In the present paper, the effect of noise intensity on stochastic parabolic equations is discussed. We focus on the effect of noise on the energy solutions of the stochastic parabolic equations. By utilising Itô's formula and the energy estimate method, we obtain the excitation indices of the energy solutions $u$ at any finite time $t$. Furthermore, we improve certain existing results in the literature by presenting a comparably simple method to show those existing results.
In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approac…
We prove the large deviation principle for the law of the solutions to a class of parabolic semilinear stochastic partial differential equations driven by multiplicative noise, in $C\big([0,T]:L^ρ(D)\big)$, where $D\subset {\mathbb R}^d$ with $d\geqslant 1$ is a bounded convex domain with smooth boundary and $ρ$ is any real, positive and large enough number. The equation has nonlinearities of polynomial growth of any order, the space variable is of any dimension, and the proof is based on the weak convergence method.
We study the asymptotic behavior of stochastic hyperbolic parabolic equations with slow and fast time scales. Both the strong and weak convergence in the averaging principe are established, which can be viewed as a functional law of large numbers. Then we study the stochastic fluctuations of the original system around its averaged equation. We show that the normalized difference converges weakly to the solution of a linear stochastic wave equation, which is a form of functional central limit theorem. We provide a unified proof for the above convergence by using the Poisson equation in Hilbert …
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations with gradient-dependent nonlinearities
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that these mild solutions exist uniquely in suitably chosen functional classes.
The stochastic parabolic equations with random potentials, driving forces and initial conditions are considered. The Wick product is used to give sense to the product of two generalized stochastic processes, and the existence and uniqueness of solutions are proved via the chaos expansion method from white noise analysis. The estimates on coefficients in the chaos expansion form of the solutions are provided.
In this work we firstly prove the well-posedness of the non-linear martingale problem related to a McKean-Vlasov stochastic differential equation with singular interaction kernel in $\mathbb{R}^d$ for $d\geq 3$. The particularity of our setting is that the McKean-Vlasov process we study interacts at each time with all its past time marginal laws by means of a singular space-time kernel. Secondly, we prove that our stochastic process is a probabilistic interpretation for the parabolic-parabolic Keller-Segel system in $\mathbb{R}^d$. We thus obtain a well-posedness result to the latter under exp…
We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $γ$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < γ- d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2γ/(γ-d/2-r)>2$ when $r>-d/2$. Mor…
In this paper, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations with ergodic structures. The limit function is represented as the viscosity solution to a fully nonlinear degenerate PDEs. Our approach is mainly based on G-stochastic analysis argument. As a byproduct, we also establish the averaging principle for stochastic differential equations driven by G-Brownian motion with two time-scales. The results extend Khasminskii's averaging principle to nonlinear case.
