TY - JOUR
AB - The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of α-Hölder drift in the recent literature the rate α/2 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 1/2 for all α>0. The result extends to Dini continuous coefficients, while in d=1 also to all bounded measurable coefficients.
AU - Dareiotis, Konstantinos
AU - Gerencser, Mate
ID - 6359
JF - Electronic Journal of Probability
TI - On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift
VL - 25
ER -
TY - JOUR
AB - We give a Wong-Zakai type characterisation of the solutions of quasilinear heat equations driven by space-time white noise in 1 + 1 dimensions. In order to show that the renormalisation counterterms are local in the solution, a careful arrangement of a few hundred terms is required. The main tool in this computation is a general ‘integration by parts’ formula that provides a number of linear identities for the renormalisation constants.
AU - Gerencser, Mate
ID - 7388
IS - 3
JF - Annales de l'Institut Henri Poincaré C, Analyse non linéaire
SN - 0294-1449
TI - Nondivergence form quasilinear heat equations driven by space-time white noise
VL - 37
ER -
TY - JOUR
AB - We give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent results of [1, 5, 11]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [3, 4, 7] applies, including in particular one‐dimensional systems with KPZ‐type nonlinearities driven by space‐time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution. The main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small.
AU - Gerencser, Mate
AU - Hairer, Martin
ID - 6028
IS - 9
JF - Communications on Pure and Applied Mathematics
TI - A solution theory for quasilinear singular SPDEs
VL - 72
ER -
TY - JOUR
AB - 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[ SIAM J. Math. Anal.34(2003) 1167–1182], for any α>0 one can find a simple 1-dimensional constant coefficient linear equation whose solution at the boundary is not α-Hölder continuous.We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on C1 domains are proved to be α-Hölder continuous up to the boundary with some α>0.
AU - Gerencser, Mate
ID - 6232
IS - 2
JF - Annals of Probability
SN - 00911798
TI - Boundary regularity of stochastic PDEs
VL - 47
ER -
TY - JOUR
AB - A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.
AU - Gerencser, Mate
AU - Gyöngy, István
ID - 301
IS - 3
JF - Stochastic Processes and their Applications
TI - A Feynman–Kac formula for stochastic Dirichlet problems
VL - 129
ER -
TY - JOUR
AB - We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and L1-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators Δ(|u|m−1u) for all m∈(1,∞), and Hölder continuous diffusion nonlinearity with exponent 1/2.
AU - Dareiotis, Konstantinos
AU - Gerencser, Mate
AU - Gess, Benjamin
ID - 65
IS - 6
JF - Journal of Differential Equations
TI - Entropy solutions for stochastic porous media equations
VL - 266
ER -
TY - JOUR
AB - 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(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) 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.
AU - Gerencser, Mate
AU - Hairer, Martin
ID - 319
IS - 3-4
JF - Probability Theory and Related Fields
SN - 01788051
TI - Singular SPDEs in domains with boundaries
VL - 173
ER -
TY - JOUR
AB - In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14, 1477–1500 (doi:10.4310/CMS.2016.v14. n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ? {4, 5, . . .}, there exist d-dimensional stochastic differential equations 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 phenomenon also arises in two (d = 2) and three (d = 3) space dimensions.
AU - Gerencser, Mate
AU - Jentzen, Arnulf
AU - Salimova, Diyora
ID - 560
IS - 2207
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
SN - 13645021
TI - On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
VL - 473
ER -
TY - JOUR
AB - Cauchy problems with SPDEs on the whole space are localized to Cauchy problems on a ball of radius R. 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 discretization, and thus is fully implementable.
AU - Gerencser, Mate
AU - Gyöngy, István
ID - 642
IS - 307
JF - Mathematics of Computation
SN - 00255718
TI - Localization errors in solving stochastic partial differential equations in the whole space
VL - 86
ER -