Related papers: Sticky-Reflected Stochastic Heat Equation Driven b…
Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the…
We consider a general class of SPDEs in $\mathbb{R}^d$ driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring…
We prove that the stochastic heat flow Caravenna Sun Zygouras (2023), Tsai (2024) is a black noise in the sense of Tsirelson (2004). As a corollary, the 2d stochastic heat equation driven by a mollified spacetime white noise becomes…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some…
We consider nonlinear parabolic SPDEs of the form $\partial_t u=-(-\Delta)^{\alpha/2} u + b(u) +\sigma(u)\dot w$, where$\dot w$ denotes space-time white noise. The functions $b$ and $\sigma$ are both locally Lipschitz continuous. Under some…
We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and…
In Rajeev (2013), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one to one correspondence between solutions of a class of finite dimensional SDEs and solutions of a class of SPDEs in…
In this paper we study the longtime dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. As a preparation for this purpose we have to show the existence and uniqueness of a cocycle…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the…
We analyze the spatial asymptotic properties of the solution to the stochastic heat equation driven by an additive L\'evy space-time white noise. For fixed time $t > 0$ and space $x \in \mathbb{R}^d$ we determine the exact tail behavior of…
Thermodynamics of nanoscale devices is an active area of research. Despite their noisy surrounding they often produce mechanical work (e.g. micro-heat engines), display rectified Brownian motion (e.g. molecular motors). This invokes…
This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices…
We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stochastic Process. Appl., 2016). To this end, we extend the notion of non-linear Young equations to a…
We propose a time-implicit, finite-element based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear…
We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations.…
In this article, we introduce a L\'evy analogue of the spatially homogeneous Gaussian noise of Dalang (1999), and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered…
The reflection coefficient is an important thermal property of materials, especially at the nanoscale, and determining this property requires solving an inverse problem based on macroscopic temperature measurements. In this manuscript, we…
We establish explicit integral tests for spatial asymptotic behaviors of fractional stochastic heat equations driven by additive L\'evy white noise. Our results indicate that fractional stochastic heat equations enjoy the so-called additive…
We derive consistent and asymptotically normal estimators for the drift and volatility parameters of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain.…