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

Chen, Zhen-Qing; Fukushima, Masatoshi

J. Math. Soc. Japan 70, No. 2, 833-852 (2018).

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.

Reviewer: René L. Schilling (Dresden)

MSC:

60J50	Boundary theory for Markov processes
60J65	Brownian motion
31C25	Dirichlet forms

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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