The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz...

Stochastic FitzHugh-Nagumo Equations

The stochastic FitzHugh-Nagumo equations are a system of stochastic partial differential equations that describes the propagation of action potentials along nerve axons. In the present work we obtain well-posedness and regularisation results for the FitzHugh-Nagumo equations with domain R^d. We begin by considering the weak critical variational...

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth...

Multiple Markov properties for fractional parabolic SPDEs

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets. <br/>Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time...

A Stochastic Parametrically-Forced NLS Equation

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse...

Stochastic Evolution Equations with Adapted Drift

In this thesis we study stochastic evolution equations in Banach spaces. We restrict ourselves to the two following cases. First, we consider equations in which the drift is a closed linear operator that depends on time and is random. Such equations occur as mathematical models in for instance mathematical finance and filtration theory. Second,...

Stochastic Differential Equations in Banach Spaces: Decoupling, Delay Equations, and Approximations in Space and Time

The thesis deals with various aspects of the study of stochastic partial differential equations driven by Gaussian noise. The approach taken is functional analytic rather than probabilistic: the stochastic partial differential equation is interpreted as an ordinary stochastic differential equation in a Banach space. The major part of the thesis...

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.

A nonlinear Schrödinger equation in L² with multiplicative white noise

In this thesis existence and uniqueness of the solution of a nonlinear Schrödinger equation in L² with multiplicative white noise is proven under some assumptions. Furthermore, pathwise L²-norm conservation of the solution is studied.

A new Eulerian Monte Carlo method for the joint velocity-scalar PDF equations in turbulent flows

In the field of turbulent combustion, Lagrangian Monte Carlo (LMC) methods (Pope, 85) have become an essential component of the probability density function (PDF) approach. LMC methods are based on stochastic particles, which evolve from prescribed stochastic ordinary differential equations (SODEs). They are used to compute the one-point...