##
**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.

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.

Reviewer: René L. Schilling (Dresden)

### Keywords:

transient reflecting Brownian motion; time change; Liouville domain; Beppo Levi space; quasi-homeomorphism; zero flux; time-change; positive continuous additive functional
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\textit{Z.-Q. Chen} and \textit{M. Fukushima}, J. Math. Soc. Japan 70, No. 2, 833--852 (2018; Zbl 1395.60093)

### References:

[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 |

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