1,720,977 research outputs found
Collisions of Lévy processes
No full textLet
X
,
Y
X,Y
be two independent Lévy processes in
R
R
. We describe simple conditions on the density functions of
X
,
Y
X,Y
which guarantee that the paths
X
(
⋅
)
,
Y
(
⋅
)
X( \cdot ),Y( \cdot )
will have uncountably many collisions almost surely.</p
Conditional Local Nondeterminism and Hausdorff Measure of Level Sets
No full textAbstractLet X be a real stochastic process. We localize S. M. Berman's formulation on the local nondeterminism of X to a fixed level. With this localized idea, we prove that, for large classes of Gaussian and Markov X, at each x the level set X(t, w) = x has infinite Hausdorff ϕ - measure (ϕ is certain measure function) for w in a set of positive probability.</jats:p
Images of Gaussian random fields: Salem sets and interior points
Full text linkLet X = {X(t), t ∈ RN} be a Gaussian random field in Rd with stationary increments. For any Borel set E ⊂ RN, we provide sufficient conditions for the image X(E) to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of X and the continuity of the local times of X on E, respectively. Our results extend and improve the previous theorems of Pitt [24] and Kahane [12, 13] for fractional Brownian motion
A LIL for occupation times of stable processes
No full textWe prove a Strassen-type law of iterated logarithms for the occupation times of an
R
d
{R^d}
-valued
(
d
≥
1
)
(d \geq 1)
stable process with the scaling property and positive density functions. An immediate application of our result is to obtain the asymptotic behavior of the occupation times of a path occupied in large spheres.</p
Multiple points of a random field
No full textWe prove that a
d
d
-dimensional random field
X
≡
{
X
(
t
)
}
t
∈
R
+
N
X \equiv {\{ X(t)\} _{t \in R_ + ^N}}
has uncountably many
r
r
-multiple points a.s. when it satisfies Pitt’s (
(
A
r
)
({A_r})
) condition [9]. Those
t
t
’s for which
X
(
t
)
X(t)
hits the multiple point can be separated by any given positive number, and multiple points can occur while
t
t
is restricted to any given "time inteval". Two corollaries concerning Gaussian fields and fields with independent increments are also presented.</p
The exact packing measure of Brownian double points
No full textLet D ⊂ R³ be the set of double points of a 3-dimensional Brownian motion. We show that, if ξ = ξ3(2, 2) is the intersection exponent of two packets of two independent Brownian motions, then almost surely, the φ-packing measure of D is zero if 0 + r −1−ξ φ(r) ξ dr < ∞, and infinity otherwise. As an important step in the proof we show up-to-constants estimates for the tail at zero of Brownian intersection local times in dimensions two and three
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