Remarks on non-linear noise excitability of some stochastic heat equations

We consider nonlinear parabolic SPDEs of the form ${\partial }_{t}u=\Delta u+\lambda \sigma \left(u\right)\dot{w}$ on the interval $(0,L)$, where $\dot{w}$ denotes space–time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $exp[\text{const}\times {\lambda }^4]$ as $\lambda \to \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result in Khoshnevisan and Kim.     

Regularities for semilinear stochastic partial differential equations

In this paper, we study the regularities of solutions to semilinear stochastic partial differential equations in general settings, and prove that the solution can be smooth arbitrarily when the data is sufficiently regular. As applications, we also study several classes of semilinear stochastic partial differential equations on abstract Wiener space, complete Riemannian manifold as well as bounded domain in Euclidean space.     

Differentiability of implicit functions: Beyond the implicit function theorem

The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S:u?y$S:u?y$ given by the equation e(y,u)=0$e(y,u)=0$. However, the IFT is not applicable when different norms are necessary for the differentiation of e w.r.t. y   and the invertibility of the partial derivative _ey(y,u)$e_y(y,u)$. We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state ∇y.     

Filtering partially observable diffusions up to the exit time from a domain

We consider a two-component diffusion process with the second component treated as the observations of the first one. The observations are available only until the first exit time of the first component from a fixed domain. We derive filtering equations for an unnormalized conditional distribution of the first component before it hits the boundary and give a formula for the conditional distribution of the first component at the first time it hits the boundary.     

Branches of positive and free boundary solutions for some singular quasilinear elliptic problems

We study the existence and multiplicity of solutions, strictly positive or nonnegative having a free boundary (the boundary of the set where the solution vanishes) of some one-dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms.     

Stochastic Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of L^p convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.     

概率方法解拟线性偏微分方程

本文用随机分析方法证明了拟线性抛物型方程ｕｔ＋ｆ（ｕ）ｕｘ、ｕｘｘ＝０,ｕ（０,ｘ）＝ｕ０（ｘ）在ｕ０有界可测,ｆ连续且ｆ＞０条件下,其解当→０时收敛于拟线性方程ｕｔ＋ｆ（ｕ）ｕｘ＝０,ｕ（０,ｘ）＝ｕ０（ｘ）的熵解,即论证了“沾性消失法”解此方程的正确性,１９５７年Ｏｌｅｉｎｉｋ曾用差分方法解决了此问题。这里用概率方法重新获得此结果。     

Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane

We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.

     

Some Results on Nonlinear Backward Stochastic Evolution Equations

Abstract

In this paper, we present some results concerning existence and uniqueness of solutions for a rather general class of nonlinear backward stochastic partial differential equations. These results are illustrated with two examples.     

随机微分方程的强解

在这篇文章中我们通过一种去掉扩散系数的变换证明了随机微分方程强解的存在唯一性.     

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.      

Conservation laws with a random source

We study the scalar conservation law with a noisy nonlinear source, namely,u _l + f(u)_x = h(u, x, t) + g(u)W(t), whereW(t) is the white noise in the time variable, and we analyse the Cauchy problem for this equation where the initial data are assumed to be deterministic. A method is proposed to construct approximate weak solutions, and we then show that this yields a convergent sequence. This sequence converges to a (pathwise) solution of the Cauchy problem. The equation can be considered as a model of deterministic driven phase transitions with a random perturbation in a system of two constituents. Finally we show some numerical results motivated by two-phase flow in porous media. This research has been supported by VISTA (a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap, Statoil) and NAVF (the Norwegian Research Council for Science and the Humanities).     

一类拟线性退化抛物方程Dirichlet问题粘性解的正则性

本文考虑下面的Ｄｉｒｉｃｈｌｅｔ问题利用粘性解理论证明了；当Ｈ，Г满足一定条件时，（ＤＰ）的粘性解ｕ（ｘ，ｔ）满足：如果ψ∈Ｃａ，ａ／２，则ｕ（ｘ，ｔ）∈Ｃａ，ａ／２，若ψ＝０，则ｕ（ｘ，ｔ）是Ｌｉｐｓｃｈｉｔｚ连续的．     

Large deviations for stochastic nonlinear beam equations

We establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels (also called stochastic nonlinear beam equations).     

An <Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>-Theory of SPDEs on Lipschitz Domains

Stochastic partial differential equations are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular, it can be negative and fractional. It is allowed that the coefficients of the equations blow up near the boundary.