Condition for intersection occupation measure to be absolutely continuous - Chen, X. (Ukrains’kyi Matematychnyi Zhurnal, 2020)
UDC 519.21Given the i.i.d. $\mathbb{R}^d$-valued stochastic processes $X_1(t),…, X_p(t),$ $p\ge 2,$ with the stationary increments, a minimal condition is provided for the occupation measure$$μ_t(B)=\int\limits _{[0,t]^p}1_B\big(X_1(s_1) - X_2(s_2),…, X_{p-1}(s_{p-1}) -$$$$- X_p(s_p)\big)ds_1… ds_p,\quad B\subset \mathbb{R}^{d(p-1)},$$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.