## General gauge theorem for multiplicative functionals.(English)Zbl 0647.60083

Let $$(X_ t,{\mathcal F}_ t,\theta_ t)$$ be a Hunt process with state space (E,$${\mathcal E})$$ and transition semigroup $$(P_ t)$$. It is resumed that there exists a reference measure m such that for some $$\lambda >0$$, the $$\lambda$$-potential $$U^{\lambda}=\int^{\infty}_{0}e^{-\lambda t}dt$$ satisfies $$U^{\lambda}(x,\cdot)\ll m$$ for each x. Given a potential V, i.e., an excessive function such that $$P_{K^ c}V\to 0$$ almost everywhere as the compact set K increases to E, there is a unique increasing additional functional A such that $$V(x)=E^ x[A_{\infty}]$$, to which is associated the multiplicative functional $$e_ A(t)=e^{A(t)}.$$
Given $$V=V^{(1)}-V^{(2)}$$ for potentials V(i), let $$A=A^{(1)}- A^{(2)}$$, and let $$e_ A$$ be defined as before. Let D be a nonempty, open subset of E and define the function $$g(x)=E^ x[e_ A(\tau_ D)]$$, where $$\tau_ D$$ is the first exit time from D; g is termed the gauge for (D,A). Various properties of g are presented:
(i) If d is relatively compact, then $$F=\{x\in D: g(x)<\infty \}$$ is absorbing relative to D and g is bounded on F;
(ii) If for some $$\lambda$$ the measures $$\{U^{D,\lambda}(x,\cdot): x\in D\}$$ are mutually equivalent, then either $$g\equiv \infty$$ on D or g is bounded on E;
(iii) If $$(P_ t)$$ is doubly Feller, D is regular and g is bounded on D, then g is continuous on E.
The authors also present various consequences of bounded gauge and a “super gauge” theorem to the effect that if the gauge for (D,A) is bounded, then so, for sufficiently small $$\epsilon$$, is that for $$(D,(1+\epsilon)A)$$.
Reviewer: A.Karr

### MSC:

 60J40 Right processes 60J45 Probabilistic potential theory 60J65 Brownian motion 35J10 Schrödinger operator, Schrödinger equation
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### References:

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