Condition for intersection occupation measure to be absolutely continuous

X.  Chen

doi:10.37863/umzh.v72i9.6278

X. Chen Univ. Tennessee, USA

DOI: https://doi.org/10.37863/umzh.v72i9.6278

Keywords: Intersection local time, occupation measure, Plancherel-Parseval theorem

Abstract

UDC 519.21

Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),\ldots, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure
$$\mu_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),\ldots, X_{p-1}(s_{p-1}) -$$

$$- X_p(s_p)\big)ds_1\ldots ds_p,\quad B\subset \mathbb{R}^{d(p-1)},$$
to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{d(p-1)}.$
An isometry identity related to the resulting density (known as intersection local time) is also established.

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