Condition for the Intersection Occupation Measure to be Absolutely Continuous

Given i.i.d. ℝd-valued stochastic processes X1(t), . . . ,Xp(t), p ≥ 2, with stationary increments, a minimal condition is provided for the occupation measure μtB=∫0t1BX1s1−X2s2…Xp−1sp−1−Xpspds1…dsp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mu}_t(B)=\underset{\left[0,t\right]}{\int }1B\left({X}_1\left({s}_1\right)-{X}_2\left({s}_2\right),\dots, {X}_{p-1}\left({s}_{p-1}\right)-{X}_p\left({s}_p\right)\right){ds}_1\dots {ds}_p, $$\end{document}B ⊂ ℝd(p−1), to be absolutely continuous with respect to the Lebesgue measure on ℝd(p−1). An isometry identity related to the resulting density (known as intersection local time) is also established.


Main Theorem
Let X(t) be a stochastic process taking values in R d with X(0) = 0 and let p t (x) (x 2 R d ) be the density function of X(t). Assume that, for any 0  s < t, (1.1) Let X 1 (t), . . . , X p (t) be independent copies of X(t). Given t 1 , . . . , t p ≥ 0 and x 2 R d(p−1) , the intersection local time ↵(t 1 , . . . , t p , x) of X 1 (t), . . . , X p (t) formally written as is defined as the Radon-Nikodym derivative of the occupation measure with respect to the Lebesgue measure on R d(p −1) . The most investigated setting is when X(t) is a Brownian motion. The criterion for the mutual intersection of independent Brownian motions was completed by Dvoretzky, Erdös, and Kakutani [3,4] in the 1950s. Their work was followed by the extensive investigations, either of the trajectory properties of the Brownian intersection local times (see, e.g., [1,7,8]) or of the extensions to some other stochastic processes (see, e.g., [2,5,6]).
A critical step in constructing the intersection local times is to establish the absolute continuity of µ t 1 ,...,tp (·) with respect to the Lebesgue measure on R d(p−1) . In the literature, this is mostly done either for Gaussian processes [2,6] or for Markov processes [5] with Gaussian/Markovian property used as the main tool. In the present paper, we do this without using the Gaussian/Markovian property.
The main result of the paper is the following theorem: and satisfies the isometry identity for any t 1 , . . . , t p ≥ 0.
Remark 1.1. In a special case where X(t) is symmetric, i.e., X(−t) d = X(t) (or p s (x) = p s (−x)), for every t ≥ 0, the isometry identity (1.7) becomes Proof of Theorem 1.1. For any measure µ on R d(p−1) , its Fourier transform is defined aŝ For any ✓ > 0, we define a random measure as follows: Note that To prove (1.5) and (1.6), all that we need is to establish the almost sure absolute continuity of µ ✓ (for some ✓ > 0) and the square integrability of the consequential density of the measure µ ✓ (·). According to the Plancherel-Parseval theorem (Theorem B.3 in [1, p. 302]), this is validated by the integrability (1.9) To establish (1.9) and isometry (1.7), we first prove that Here and elsewhere, we follow the convention that λ 0 = λ p = 0. Therefore, We take expectations on both sides. By the independence of X 1 , . . . , X p and by the increment stationarity given in (1.1), we get and R is a Bernoulli random variable independent of X(t) with the distribution Integrating both sides, we find where the last step follows from the substitution The following steps are required to justify the applicability of the Fubini theorem. Note that we have |Q t (λ)|  t 2 /2. In particular, for any " > 0, This justifies the application of the Fubini theorem in the following way: identity matrix) and the last step follows from the inverse Fourier transform. We now let " ! 0 + on both sides. First, by using (1.11), we can show that with proper substitution of the variables. By virtue of the monotonic convergence, the left-hand side increases to regardless of finiteness or infiniteness of the limit.
We complete this section by the following comment: The density ↵(t 1 , . . . , t p , x) addressed in Theorem 1.1 exists only in the form of equivalent class; this is a fact that brings some inconvenience, when it comes to application. Thus, it becomes ambiguous to talk about ↵(t 1 , . . . , t p , 0) for given t 1 , . . . , t p because ↵(t 1 , . . . , t p , 0) represents a class of random variables such that any two members of this class are equal to each other with probability 1. The problem is to find a continuous modification of ↵(t 1 , . . . , t p , x). A standard procedure according to the Kolmogorov extension theory requires the local Hölder type of moment continuity for some m > 0 and β > d + 1. This cannot be achieved without additional assumptions. In the case where X(t) is Gaussian, e.g., (1.15) can be installed under certain nonlocal determinism conditions (Theorem 2.8 in [6]).

Applications to Gaussian Processes
Let X(t) be an R d -valued stochastic process satisfying our pointwise increment-stationarity given by (1.1). In addition, assume that X(t) is pointwise Gaussian: Consequently, all statements in Theorem 1.1 hold under (2.2).
Hence, by condition (1.14), we get In the remaining part of this section, we consider two examples.