89 research outputs found
Non-linear Rough Heat Equations
This article is devoted to define and solve an evolution equation of the form
, where stands for the Laplace operator
on a space of the form , and is a finite dimensional
noisy nonlinearity whose typical form is given by , where each is a
-H\"older function generating a rough path and each is a smooth
enough function defined on . The generalization of the usual
rough path theory allowing to cope with such kind of systems is carefully
constructed.Comment: 36 page
Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions
In this paper we obtain Gaussian-type lower bounds for the density of
solutions to stochastic differential equations (SDEs) driven by a fractional
Brownian motion with Hurst parameter . In the one-dimensional case with
additive noise, our study encompasses all parameters , while the
multidimensional case is restricted to the case . We rely on a mix of
pathwise methods for stochastic differential equations and stochastic analysis
tools.Comment: Published at http://dx.doi.org/10.1214/14-AOP977 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Numerical Schemes for Rough Parabolic Equations
This paper is devoted to the study of numerical approximation schemes for a
class of parabolic equations on (0, 1) perturbed by a non-linear rough signal.
It is the continuation of [8, 7], where the existence and uniqueness of a
solution has been established. The approach combines rough paths methods with
standard considerations on discretizing stochastic PDEs. The results apply to a
geometric 2-rough path, which covers the case of the multidimensional
fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201
Stochastic evolution equations driven by Liouville fractional Brownian motion
Let H be a Hilbert space and E a Banach space. We set up a theory of
stochastic integration of L(H,E)-valued functions with respect to H-cylindrical
Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in
the interval (0,1). For Hurst parameters in (0,1/2) we show that a function
F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical
Liouville fBm if and only if it is stochastically integrable with respect to an
H-cylindrical fBm with the same Hurst parameter. As an application we show that
second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by
space-time noise which is white in space and Liouville fractional in time with
Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous
both and space.Comment: To appear in Czech. Math.
Regularity of the Solutions to SPDEs in Metric Measure Spaces
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernel estimates. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4
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