Schedule for: 22w5172 - Theory and Computational Methods for SPDEs (Online)

We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplicative than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order $2/d$, where $d$ is the spatial dimension, the largest peaks will be of the same order for both additive and multiplicative noise, which is in sharp contrast to the behavior of the solution under Gaussian noise. However, in this case, a closer inspection reveals a second layer of peaks, beneath the largest peaks, that is exclusive to multiplicative noise and that can be observed by sampling the solution on the lattice. This is based on joint work with Péter Kevei (Szeged).

In this talk, we introduce some tools which are needed for the stochastic analysis with respect to a L\'evy noise, using the random field approach introduced by Walsh (1986). We consider the case of a L\'evy white noise, with possibly infinite variance (such as the $\alpha$-stable L\'evy noise). The focus will be on the stochastic wave equation with this type of noise, on the entire space $\mathbb{R}^d$, in dimension is $d=1$ or $d=2$. In this equation, the noise is multiplied by a Lipschitz function $\sigma(u)$ of the solution. We will show that the solution exists and has a c\`adl\`ag modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.

I will present the properties of a recently proposed numerical scheme for the temporal discretization of parabolic semilinear one-dimensional SPDEs driven by additive space-time white noise. The scheme is a modification of the standard semi-implicit Euler method. I will show the main benefits of the new method compared with that standard scheme: it preserves the spatial regularity of the solution for any choice of the time-step size, and it provides error estimates in total variation distance for the approximation of the invariant distribution under appropriate conditions. Arxiv preprint reference: http://arxiv.org/abs/2203.10598

We study the asymptotic speed of a random front for solutions to stochastic reaction-diffusion equations with a Wright-Fisher noise proportional to σ. Under some conditions on the drift, we show the existence of the speed of the front and derive its asymptotics depending on σ. This talk is based on joint works with C. Mueller, L. Ryzhik, C. Barnes and Z. Sun.

In this talk we will deal with the approximation of stochastic PDEs, in spatial dimension one, of the form \begin{equation} \partial_t u = \Delta u + f(u) + \xi, \qquad u(0, x)=u_0(x), \qquad (t, x) \in [0, 1] \times \mathbb{T}, \end{equation} where $\xi$ is a space-time white noise on $[0, 1] \times \mathbb{T}$ and $f: \mathbb{R} \to \mathbb{R}$. While the approximation of the solution of such equations has been extensively studied in the case that $f$ is Lipschitz continuous, or at least one-sided Lipschitz, very few results were available for less regular $f$. In this talk we will show that the rate of convergence of the fully discrete, explicit in time, finite difference scheme is $1/2$ in space and $1/4$ in time, even for merely bounded, measurable $f$. The proof relies on the regularisation effect of the noise. To exploit and quantify this effect we use stochastic sewing techniques. This is joint work with Oleg Butkovsky and Máté Gerencsér.

Consider the stochastic heat equation on $\mathbb{R}^d$, \begin{equation*} \frac{\partial u}{\partial t} = \frac{1}{2}\Delta u + b(u) + \sigma(u)\dot{W}\,, \end{equation*} where $\dot{W}$ is a centered Gaussian noise which is white in time and colored in space, the initial condition is assumed to be a function with compact support. The functions $b(z)$ and $\sigma(z)$ are locally Lipschitz and as $|z|\to \infty$, $|b(z)| = o(|z|\log |z|)$ and $|\sigma(z)| = o (|z|(\log |z|)^{\alpha})$ for some $0 < \alpha <\frac{1}{2}$. We show that under improved Dalang's condition, there is a unique global solution. This is based on joint work with Le Chen.

We will discuss, various time and space numerical schemes for various models in fluid flow models. We will prove their convergence and discuss some convergence rates.

Scattering problems have been widely investigated for time-harmonic waves, including acoustic, elastic, and electromagnetic waves, due to their significant applications in elastography, remote sensing, nondestructive testing, etc. This talk is concerned with an inverse random potential scattering problem for the stochastic elastic wave equation, where the potential is modeled as a microlocally isotropic Gaussian random field. For the direct scattering problem, the well-posedness is established and the regularity of the solution is obtained by developing the unique continuation theorem for the rough potential case. For the inverse scattering problem, the statistical strength of the random potential is shown to be uniquely determined by the high frequency limit of the amplitude of the scattered waves averaged over the frequency band from almost surely a single realization.

Second order possibly degenerate stochastic PDEs and stochastic integro-differential equations will be considered. Known existence and uniqueness theorems will be revisited and improved for a class of these equations. If time permits numerical approximations of these equations will be discussed. The talk is based on recent results obtained in collaboration with Alexander Davie and Fabian Germ.

We consider semilinear stochastic evolution equations on Hilbert spaces with multiplicative Wiener noise and linear drift term of the type $A + \varepsilon G$, with $A$ and $G$ maximal monotone operators and $\varepsilon$ a "small" parameter, and study the differentiability of mild solutions with respect to $\varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the sense that its domain can be strictly contained in the domain of $A$. We also discuss the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions as $\varepsilon \to 0$, with control on the remainder. These results are then applied to a parabolic perturbation of the Musiela SPDE modeling the dynamics of forward rates. The talk is partly based on joint work with S. Albeverio and E. Mastrogiacomo.

This is joint work with Davar Khoshnevisan and Kunwoo Kim. We discuss solutions $u:[0,\infty)\times\mathbf{R}\times\Omega\to[0,\infty)$ to the stochastic heat equation with multiplicative noise \begin{equation*} \partial_tu = \partial_x^2u + \sigma(u)\dot{W} \end{equation*} where $\dot{W}=\dot{W}(t,x)$ is spacetime white noise, and the initial function $u(0,x)$ is nonnegative but not identically zero. We assume that $\sigma(u)$ is approximately linear, so there exist $c,C>0$ such that $c\le\sigma(u)\le C$ for all $u\ge0$. If $\sigma(u)=u$, then we have the well-known parabolic Anderson model, whose solutions forms tall and widely separated peaks when $u(0,x)=1$. Although the peaks have been extensively studied, much less is known about the regions between peaks, which we call valleys. For $\sigma(u)$ satisfying the condition in the previous paragraph and for $u(0,x)=1$, we estimate the size of the valleys, as well as the size of the solution $u$ within a valley.

We study slow-fast systems of coupled equations from fluid dynamics, where the fast component is perturbed by additive noise. We prove that, under a suitable limit of infinite separation of scales, the slow component of the system converges in law to a solution of the initial equation perturbed with transport noise, and subject to the influence of an additional Itô-Stokes drift. The obtained limit equation is very similar to turbulent models derived heuristically. Our results apply to the Navier-Stokes equations in dimension $d = 2,3$; the Surface Quasi-Geostrophic equations in dimension $d = 2$; and the Primitive equations in dimension $d = 2, 3$.

The sample-function regularity of the random-field solution to a stochastic partial differential equation (SPDE) depends naturally on the roughness of the external noise, as well as on the properties of the underlying integro-differential operator that is used to define the equation. In this talk, we consider parabolic and hyperbolic SPDEs on $(0\,,\infty)\times\mathbb{R}^d$ of the form $\partial_t u = \mathcal{L}u + g(u) + \dot{F}$ and $\partial^2_t u = \mathcal{L}u + c + \dot{F}$, with suitable initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$ that is white in its time variable and correlated in its space variable, and driven by the generator $\mathcal{L}$ of a genuinely $d$-dimensional Lévy process $X$. We find optimal Hölder conditions for the respective random-field solutions to these SPDEs. Our conditions are stated in terms of indices that describe thresholds on the integrability of some functionals of the characteristic exponent of the process $X$ with respect to the spectral measure of the spatial covariance of $\dot{F}$. Those indices are suggested by the results of Sanz-Solé and Sarrà (2000, 2002) on the particular case that $\mathcal{L}$ is the Laplace operator on $\mathbb{R}^d$. This talk is based on joint work with Marta Sanz-Solé.