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Zbl 0981.60072
Rosen, Jay
Dirichlet processes and an intrinsic characterization of renormalized intersection local times.
(English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 37, No.4, 403-420 (2001). ISSN 0246-0203

The intersection local time of order $n$, $n= 2,3,4,\dots$, is the amount of time'' a stochastic process $\{X_t\}_{t\ge 0}$ spends at $n$-fold intersections of its path. Ideally, one would take a sequence of functions $f_\varepsilon(x)$ of type $\delta$, write $$\alpha_{n,\varepsilon}(\mu, t):= \iint_{0\le t_1\le\cdots\le t_n\le t} f_\varepsilon(X_{t_1}- x) \prod^n_{j=2} f_\varepsilon(X_{t_j}- X_{t_{j-1}}) dt_1\cdots dt_n\mu(dx)\tag{*}$$ 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 $\{X_t\}_{t\ge 0}$. In order to ensure that this procedure works, the author considers symmetric stable Lévy processes $\{X_t\}_{t\ge 0}$ (index $>1$) on $\bbfR^2$. The ultimate aim is to find an intrinsic natural characterization of the additive functionals $\gamma_n(\mu, t)$. Denote by $\{Y_t\}_{t\ge 0}$ the process $\{X_t\}_{t\ge 0}$ killed at an independent exponential time $\lambda$. Write $L^\mu_t$ for the continuous additive functional with Revuz measure $\mu$. Then $$U^1\mu(X_t)= \bbfE^x(L^\mu_t\mid{\cal F}_t)- L^\mu_{t\wedge\lambda}$$ is the $1$-potential of $\mu$. The main result of the paper states that $\{Y_t\}_{t\ge 0}$ admits a continuous $\gamma_n(\mu, t)$ with zero quadratic variation (!) if $U^1\mu(x)$ is bounded and if for some $\zeta>0$ $$\sup_x \int_{|x|\le 1}|x- y|^{(1- 2n)(2-\beta)-\zeta}\mu(dy)< \infty.$$ In this case the random additive measure-valued process $\pi^{\mu, n-1}_t(A):= \gamma_{n-1}(\text{\bf 1}_A\cdot\mu, t)$ has a continuous version and satisfies the Doob-Meyer-type decomposition $$U^1 \pi^{\mu,n- 1}_1(Y_t)= \bbfE^x(\gamma_n(\mu, \lambda)\mid{\cal F}_t)- \gamma_n(\mu, t).$$ 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. {\it 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 $\bbfR^2$. This is based on the fact that already for Brownian motion in $\bbfR^3$ 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 $(**)$.
[René L.Schilling (Brighton)]
MSC 2000:
60G51 Processes with independent increments
60G44 Martingales with continuous parameter
60G12 General second order processes

Keywords: intersection local time; continuous additive functional; Lévy process; stable process; Dirichlet process; Doob-Meyer decomposition; quadratic variation

Citations: Zbl 0779.60068; Zbl 0644.60078

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