8,154 research outputs found
On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling
The characteristic functional is the infinite-dimensional generalization of
the Fourier transform for measures on function spaces. It characterizes the
statistical law of the associated stochastic process in the same way as a
characteristic function specifies the probability distribution of its
corresponding random variable. Our goal in this work is to lay the foundations
of the innovation model, a (possibly) non-Gaussian probabilistic model for
sparse signals. This is achieved by using the characteristic functional to
specify sparse stochastic processes that are defined as linear transformations
of general continuous-domain white noises (also called innovation processes).
We prove the existence of a broad class of sparse processes by using the
Minlos-Bochner theorem. This requires a careful study of the regularity
properties, especially the boundedness in Lp-spaces, of the characteristic
functional of the innovations. We are especially interested in the functionals
that are only defined for p<1 since they appear to be associated with the
sparser kind of processes. Finally, we apply our main theorem of existence to
two specific subclasses of processes with specific invariance properties.Comment: 24 page
Unified View on L\'evy White Noises: General Integrability Conditions and Applications to Linear SPDE
There exists several ways of constructing L\'evy white noise, for instance
are as a generalized random process in the sense of I.M. Gelfand and N.Y.
Vilenkin, or as an independently scattered random measure introduced by B.S.
Rajput and J. Rosinski. In this article, we unify those two approaches by
extending the L\'evy white noise, defined as a generalized random process, to
an independently scattered random measure. We are then able to give general
integrability conditions for L\'evy white noises, thereby maximally extending
their domain of definition. Based on this connection, we provide new criteria
for the practical determination of this domain of definition, including
specific results for the subfamilies of Gaussian, symmetric--stable,
Laplace, and compound Poisson noises. We also apply our results to formulate a
general criterion for the existence of generalized solutions of linear
stochastic partial differential equations driven by a L\'evy white noise.Comment: 43 page
Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes
The fractional Laplacian commutes with the primary
coordination transformations in the Euclidean space \RR^d: dilation,
translation and rotation, and has tight link to splines, fractals and stable
Levy processes. For , its inverse is the classical Riesz potential
which is dilation-invariant and translation-invariant. In this work,
we investigate the functional properties (continuity, decay and invertibility)
of an extended class of differential operators that share those invariance
properties. In particular, we extend the definition of the classical Riesz
potential to any non-integer number larger than and
show that it is the unique left-inverse of the fractional Laplacian
which is dilation-invariant and
translation-invariant. We observe that, for any and
, there exists a Schwartz function such that is not -integrable. We then introduce the new unique left-inverse
of the fractional Laplacian with the
property that is dilation-invariant (but not
translation-invariant) and that is -integrable for any
Schwartz function . We finally apply that linear operator
with to solve the stochastic partial differential equation
with white Poisson noise as its driving term
.Comment: Advances in Computational Mathematics, accepte
Generating Sparse Stochastic Processes Using Matched Splines
We provide an algorithm to generate trajectories of sparse stochastic
processes that are solutions of linear ordinary differential equations driven
by L\'evy white noises. A recent paper showed that these processes are limits
in law of generalized compound-Poisson processes. Based on this result, we
derive an off-the-grid algorithm that generates arbitrarily close
approximations of the target process. Our method relies on a B-spline
representation of generalized compound-Poisson processes. We illustrate
numerically the validity of our approach
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