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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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