## 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.

### MSC:

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

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