One-point extensions of Markov processes by darning

Abstract

This paper is a continuation of the works by Fukushima-Tanaka (Ann Inst Henri Poincaré Probab Stat 41: 419-459, 2005) and Chen-Fukushima-Ying (Stochastic Analysis and Application, p.153-196. The Abel Symposium, Springer, Heidelberg) on the study of one-point extendability of a pair of standard Markov processes in weak duality. In this paper, general conditions to ensure such an extension are given. In the symmetric case, characterizations of the one-point extensions are given in terms of their Dirichlet forms and in terms of their L 2 -infinitesimal generators. In particular, a generalized notion of flux is introduced and is used to characterize functions in the domain of the L 2 -infinitesimal generator of the extended process. An important role in our investigation is played by the α-order approaching probability u α .

Flux and Lateral Conditions for Symmetric Markov Processes

“…The absorbed Brownian motion X 0 is symmetric with respect to the Lebesgue measure m but it can also be viewed as an m-symmetric diffusion on E 0 as has been observed in [4] already. Let ( E 0 , F) and ( E ref , ( F 0 ) ref a ) be the Dirichlet form of X 0 on L 2 (E 0 ; m) and its active reflected Dirichlet space, respectively.…”

“…The first way is to add to F 0 the space of all harmonic functions on E 0 having finite Dirichlet integrals by using the equilibrium measures ( [16] and [3]) or the energy functional ( [6] and [4]), while the second way is to consider the space of all functions on E 0 with finite Dirichlet integrals by using the energy measures of u ∈ F loc (see [15] and [3]). Note that the results in [3] and [15,16] are applicable here for our quasi-regular Dirichlet form (E 0 , F 0 ) due to its quasi-homeomorphism (see [7]) to a transient regular Dirichlet form on a locally compact metric space.…”

“…We now recall the definitions, with some improvements over those in [6] and [4]. In particular, we will not apply a quasi-homeomorphism to X 0 and (E 0 , F 0 ) rendering them into a Hunt process and a regular Dirichlet form except for the proof of Lemmas 2.2 and 2.4(ii).…”

“…(1.2) This question belongs to the boundary problem of Markov processes, dating back to Feller [9] where the problem was raised after his discovery of the most general boundary conditions for the one-dimensional diffusions [8]. When F is a singleton {a}, such a problem was studied in [4] at the infinitesimal generator level. In [4, Section 4], we give a characterization of the L 2 -generator of X via a lateral condition described in terms of flux of X 0 at the point a.…”

“…and we can apply [4,Theorem 4.10] to construct a unique m 1 -symmetric diffusion X 1 on R + extending X 01 by darning the hole a 1 with entrance law μ 1 t determined by ∞ 0 μ 1 t dt = m 1 .…”