Stochastic partial differential equations driven by Lévy white noises

We study various aspects of stochastic partial differential equations driven by Lévy white noise. This driving noise, which is a generalization of Gaussian white noise, can be viewed either as a generalized random process or as an independently scattered random measure. After unifying these approaches and establishing appropriate stochastic integral representations, we show that a necessary and sufficient condition for a Lévy white noise to have values in the space of tempered Schwartz distributions, is that the underlying Lévy measure have a positive absolute moment. In the case of a linear stochastic partial differential equation with a general differential operator and driven by a symmetric pure jump Lévy white noise, we show that when the mild solution is locally Lebesgue integrable, then it is equal to the generalized solution, and that a random field representation exists for the generalized solution if and only if the fundamental solution of the operator has certain integrability properties. In that case, we show that the random field representation is equal to the mild solution. For this purpose, a new stochastic Fubini theorem is proved. These results are applied to the linear stochastic heat and wave equations driven by a symmetric alpha-stable noise. We then study the non-linear stochastic heat equation driven by a general type of Lévy white noise, possibly with heavy tails and non-summable small jumps. Our framework includes in particular the alpha-stable noise. In the case of the equation on the whole space, we show that the law of the solution that we construct does not depend on the space variable. Then we show in various domains D that the solution u to the stochastic heat equation is such that t -> u(t,·) has a càdlàg version in a fractional Sobolev space of order r < -d /2. Finally, we show that the partial functions have a continuous version under some optimal moment conditions. In the alpha-stable case, we show that for the choices of alpha for which this moment condition is not satisfied, the sample paths of the partial functions are unbounded on any non-empty open subset.

Path properties of the solution to the stochastic heat equation with Levy noise

Carsten Hao Ye Chong, Robert Dalang, Thomas Marie Jean-Baptiste Humeau

We consider sample path properties of the solution to the stochastic heat equation, in Rd or bounded domains of Rd, driven by a Levy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a cadlag modification in fractional Sobolev spaces of index less than - Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the Levy noise such that noises with a smaller index entail continuous sample paths, while Levy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Levy noises, and to light- as well as heavy-tailed jumps.

SPRINGER2019

Normal approximation of the solution to the stochastic heat equation with Levy noise

Carsten Hao Ye Chong

Given a sequence L & x2d9;epsilon of Levy noises, we derive necessary and sufficient conditions in terms of their variances sigma 2(epsilon) such that the solution to the stochastic heat equation with noise sigma(epsilon)-1L & x2d9;epsilon converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of Asmussen and Rosinski (J Appl Probab 38(2):482-493, 2001), Cohen and Rosinski (Bernoulli 13(1):195-210, 2007) for finite-dimensional Levy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as cadlag processes with values in a Sobolev space of negative real order.

SPRINGER2020

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

EPFL2016