Approximation of nonlinear stochastic partial differential equations by a kernel-based collocation method

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.

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 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, ...).

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

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.

Fully nonlinear stochastic partial differential equations

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.

In the present work, we establish the approximation via modulation or amplitude equations of the nonlinear stochastic partial differential equation (SPDE) driven by cylindrical $\alpha$-stable Lévy processes. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to simpler finite dimensional effective dynamics. In the presence of a multiplicative stable Lévy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where due to averaging effects nondominant patterns are uniformly small in time, large jumps in the Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account.

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

A theoretical solution framework to the nonlinear stochastic partial differential equations (SPDE) of the kinematic wave and diffusion wave models of overland flows under stochastic inflows/outflows, stochastic surface roughness field and stochastic state of flows was obtained. This development was realized by means of an eigenfunction representation of the time-space overland flow depths, and by transforming the problem into the phase space. By using Van Kampen's lemma and the cumulant expansion theory of Kubo-Van Kampen-Fox, the deterministic partial differential equation (PDE) for the evolutionary probability density function (pdf) of overland flow depths was finally obtained. Once this deterministic PDE is solved for the time-varying pdf of overland flow depths, then the time-space varying pdf of overland flow depths can be obtained by a transformation given in the text. In this solution framework it is possible to incorporate the stochastic dynamic behavior of the parameters and of the forcing functions of the overland flow process. For example, not only the individual rainfall duration and fluctuating rain intensity characteristics but also the sequential behavior of rainfall patterns is incorporated into the evolutionary probability density function of overland flow depths.

We prove uniqueness and existence of certain nonlinear stochastic partial differential equations (SPDEs) of divergence type defined on C 1-domains. Some L p and Holder estimates of the solution and its gradient are also obtained.

Talagrand’s transportation inequality for SPDEs with locally monotone drifts

Since Pardoux and Peng firstly studied the following nonlinear backward stochastic differential equations in 1990. The theory of BSDE has been widely studied and applied, especially in the stochastic control, stochastic differential games, financial mathematics and partial differential equations. In 1994, Pardoux and Peng came up with backward doubly stochastic differential equations to give the probabilistic interpretation for stochastic partial differential equations. Backward doubly stochastic differential equations theory has been widely studied because of its importance in stochastic partial differential equations and stochastic control problems. In this article, we will study the theory of doubly stochastic systems and related topics further.