning the Hausdorff measure of the set of multiple points of a d-dimensiohal Brownian motion. Let X = (X_t, t ≥ 0) denote a standard two-dimensional Brownian motion and, for every integer k ≥ 1, let M_k, denote the set of k-multiple points of X (a point z is said to be k-multiple if there exist k distinct times \(0 \leqslant t_1 < \ldots < t_k \) such that \(X_{t_1 } = \ldots = X_{t_k } = z \)). A canonical measure on M_k, can be constructed as follows. Set: The intersection local time of X with itself, at the order k, is the Radon measure on J _k formally defined by:

$${\alpha _{\text{k}}}({\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}}) = {\delta _{(0)}}({{\text{X}}_{{{\text{t}}_1}}} - {{\text{X}}_{{{\text{t}}_2}}})...{\delta _{(0)}}({{\text{X}}_{{{\text{t}}_{{\text{k}} - 1}}}} - {{\text{X}}_{{{\text{t}}_{\text{k}}}}})\;{\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}} $$

where δ₍₀₎ denotes the Dirac measure at 0 in ℝ². A precise definition of α _k may be found in Rosen [7] or Dynkin [2]. As the previous formal definition suggests, the measure α _k is supported on the set \(\left\{ {({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}});{{\text{X}}_{{{{\text{t}}_{{1}}}}}} = ... = {{\text{X}}_{{{{\text{t}}_{\text{k}}}}}}} \right\} \) of k-multiple times. Let ℓ_k denote the image measure of α_k by the mapping \(({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}}) \to {{\text{X}}_{{{{\text{t}}_{{1}}}}}} \). It follows that ℓ_k is supported on M_k. Notice that ℓ is not a Radon measure, but is a countable sum of finite measures.

Jean-François Le Gall

On a Stability Property of Harmonic Measures

Abstract

The purpose of this note is to prove that the exit distributions of a diffusion from a somewhat smooth domain are stable under a large class of perturbations. That this need not be so in general had been observed already by M.V. Keldyš [5], who constructed a Jordan domain D containing the origin, with the property that if D _n ,n ≥ 1, is any sequence of smooth domains such that

$$ \overline D \subset {D_n} \subset {D_{{n - 1}}}\;{\text{and}}\;\overline D = \mathop{ \cap }\limits_n {D_n} $$

(1)

then the classical harmonic measures h _Dn (0,dy),n ≥ 1, do not converge weakly to the harmonic measure h _D (0,dy). Of course, these measures are the exit distributions of Brownian motion starting from the origin.

Peter March

Behaviour of Excessive Functions of Certain Diffusions under the Action of the Transition Semi-Group

Abstract

In earlier papers [4],[5], it has been shown that under certain analytic conditions concerning its potential kernel, a strong Markov process, which is transient and with continuous sample paths, has all of its excessive harmonic functions, which are not identically infinite, continuous. Also, it has been shown that under the same conditions the excessiveness of harmonic functions of the process is automatic. In this paper we are studying the behaviour of excessive functions of the process under the action of the transition semi-group of the process. For example, all excessive functions for the Brownian motion semi-group are transformed into continuous functions by the semi-group. It seems that even this classical case does not appear in the literature. This will be shown below under a more general setting.

Z. R. Pop-Stojanović

A Maximal Inequality

Abstract

Let X be a uniformly integrable, cadlag non-negative regular supermartingale. Such a process X has the representation

$$ {X_t} = E\left[ {{A_{{\infty + }}} - {A_t}\left| {{F_t}} \right.} \right] $$

(1)

where A_t is continuous and increasing on the half open interval [0,∞), A_o = 0 and A may assign mass to ∞ which is just A_∞+ - A_∞ where \( {{\text{A}}_{\infty }} = \mathop{{\lim }}\limits_{{{\text{t}} \uparrow \infty }} {{\text{A}}_t} \). Then we have the maximal inequality.

K. Murali Rao

Some Results for Functions of Kato Class in Domains of Infinite Measure

Abstract

In this note we extend the gauge theorem to certain sets of infinite measure provided they are “Small” at infinity. For such sets when the gauge is bounded we show that the Schrödinger-Green Kernel is weakly compact in L’ and compact in L^p for 1<p^<∞.

Murali K. Rao

Some Properties of Invariant Functions of Markov Processes

Abstract

Let X = (Ω, F, F_t, X_t, θ_t, P^x) be a right process on a Lusin topological state space E with Borel field B. A point Δ ∈ E will serve as cemetery point. Let P_t and U^a denote the semigroup and resolvent of X. We suppose X is a Borel right process; in particular, U^a f ∈ B⁺ whenever f ∈ B⁺. We restrict our attention to transient Borel right processes throughout this paper, so there is a strictly positive B-measurable function q so that Uq ≤ 1.

Wu Rong

Right Brownian Motion and Representation of Initial Problem

Abstract

Let {X _t ⁺ : t > 0} be the right Brownian motion on [0, ∞) determined by the transition density: for x,y ∈ [0,∞).

$$ {p^{ + }}(t;x,y) = \left\{ {\begin{array}{*{20}{c}} {\frac{y}{{x\sqrt {{2\pi t}} }}[\exp ( - |x - y{|^2}/2t) - \exp ( - |x + y{|^2}/2t)],\quad x > 0} \\ {\sqrt {{\frac{2}{\pi }}} \;\frac{{{y^2}}}{{{t^{{3/2}}}}}\;\exp ( - {y^2}/2t),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x = 0} \\ \end{array} } \right. $$

(1)

This is a Markov process having the tendency moving to the right direction. 0 can be a starting point, but is never reached, i.e., {0} is a polar set.

Z. Zhao

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