Condition for the Intersection Occupation Measure to be Absolutely Continuous

Given i.i.d. ℝ^d-valued stochastic processes X₁(t), . . . ,X_p(t), p ≥ 2, with stationary increments, a minimal condition is provided for the occupation measure

${\mu}_t(B)=\underset{\left[0,t\right]}{\int }1B\left({X}_1\left({s}_1\right)-{X}_2\left({s}_2\right),\dots, {X}_{p-1}\left({s}_{p-1}\right)-{X}_p\left({s}_p\right)\right){ds}_1\dots {ds}_p,$ B ⊂ ℝ^d(p−1),

to be absolutely continuous with respect to the Lebesgue measure on ℝ^d(p−1). An isometry identity related to the resulting density (known as intersection local time) is also established.