38 research outputs found
Boundary regularity of stochastic PDEs
The boundary behaviour of solutions of stochastic PDEs with Dirichlet
boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as
shown by Krylov, for any one can find a simple -dimensional
constant coefficient linear equation whose solution at the boundary is not
-H\"older continuous.
We obtain a positive counterpart of this: under some mild regularity
assumptions on the coefficients, solutions of semilinear SPDEs on domains
are proved to be -H\"older continuous up to the boundary with some
.Comment: 29 page
On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift
The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate
SDEs with irregular drift coefficients is considered. In the case of
-H\"older drift in the recent literature the rate was proved
in many related situations. By exploiting the regularising effect of the noise
more efficiently, we show that the rate is in fact arbitrarily close to
for all . The result extends to Dini continuous coefficients, while
in also to all bounded measurable coefficients.Comment: In version 2, we have dropped the -condition that was imposed on
the drift in the -dimensional case and the result now is stated for
bounded measurable drif
Singular SPDEs in domains with boundaries
We study spaces of modelled distributions with singular behaviour near the
boundary of a domain that, in the context of the theory of regularity
structures, allow one to give robust solution theories for singular stochastic
PDEs with boundary conditions. The calculus of modelled distributions
established in Hairer (Invent. Math. 198, 2014) is extended to this setting. We
formulate and solve fixed point problems in these spaces with a class of
kernels that is sufficiently large to cover in particular the Dirichlet and
Neumann heat kernels. These results are then used to provide solution theories
for the KPZ equation with Dirichlet and Neumann boundary conditions and for the
2D generalised parabolic Anderson model with Dirichlet boundary conditions.
In the case of the KPZ equation with Neumann boundary conditions, we show
that, depending on the class of mollifiers one considers, a "boundary
renormalisation" takes place. In other words, there are situations in which a
certain boundary condition is applied to an approximation to the KPZ equation,
but the limiting process is the Hopf-Cole solution to the KPZ equation with a
different boundary condition.Comment: 53 pages, Minor reviso
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
In the recent article [Jentzen, A., M\"uller-Gronbach, T., and Yaroslavtseva,
L., Commun. Math. Sci., 14(6), 1477--1500, 2016] it has been established that
for every arbitrarily slow convergence speed and every natural number there exist -dimensional stochastic differential equations
(SDEs) with infinitely often differentiable and globally bounded coefficients
such that no approximation method based on finitely many observations of the
driving Brownian motion can converge in absolute mean to the solution faster
than the given speed of convergence. In this paper we strengthen the above
result by proving that this slow convergence phenomena also arises in two
() and three () space dimensions.Comment: 25 page
Correlation bound for distant parts of factor of IID processes
We study factor of i.i.d. processes on the -regular tree for .
We show that if such a process is restricted to two distant connected subgraphs
of the tree, then the two parts are basically uncorrelated. More precisely, any
functions of the two parts have correlation at most ,
where denotes the distance of the subgraphs. This result can be considered
as a quantitative version of the fact that factor of i.i.d. processes have
trivial 1-ended tails.Comment: 18 pages, 5 figure
Localization errors in solving stochastic partial differential equations in the whole space
Cauchy problems with SPDEs on the whole space are localized to Cauchy
problems on a ball of radius . This localization reduces various kinds of
spatial approximation schemes to finite dimensional problems. The error is
shown to be exponentially small. As an application, a numerical scheme is
presented which combines the localization and the space and time
discretisation, and thus is fully implementable.Comment: Some details added; published versio
Strong convergence of parabolic rate of discretisations of stochastic Allen-Cahn-type equations
Consider the approximation of stochastic Allen-Cahn-type equations (i.e.
-dimensional space-time white noise-driven stochastic PDEs with polynomial
nonlinearities such that ) by a fully discrete
space-time explicit finite difference scheme. The consensus in literature,
supported by rigorous lower bounds, is that strong convergence rate with
respect to the parabolic grid meshsize is expected to be optimal. We show that
one can reach almost sure convergence rate (and no better) when measuring
the error in appropriate negative Besov norms, by temporarily `pretending' that
the SPDE is singular.Comment: 28 page
Stochastic PDEs with extremal properties
We consider linear and semilinear stochastic partial differential equations that in
some sense can be viewed as being at the "endpoints" of the classical variational
theory by Krylov and Rozovskii [25]. In terms of regularity of the coeffcients,
the minimal assumption is boundedness and measurability, and a unique L2-
valued solution is then readily available. We investigate its further properties,
such as higher order integrability, boundedness, and continuity. The other class
of equations considered here are the ones whose leading operators do not satisfy
the strong coercivity condition, but only a degenerate version of it, and therefore
are not covered by the classical theory. We derive solvability in Wmp spaces and also discuss their numerical approximation through finite different schemes