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Dirichlet processes and an intrinsic characterization of renormalized intersection local times. (English) Zbl 0981.60072

The intersection local time of order \(n\), \(n= 2,3,4,\dots\), is the amount of “time” a stochastic process \(\{X_t\}_{t\geq 0}\) spends at \(n\)-fold intersections of its path. Ideally, one would take a sequence of functions \(f_\varepsilon(x)\) of type \(\delta\), write \[ \alpha_{n,\varepsilon}(\mu, t):= \iint_{0\leq t_1\leq\cdots\leq t_n\leq t} f_\varepsilon(X_{t_1}- x) \prod^n_{j=2} f_\varepsilon(X_{t_j}- X_{t_{j-1}}) dt_1\cdots dt_n\mu(dx)\tag{\(*\)} \] and let \(\varepsilon\to 0\) to get the weighted (w.r.t. the measure \(\mu\)) total time (up to epoch \(t\)) the process spends at the points \(X_{t_1}=\cdots= X_{t_n}= x\). Since the limit \(\varepsilon\to 0\) does not exist, one needs to look at renormalized local times \(\gamma_n(\mu, t)= \lim_{\varepsilon\to 0} \gamma_{n,\varepsilon}(\mu, t)\) with \[ \gamma_{n,\varepsilon}(\mu, t)= \sum^{n-1}_{k=0} (-1)^k{n-1\choose k} (u^1_\varepsilon(0))^k \alpha_{n-k,\varepsilon}(\mu, t).\tag{\(**\)} \] Here \(u^1_\varepsilon= f_\varepsilon* u^1\) and \(u^1\) is the \(1\)-potential density of \(\{X_t\}_{t\geq 0}\). In order to ensure that this procedure works, the author considers symmetric stable Lévy processes \(\{X_t\}_{t\geq 0}\) (index \(>1\)) on \(\mathbb{R}^2\). The ultimate aim is to find an intrinsic natural characterization of the additive functionals \(\gamma_n(\mu, t)\). Denote by \(\{Y_t\}_{t\geq 0}\) the process \(\{X_t\}_{t\geq 0}\) killed at an independent exponential time \(\lambda\). Write \(L^\mu_t\) for the continuous additive functional with Revuz measure \(\mu\). Then \[ U^1\mu(X_t)= \mathbb{E}^x(L^\mu_t\mid{\mathcal F}_t)- L^\mu_{t\wedge\lambda} \] is the \(1\)-potential of \(\mu\). The main result of the paper states that \(\{Y_t\}_{t\geq 0}\) admits a continuous \(\gamma_n(\mu, t)\) with zero quadratic variation (!) if \(U^1\mu(x)\) is bounded and if for some \(\zeta>0\) \[ \sup_x \int_{|x|\leq 1}|x- y|^{(1- 2n)(2-\beta)-\zeta}\mu(dy)< \infty. \] In this case the random additive measure-valued process \(\pi^{\mu, n-1}_t(A):= \gamma_{n-1}(\mathbf{1}_A\cdot\mu, t)\) has a continuous version and satisfies the Doob-Meyer-type decomposition \[ U^1 \pi^{\mu,n- 1}_1(Y_t)= \mathbb{E}^x(\gamma_n(\mu, \lambda)\mid{\mathcal F}_t)- \gamma_n(\mu, t). \] In particular, \(U^1\pi^{dx,n-1}_t\) is a Dirichlet process. This paper generalizes and extends previous results for Brownian motion (where \(\beta= 2\)), see e.g. J.-F. Le Gall [in: Probabilités. Lect. Notes Math. 1527, 111-235 (1992; Zbl 0779.60068)]. The author conjectures that the present results are limited to \(\mathbb{R}^2\). This is based on the fact that already for Brownian motion in \(\mathbb{R}^3\) the renormalized intersection local time \(\gamma_2(\mu, t)\) does not exist, cf. the author [Ann. Probab. 16, No. 1, 75-79 (1988; Zbl 0644.60078)]. The proof of the theorem is quite technical and rests on several mainly analytic lemmata which are used to restate and bound the expressions \((*)\) and \((**)\).

MSC:

60J55 Local time and additive functionals
60G51 Processes with independent increments; Lévy processes
60G44 Martingales with continuous parameter
60G12 General second-order stochastic processes
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