# Stochastics and Partial Differential Equations: Analysis and Computations

## List of Papers (Total 59)

#### A reflected moving boundary problem driven by space–time white noise

We study a system of two reflected SPDEs which share a moving boundary. The equations describe competition at an interface and are motivated by the modelling of the limit order book in financial markets. The derivative of the moving boundary is given by a function of the two SPDEs in their relative frames. We prove existence and uniqueness for the equations until blow-up, and...

#### Electro-rheological fluids under random influences: martingale and strong solutions

We study generalised Navier–Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent $$p=p(\omega ,t,x)$$ (as a result of...

#### Some results on the penalised nematic liquid crystals driven by multiplicative noise: weak solution and maximum principle

In this paper, we prove several mathematical results related to a system of highly nonlinear stochastic partial differential equations (PDEs). These stochastic equations describe the dynamics of penalised nematic liquid crystals under the influence of stochastic external forces. Firstly, we prove the existence of a global weak solution (in the sense of both stochastic analysis...

#### Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

We study the random conductance model on the lattice $${\mathbb {Z}}^d$$, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity...

#### Iterated stochastic integrals in infinite dimensions: approximation and error estimates

Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden et al. (Stoch Anal Appl 10(4):431–441, 1992.  https://doi.org/10.1080/07362999208809281) and by Wiktorsson (Ann Appl Probab 11(2):470–487, 2001.  https...

#### Recursive formulae in regularity structures

We construct renormalised models of regularity structures by using a recursive formulation for the structure group and for the renormalisation group. This construction covers all the examples of singular SPDEs which have been treated so far with the theory of regularity structures and improves the renormalisation procedure based on Hopf algebras given in Bruned–Hairer–Zambotti...

#### Strong solutions to SPDEs with monotone drift in divergence form

We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not...

#### Weak universality of dynamical $$\Phi ^4_3$$ : 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...

#### Exponential integrators for nonlinear Schrödinger equations with white noise dispersion

This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our...

#### Itô formula for processes taking values in intersection of finitely many Banach spaces

Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale y(t) from Gyöngy and Krylov (Stochastics 6(3):153–173, 1982) to the case \begin{aligned} \sum _{i=1}^m \int _{(0,t]} v_i^{*}(s)\,dA(s) + h(t)=:y(t)\in V \quad dA\times {\mathbb {P}}\text {-a.e.}, \end{aligned}where A is an increasing right-continuous adapted process...

#### A solution theory for a general class of SPDEs

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations...

#### Eikonal equations and pathwise solutions to fully non-linear SPDEs

We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.

#### Sublinear growth of the corrector in stochastic homogenization: optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and “essentially Gaussian” coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the...

#### Symmetrization of exterior parabolic problems and probabilistic interpretation

We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (Trans Am Math Soc 352(4):1759–1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric...

#### Optimal rate of convergence for stochastic Burgers-type equations

Recently, a solution theory for one-dimensional stochastic PDEs of Burgers type driven by space-time white noise was developed. In particular, it was shown that natural numerical approximations of these equations converge and that their convergence rate in the uniform topology is arbitrarily close to $$\frac{1}{6}$$. In the present article we improve this result in the case of...

#### Convergence of tamed Euler schemes for a class of stochastic evolution equations

We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done by using the recently developed tamed Euler method, which employs a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization.

#### Splitting up method for the 2D stochastic Navier–Stokes equations

In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier–Stokes equations on the torus suggested by the Lie–Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given for...

#### Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case

We continue the development of the theory of pathwise stochastic entropy solutions for scalar conservation laws in $${\mathbb {R}}^N$$ with quasilinear multiplicative “rough path” dependence by considering inhomogeneous fluxes and a single rough path like, for example, a Brownian motion. Following our previous note where we considered spatially independent fluxes, we introduce...

#### Consistency and stability of a Milstein–Galerkin finite element scheme for semilinear SPDE

We present an abstract concept for the error analysis of numerical schemes for semilinear stochastic partial differential equations (SPDEs) and demonstrate its usefulness by proving the strong convergence of a Milstein–Galerkin finite element scheme. By a suitable generalization of the notion of bistability from Beyn and Kruse (Discrete Contin Dyn Syst Ser B 14(2), 389–407 2010...

#### On stochastic finite difference schemes

Finite difference schemes in the spatial variable for degenerate stochastic parabolic PDEs are investigated. Sharp estimates on the rate of $$L_p$$ and almost sure convergence of the finite difference approximations are presented and results on Richardson extrapolation are established for stochastic parabolic schemes under smoothness assumptions.

#### A note on error estimation for hypothesis testing problems for some linear SPDEs

The aim of the present paper is to estimate and control the Type I and Type II errors of a simple hypothesis testing problem of the drift/viscosity coefficient for stochastic fractional heat equation driven by additive noise. Assuming that one path of the first $$N$$ Fourier modes of the solution is observed continuously over a finite time interval $$[0,T]$$, we propose a new...