P. J. Fitzsimmons

Markovian Bridges: Construction, Palm Interpretation, and Splicing

By a Markovian bridge we mean a process obtained by conditioning a Markov process X to start in some state x at time 0 and arrive at some state z at time t. Once the definition is made precise, we call this process the (x, t, z)-bridge derived from X. Important examples are provided by Brownian and Bessel bridges, which have been extensively studied and find numerous applications. See for example [PY1,SW,Sa,H,EL,AP,BP].It is part of Markovian folklore that the right way to define bridges in some generality is by a suitable Doob h -transform of the space-time process. This method was used by Getoor and Sharpe [GS4] for excursion bridges, and by Salminen [Sa] for one-dimensional diffusions, but the idea of using h-transforms to construct bridges seems to be much older. Our first object in this paper is to make this definition of bridges precise in a suitable degree of generality, with the aim of dispelling all doubts about the existence of clearly defined bridges for nice Markov processes. This we undertake in Section 2. In Section 3 we establish a conditioning formula involving bridges and continuous additive functionals of the Markov process. This formula can be found in [RY, Ex. (1.16) of Ch. X, p.378] under rather stringent continuity conditions. One of our goals here is to prove the formula in its “natural” setting. We apply the conditioning formula in Section 4 to show how Markovian bridges are involved in a family of Palm distributions associated with continuous additive functionals of the Markov process. This generalizes an approach to bridges suggested in a particular case by Kallenberg [K1], and connects this approach to the more conventional definition of bridges adopted here.

Construction and regularity of measure-valued markov branching processes

A construction is given for a general class of measure-valued Markov branching processes. The underlying spatial motion process is an arbitrary Borel right Markov process, and state-dependent offspring laws are allowed. It is shown that such processes are Hunt processes in the Ray weak* topology, and have continuous paths if and only if the total mass process is continuous. The entrance spaces of such processes are described explicitly.

Transition density estimates for Brownian motion on affine nested fractals

A class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals. For these fractals estimates on the transition density for the Brownian motion are obtained using the associated Dirichlet form. An upper bound for the diagonal can be found using a Nash-type inequality, then probabilistic techniques are used to obtain the off-diagonal bound. The approach differs from previous treatments as it uses only the Dirichlet form and no estimates on the resolvent. The bounds obtained are expressed in terms of an intrinsic metric on the fractal.

Absolute continuity of symmetric Markov processes

We study Girsanov’s theorem in the context of symmetric Markov processes, extending earlier work of Fukushima–Takeda and Fitzsimmons on Girsanov transformations of “gradient type.” We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward–backward martingale method of Lyons–Zheng, to cover the case of processes with jumps.

Excessive measures and Markov processes with random birth and death

On utilise un processus markovien continu a droite stationnaire defini sur un intervalle de temps aleatoire, comme outil naturel pour developper plusieurs aspects de la theorie des mesures excessives

Kac's moment formula and the Feynman{Kac formula for additive functionals of a Markov process

Mark Kac introduced a method for calculating the distribution of the integral Av=[integral operator]0Tv(Xt) dt for a function v of a Markov process (Xt, t[greater-or-equal, slanted]0) and a suitable random time T, which yields the Feynman-Kac formula for the moment-generating function of Av. We review Kacs method, with emphasis on an aspect often overlooked. This is Kacs formula for moments of Av, which may be stated as follows. For any random time T such that the killed process (Xt, 0[less-than-or-equals, slant]t

On the Quasi-regularity of Semi-Dirichlet Forms

We prove that if a right Markov process is associated with a semi-Dirichlet form, then the form is necessarily quasi-regular. As applications, we develop the theory of Revuz measures in the semi-Dirichlet context and we show that quasi-regularity is invariant with respect to time change.

Intersections and limits of regenerative sets

SummaryRegenerative subsets of ℝ constitute an analog of classical renewal processes. Limits and intersections of independent regenerative sets are discussed. These ideas are related to the usual quantities associated with subordinators.

The set of real numbers left uncovered by random covering intervals

SummaryRandom covering intervals are placed on the real line in a Poisson manner. Lebesgue measure governs their (random) locations and an arbitrary measure μ governs their (random) lengths. The uncovered set is a regenerative set in the sense of Hoffmann-Jørgensens generalization of regenerative phenomena introduced by Kingman. Thus, as has previously been obtained by Mandelbrot, it is the closure of the image of a subordinator —one that is identified explicitly. Well-known facts about subordinators give Shepps necessary and sufficient condition on μ for complete coverage and, when the coverage is not complete, a formula for the Hausdorff dimension of the uncovered set. The method does not seem to be applicable when the covering is not done in a Poisson manner or if the line is replaced by the plane or higher dimensional space.

Stochastic calculus for symmetric Markov processes

Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Ito formula for Dirichlet processes is obtained.