## The Annals of Probability

- Ann. Probab.
- Volume 36, Number 2 (2008), 594-622.

*L*^{p} moduli of continuity of Gaussian processes and local times of symmetric Lévy processes

Michael B. Marcus and Jay Rosen

#### Abstract

Let *X*={*X*(*t*), *t*∈*R*_{+}} be a real-valued symmetric Lévy process with continuous local times {*L*_{t}^{x}, (*t*, *x*)∈*R*_{+}×*R*} and characteristic function *Ee*^{iλX(t)}=*e*^{−tψ(λ)}. Let

If *σ*_{0}^{2}(*h*) is concave, and satisfies some additional very weak regularity conditions, then for any *p*≥1, and all *t*∈*R*_{+},

for all *a*, *b* in the extended real line almost surely, and also in *L*^{m}, *m*≥1. (Here *η* is a normal random variable with mean zero and variance one.)

This result is obtained via the Eisenbaum Isomorphism Theorem and depends on the related result for Gaussian processes with stationary increments, {*G*(*x*), *x*∈*R*^{1}}, for which *E*(*G*(*x*)−*G*(*y*))^{2}=*σ*_{0}^{2}(*x*−*y*);

for all *a*, *b*∈*R*^{1}, almost surely.

#### Article information

**Source**

Ann. Probab., Volume 36, Number 2 (2008), 594-622.

**Dates**

First available in Project Euclid: 29 February 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1204306961

**Digital Object Identifier**

doi:10.1214/009117907000000277

**Mathematical Reviews number (MathSciNet)**

MR2393991

**Zentralblatt MATH identifier**

1260.60156

**Subjects**

Primary: 60J55: Local time and additive functionals 60G15: Gaussian processes 60G17: Sample path properties

**Keywords**

Gaussian processes local times Levy processes

#### Citation

Marcus, Michael B.; Rosen, Jay. L p moduli of continuity of Gaussian processes and local times of symmetric Lévy processes. Ann. Probab. 36 (2008), no. 2, 594--622. doi:10.1214/009117907000000277. https://projecteuclid.org/euclid.aop/1204306961