Reflections at infinity of time changed RBMs on a domain with Liouville branches. (English) Zbl 1395.60093

Let \(\mathbb D(u,v) := \int_D \nabla u \nabla v\,dx\) be the classical Dirichlet form defined for functions \(u,v\in H^1(D) = \{u\in L^2(D;dx) : \nabla u\in L^2(D;dx) \}\) on a domain \(D\subset\mathbb R^d\), and let \(\mathrm{BL}(D) := \{u\in L^2_{\mathrm{loc}}(D;dx) : \nabla u\in L^2(D; dx)\}\) be Beuerling-and-Deny’s Beppo Levi space. The set \(D\) is called Liouville domain if any \(u\in \mathrm{BL}(D)\) has a unique decomposition \(u=u_0+c\) where \(c\in\mathbb R\) is a constant and \(u_0\) is an element of the extended Dirichlet space \(H_e^1(D)\) (we assume that the Dirichlet form is transient). Examples of Liouville sets are \(D=\mathbb R^d\) if \(d\geq 3\), and all unbounded uniform domains in \(\mathbb R^d\), \(d\geq 3\), which (in a suitable way) do not get “too narrow” at infinity. A set \(D\subset\mathbb R^d\) with Lipschitz boundary has \(N\) Liouville branches, if \(\mathbb R^d\setminus D\) can be written as a union \(\bigcup_i^N C_i\) of unbounded Liouville domains whose closures are mutually disjoint and whose boundaries are Lipschitz. Denote by \(\partial_i\) the point at infinity of \(C_i\). Then \(\overline D^* := \overline D \cup\{\partial_1,\dots,\partial_N\}\) can be understood as the \(N\)-point compactification of \(D\). Finally, let \(\phi_i(x) = \mathbb P^x(\lim_{t\to\infty} W_t = \partial_i)\) the probability that (the necessarily transient) Brownian motion \(W\), started at \(x\), converges to \(\partial_i\). Because of transience, one can make a time change using the inverse of the additive functional \(A_t = \int_0^t f(W_s)\,ds\), \(f\in L^1(D,dx)\), \(f>0\) on \(\overline D\), and one has \(A_\infty < \infty\) a.s. Thus, the process \(X_t := W_{A_t^{-1}}\) explodes (i.e.converges) towards \(\partial_i\), \(i=1,\dots,N\), in finite time. Therefore it makes sense to ask for extensions of \(X\) beyond its original life-time.
The main result of the paper states that there exists a unique extension of the process \(X\) to \(\overline D^*\) and of the associated Dirichlet form. The extension of the form is \((\mathcal E^*,\mathcal F^*)\) and it is defined on \(L^2(\overline D^*,dm) = L^2(D,dm)\) with \(m(dx) = f(x)\,dx\). Moreover, the extended Dirichlet space is given by \(\mathcal E^* = \frac 12\mathbb D\), \(\mathcal F_e^*=H_e^1(D)\oplus\mathrm{span}\{\phi_i : i=1,\dots,N\}\) and this is just the Beppo Levi space \(\mathrm{BL}(D)\). Consequently, the original process has at most \(N\) essentially different extensions by reflection at the points \(\{\partial_1,\dots,\partial_N\}\) (that is, we reflect at exactly \(k\in\{1,\dots,N\}\) of the points). Moreover, the generator of the extended form is identified as \(\mathcal Au = \frac 1f \frac 12 \Delta u\) a.e.on \(D\) where \(\Delta\) is the Laplace operator (defined in a distributional sense) on the functions \(u\in \mathrm{BL}(D)\cap L^2(D,m)\) and with the lateral condition that \[ \frac 12\mathbb D(u,\phi_i) + \frac 12\int_D \Delta u\phi_i\,dx = 0,\quad i=1,\dots N, \] (this is the generator for reflection at all \(\partial_i\)’s, the other extensions have similarly looking generators). Most of the results of the paper can be extended to general positive continuous additive functionals (not necessarily with a density \(f\)) using the machinery of quasi-homeomorphisms of Dirichlet forms.


60J50 Boundary theory for Markov processes
60J65 Brownian motion
31C25 Dirichlet forms
Full Text: DOI arXiv Euclid


[1] N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter, 1991. · Zbl 0748.60046
[2] M. Brelot, Etude et extension du principe de Dirichlet, Ann. Inst. Fourier, 5 (1953/54), 371-419. · Zbl 0067.33002
[3] Z.-Q. Chen, On reflecting diffusion processes and Skorokhod decompositions, Probab. Theory Relat. Fields, 94 (1993), 281-315. · Zbl 0767.60074
[4] Z.-Q. Chen and M. Fukushima, On unique extension of time changed reflecting Brownian motions, Ann. Inst. Henri Poincaré Probab. Statist., 45 (2009), 864-875.
[5] Z.-Q. Chen and M. Fukushima, One-point reflections, Stochastic Process Appl., 125 (2015), 1368-1393. · Zbl 1327.60151
[6] Z.-Q. Chen, Z.-M. Ma and M. Röckner, Quasi-homeomorphisms of Dirichlet forms, Nagoya Math. J., 136 (1994), 1-15. · Zbl 0811.31002
[7] J. Deny and J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953/54), 305-370. · Zbl 0065.09903
[8] X. Fang, M. Fukushima and J. Ying, On regular Dirichlet subspaces of \(H^1(I)\) and associated linear diffusions, Osaka J. Math., 42 (2005), 27-41. · Zbl 1068.60093
[9] M. Fukushima, On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan, 21 (1969), 58-93. · Zbl 0181.20903
[10] M. Fukushima, From one dimensional diffusions to symmetric Markov processes, Stochastic Process Appl., 120 (2010), 590-604. · Zbl 1221.60113
[11] M. Fukushima, On general boundary conditions for one-dimensional diffusions with symmetry, J. Math. Soc. Japan, 66 (2014), 289-316. · Zbl 1296.60211
[12] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Grruyter, 1994, 2nd Edition, 2010. · Zbl 0838.31001
[13] M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Relat. Fields, 106 (1996), 521-557. · Zbl 0867.60047
[14] A. Grigor’yan and L. Saloff-Coste, Heat kernels on manifolds with ends, Ann. Inst. Fourier, Grenoble 59 (2009), 1917-1997. · Zbl 1239.58016
[15] D. A. Herron and P. Koskella, Uniform, Sobolev extension and quasiconformal circle domains, J. Anal. Math., 57 (1991), 172-202. · Zbl 0776.30014
[16] P. W. Jones, Quasiconformal mappings and extendibility of functions in Sobolev spaces, Acta Math., 147 (1981), 71-88. · Zbl 0489.30017
[17] Y. Kuz’menko and S. Molchanov, Counterexamples to Liouville-type theorems, Moscow Univ. Math. Bull., 34 (1979), 35-39.
[18] R. G. Pinsky, Transience/recurrence for normally reflected Brownian motion in unbounded domains, Ann. Probab., 37 (2009), 676-686. · Zbl 1169.60018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.