Condition for intersection occupation measure to be absolutely continuous

  • X. Chen Univ. Tennessee, USA
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.

References

X. Chen, Random Walk intersections: large deviations and related topics. Math. Surveys and Monogr., 157, Amer.Math. Soc., Providence (2009), https://doi.org/10.1090/surv/157 DOI: https://doi.org/10.1090/surv/157

X. Chen, W. V. Li, J. Rosinski, Q.-M. Shao, ´ Large deviations for local times and intersection local times of fractional Brownian motions and Riemann – Liouville processes, Ann. Probab., 39, 727 – 778 (2011), https://doi.org/10.1214/10-AOP566 DOI: https://doi.org/10.1214/10-AOP566

A. Dvoretzky, P. Erdös, S. Kakutani, ¨ Double points of paths of Brownian motions in $n$-space, Acta Sci. Math. (Szeged), 12, 75 – 81 (1950).

A. Dvoretzky, P. Erdös, S. Kakutani, ¨ Multiple points of paths of Brownian motions in the plane, Bull. Res. Counceil Israel, 3, 364 – 371 (1954).

E. B. Dynkin, Additive functionals of several time-reversable Markov processes, J. Funct. Anal., 42, No. 1, 64 – 101 (1981).

D. Geman, J. Horowitz, J. Rosen, A local time analysis of intersection of Brownian paths in the plane, Ann. Probab., 12, No. 1, 86 – 107 (1984).

J.-F. Le Gall, Propriétés d'intersection des marches aléatoires. I. Convergence vers le temps local d'intersection. (French) , Commun. Math. Phys., 104, 471 – 507 (1986), http://projecteuclid.org/euclid.cmp/1104115088

J.-F. Le Gall, Propriétés d'intersection des marches aléatoires. II. Étude des cas critiques. (French) , Commun. Math. Phys., 104, 509 – 528 (1986), http://projecteuclid.org/euclid.cmp/1104115089

Published
22.09.2020
How to Cite
Chen, X. “Condition for Intersection Occupation Measure to Be Absolutely Continuous ”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 9, Sept. 2020, pp. 1304-12, doi:10.37863/umzh.v72i9.6278.
Section
Research articles