3,874 research outputs found
Sharp convergence of nonlinear functionals of a class of Gaussian random fields
We present a self-contained proof of a uniform bound on multi-point
correlations of trigonometric functions of a class of Gaussian random fields.
It corresponds to a special case of the general situation considered in
[Hairer-Xu], but with improved estimates. As a consequence, we establish
convergence of a class of Gaussian fields composite with more general
functions. These bounds and convergences are useful ingredients to establish
weak universalities of several singular stochastic PDEs.Comment: 22 page
A simple proof of distance bounds for Gaussian rough paths
We derive explicit distance bounds for Stratonovich iterated integrals along
two Gaussian processes (also known as signatures of Gaussian rough paths) based
on the regularity assumption of their covariance functions. Similar estimates
have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of
our argument is that we obtain the bound for the third level iterated integrals
merely based on the first two levels, and this reflects the intrinsic nature of
rough paths. Our estimates are sharp when both covariance functions have finite
1-variation, which includes a large class of Gaussian processes.
Two applications of our estimates are discussed. The first one gives the a.s.
convergence rates for approximated solutions to rough differential equations
driven by Gaussian processes. In the second example, we show how to recover the
optimal time regularity for solutions of some rough SPDEs.Comment: 20 pages, updated abstract and introductio
Inverting the signature of a path
The aim of this article is to develop an explicit procedure that enables one
to reconstruct any path (at natural parametrization) from its signature.
We also explicitly quantify the distance between the reconstructed path and the
original path in terms of the number of terms in the signature that are used
for the construction and the modulus of continuity of the derivative of the
path. A key ingredient in the construction is the use of a procedure of
symmetrization that separates the behavior of the path at small and large
scales.Comment: 31 pages; minor change
Hyperbolic development and inversion of signature
We develop a simple procedure that allows one to explicitly reconstruct any
piecewise linear path from its signature. The construction is based on the
development of the path onto the hyperbolic space.Comment: Revised; 19 pages. We splitted our older article (arXiv:1406.7833)
into two independent ones; this is one of the
Weak universality of dynamical : non-Gaussian noise
We consider a class of continuous phase coexistence models in three spatial
dimensions. The fluctuations are driven by symmetric stationary random fields
with sufficient integrability and mixing conditions, but not necessarily
Gaussian. We show that, in the weakly nonlinear regime, if the external
potential is a symmetric polynomial and a certain average of it exhibits
pitchfork bifurcation, then these models all rescale to near their
critical point.Comment: 37 pages; updated introduction and reference
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