Parabolic SPDEs Research Articles

On the density for the solution of a Burgers-type SPDE

We study the existence and regularity of densities for the solution of a class of parabolic SPDEs of Burgers type introduced by Gyöngy in a recent paper. In the case of regular, bounded coefficients, we show the existence of a smooth density and, as a consequence, we obtain the existence of the density for the classical stochastic Burgers equation.

Asymptotic Behaviour of the Density in a Parabolic SPDE

Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$. In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.

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 Lp 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.

Finite element and difference approximation of some linear stochastic partial differential equations

Difference and finite element methods are described, analyzed, and tested for numerical solution of linear parabolic and elliptic SPDEs driven by white noise. Weak and integral formulations of the stochastic partial differential equations are approximated, respectively, by finite element and difference methods. The white noise processes are approximated by piecewise constant random processes to facilitate convergence proofs for the finite element method. Error analyses of the two numerical methods yield estimates of convergence rates. Computational experiments indicate that the two numerical methods have similar accuracy but the finite element method is computationally more efficient than the difference method

On Quasi-linear parabolic SPDEs with non-Lipschitz coefficients

On Quasi-linear parabolic SPDEs with non-Lipschitz coefficients

Uniform large deviations for parabolic SPDEs and applications

Let C α([0, T] × [0, 1]) denote the set of functions f( t, x) which are α-Hölder continuous in t and 2α-Hölder continuous in x. For 0 < α < 1 4 we prove a large deviation principle in a separable subset of C α([0, T] × [0, 1]) for the solution X ε ϕ ( t, x) to a parabolic stochastic partial differential equation perturbed by a small non-linear white noise, uniformly when the initial condition ϕ belongs to a compact subset of C 2α,0([0, 1]) . This does not require any boundedness or non-degeneracy on the coefficients, and is applied to deduce asymptotics for the exit time of X ε ϕ ( t,1) from a bounded domain of C 2α,0([0, 1]) .

Approximation of the density of a solution of a nonlinear SDE — application to parabolic SPDEs

This paper studies the approximation of the density P t, x ( y) of the solution of the nonlinear limit-problem of a system of weakly interacting SDE's via a convolution of the empirical measure of the system with a family of smooth mollifiers. The method, which mainly uses coupling techniques and Malliavin calculus, is also applied to the case of nonlinear white-noise driven parabolic SPDEs.

Propagation of chaos and fluction for a system of weakly interacting white noise driven parabolic spde's

We investigate the asymptotic behaviour of a system of N weakly interacting parabolic SPDE's when N become large.In particular we study a law of number (propagation of chaos) and the central limit phenomenon (fluctuations). To do so we use in the first part a martingale problem approach and in the second subtle estimates of the solution and of the fluctuations field

Stochastic flows for nonlinear second-order parabolic SPDE

The existence of stochastic flows in $L^2$ -spaces is proved for a stochastic reaction ] diffusion equation of second order in a bounded domain.

On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE's

We investigate asymptotic properties of the maximum likelihood estimators for parameters occurring in parabolic SPDEs of the form $$du(t,x) = (A_0 + \theta A_1 )u(t,x)dt + dW(t,x),$$ whereA 0 andA 1 are partial differential operators andW is a cylindrical Brownian motion. We introduce a spectral method for computing MLEs based on finite dimensional approximations to solutions of such systems, and establish criteria for consistency, asymptotic normality and asymptotic efficiency as the dimension of the approximation goes to infinity. We derive the asymptotic properties of the MLE from a condition on the order of the operators. In particular, the MLE is consistent if and only if ord(A 1)≧1/2(ord(A 0+θA 1)−d).

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.

Uniqueness and absolute continuity of weak solutions for parabolic SPDE's

The paper presents necessary and sufficient conditions for the absolute continuity of measures generated by infinite-dimensional martingale problems. This result is applied to the study of the existence and uniqueness of weak solutions to nonlinear parabolic SPDE's. The paper also addresses the problem of stochastic integration with respect to a martingale in a quasi-complete locally convex topological vector space.

White Noise Driven Parabolic SPDEs with Measurable Drift

We prove existence and uniqueness of the solution of a white noise driven parabolic SPDE, in case the drift is measurable and satisfies a "one sided linear growth condition," and the diffusion coefficient is nondegenerate, has a locally Lipschitz derivative, and satisfies a linear growth condition. The proof combines arguments similar to those of Gyöngy and Pardoux together with an estimate of the density of the solution of the equation without drift, which is obtained with the help of the Malliavin calculus.

Hautus condition for the pathwise stabilizability of an infinite dimensional stochastic system

In this paper the pathwise stabilizability of a linear infinite dimensional stochastic differential equation through anticipative controls is studied under some commutativity assumptions. The generalization of the deterministic Hautus condition is given and it is applied to a concrete parabolic SPDE

On quasi-linear stochastic partial differential equations

We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.

Absolute Continuity of the Law of the Solution of a Parabolic SPDE

Let {u(t, x); t ≥ 0, 0 < x < 1} denote the solution of a white noise driven parabolic stochastic partial differential equation with Dirichlet boundary conditions. Using Malliavin′s calculus, we give a necessary and sufficient condition for the law of the r.v. u(t, x) to a possess a density.