A Regularized Dean--Kawasaki Model: Derivation and Analysis

The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply to approximations such as explicit finite difference schemes and Trotter-Kato type mixing formulas. The irregular time dependence disrupts the usual methods from the classical viscosity theory for creating schemes that are both monotone and convergent, an obstacle that cannot be overcome by incorporating higher order correction terms, as is done for numerical approximations of stochastic or rough ordinary differential equations. The novelty here is to regularize those driving paths with non-trivial quadratic variation in order to guarantee both monotonicity and convergence. We present qualitative and quantitative results, the former covering a wide variety of schemes for second-order equations. An error estimate is established in the Hamilton-Jacobi case, its merit being that it depends on the path only through the modulus of continuity, and not on the derivatives or total variation. As a result, it is possible to choose a regularization of the path so as to obtain efficient rates of convergence. This is demonstrated in the specific setting of equations with multiplicative white noise in time, in which case the convergence holds with probability one. We also present an example using scaled random walks that exhibits convergence in distribution.

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, we present a new idea to approximate high-dimensional nonlinear stochastic partial differential equations (SPDEs) by a kernel-based collocation method, which is a meshfree approximation method. A reproducing kernel is used to construct an approximate basis. The kernel-based collocation solution is a linear combination of the reproducing kernel with the differential and boundary operators of SPDEs at the given collocation points placed in the related high-dimensional domains. Its random expansion coefficients are computed by a random optimisation problem with constraint conditions induced by the nonlinear SPDEs. For a fixed kernel function, the convergence of kernel-based collocation solutions only depends on the fill distance of the chosen collocation points for the bounded domain of SPDEs. The numerical experiments of Sobolev-spline kernels for Klein-Gordon SPDEs show that the kernel-based collocation method produces the well-behaved approximate probability distributions of the SPDE solutions.

The objective of this article is to apply the splitting-up method to the existence theorem for nonlinear stochastic partial differential equations. With use of this method an approximating sequence is constructed. By the compactness argument a convergent subsequence can be extracted, and this fact provides the solution.

We consider nonlinear parabolic evolution equations of the form $\partial_{t}u=F(t,x,Du,D^{2}u) $, subject to noise of the form $H(x,Du) \circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a multidimensional Brownian motion. Motivated by the essentially pathwise results of [Lions, P.-L. and Souganidis, P.E.; Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris S\'{e}r. I Math. 326 (1998), no. 9] we propose the use of rough path analysis [Lyons, T. J.; Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), no. 2, 215--310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, ...).

Fully nonlinear stochastic partial differential equations

ABSTRACTStochastic centre manifolds theory are crucial in modelling the dynamical behaviour of complex systems under stochastic influences. The existence of stochastic centre manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space–time white noise.

Nonlinear stochastic integral equations in the plane

We consider conditional McKean–Vlasov stochastic differential equations (SDEs), as the ones arising in the large-system limit of mean field games and particle systems with mean field interactions when common noise is present. The conditional time-marginals of the solutions to these SDEs are governed by non-linear stochastic partial differential equations (SPDEs) of the second order, whereas their laws satisfy Fokker–Planck equations on the space of probability measures. Our paper establishes two superposition principles: The first asserts that any solution of the SPDE can be lifted to a solution of the conditional McKean–Vlasov SDE, and the second guarantees that any solution of the Fokker–Planck equation on the space of probability measures can be lifted to a solution of the SPDE. We use these results to obtain a mimicking theorem which shows that the conditional time-marginals of an Itô process can be emulated by those of a solution to a conditional McKean–Vlasov SDE with Markovian coefficients. This yields, in particular, a tool for converting open-loop controls into Markovian ones in the context of controlled McKean–Vlasov dynamics.

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher–KPP equations, stochastic Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand equations. By implementing the theory of C 0 − semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove the existence and uniqueness of solutions for this class of stochastic partial differential equations.

Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm

In this chapter, we prove the uniqueness for the solution to the filtering equation. To this end, we consider an interacting particle system whose weighted empirical distribution process satisfies the filtering equation. Note that the filtering equation is a non-linear stochastic partial differential equation (SPDE). The main idea in proving the uniqueness in this chapter is to show that the uniqueness of a non-linear SPDE is implied by that of the infinite system of ordinary stochastic differential equations and that of a corresponding linear SPDE, which follows from the same arguments as those in the previous chapter. The uniqueness for the system is obtained by a truncation argument.

Introduction to Superprocesses Superprocesses in Random Environments Linear SPDEs Particle Representations for a Class of Nonlinear SPDEs Stochastic Log-Laplace Equation SPDEs for Density Fields of the Superprocesses in Random Environments Backward Doubly Stochastic Differential Equations From SPDEs to BSDEs.

We study nonlinear parabolic stochastic partial differential equations with\nWick-power and Wick-polynomial type nonlinearities set in the framework of\nwhite noise analysis. These equations include the stochastic Fujita equation,\nthe stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation\namong many others. By implementing the theory of $C_0-$semigroups and evolution\nsystems into the chaos expansion theory in infinite dimensional spaces, we\nprove existence and uniqueness of solutions for this class of SPDEs. In\nparticular, we also treat the linear nonautonomous case and provide several\napplications featured as stochastic reaction-diffusion equations that arise in\nbiology, medicine and physics.\n