We study the nonlinear stochastic heat equation in spatial domain $\mathbb{R}$, driven by space–time white noise. A central special case is parabolic Anderson model. The initial condition taken to be a measure on such as Dirac delta function, but this may also have noncompact support and even nontempered (e.g., with exponentially growing tails). Existence uniqueness of random field solution proved without appealing Gronwall's lemma, keeping tight control over moments Picard iteration scheme....

We establish the strong comparison principle and strict positivity of solutions to following nonlinear stochastic heat equation on $\mathbb{R}^{d}$ \[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that white in time $\rho$ Lipschitz continuous. These results are obtained under condition...

We study the nonlinear stochastic time-fractional diffusion equations in spatial domain $\mathbb {R}$, driven by multiplicative space-time white noise. The fractional index $\beta$ varies continuously from $0$ to $2$. case $\beta =1$ (resp. =2$) corresponds heat wave) equation. cases \in \:]0,1[\:$ and \:]1,2[\:$ are called slow fast equations, respectively. Existence uniqueness of random field solutions with measure-valued initial data, such as Dirac delta measure, established. Upper bounds...

In this paper, we establish a necessary and sufficient condition for the existence regularity of density solution to semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under mild cone diffusion coefficient, smooth joint at multiple points. The tool use is Malliavin calculus. main ingredient prove that solutions related partial differential have negative moments all orders. Because cannot <inline-formula content-type="math/mathml"> <mml:math...

To better understand the factors facilitating or impeding translation of “promising controversies” (Taylor &amp; Snoddon, 2013, p. 439) plurilingualism theory into meaningful practices, this article presents a synthesis 30 empirical studies on plurilingual pedagogy as enacted and experienced by educators learners in various global K–12 postsecondary contexts. Informed (Coste et al., 1997/2009; Council Europe, 2020), forefronts key contributions it fosters: (a) students’ development (and...

Dans ce papier, nous montrons un principe de comparaison trajectoriel pour l'équation la chaleur stochastique, fractionnaire, nonlinéaire sur $\mathbb{R}$ avec une donnée initiale à valeur mesure. Nous donnons des estimations quantitatives proximité zéro d'une solution. Ces résultats étendent le Mueller stochastique et permettent considérer données initiales plus générales telles que mesures Dirac queue lourde qu'une croissance exponentielle linéaire en ${\pm}\infty$. généralisent travail...

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables , where is a general Gaussian noise . The existence uniqueness solution, moment bounds solution are obtained by using fundamental solutions corresponding deterministic counterpart represented Fox H-functions. Along way, we obtain some new properties solutions.

Soit {u(t,x)}t≥0,x∈Rd la solution d'une équation de chaleur stochastique non-linéaire d-dimensionnelle, perturbée par un bruit gaussien, blanc en temps et avec une covariance homogène espace donnée mesure Borel finie qui satisfait condition Dalang. Nous démontrons deux théorèmes limite centrale fonctionnels pour des champs d'occupation forme N−d∫Rdg(u(t,x))ψ(x/N)dx quand N→∞, où g est function lipschitzienne sur Rd ψ∈L2(Rd). La preuve utilise inegalités type Poincaré, le calcul Malliavin,...

Consider a parabolic stochastic PDE of the form ∂tu=1 2Δu+σ(u)η, where u=u(t,x) for t≥0 and x∈Rd, σ:R→R is Lipschitz continuous non random, η centered Gaussian noise that white in time colored space, with possibly-signed homogeneous spatial correlation f. If, addition, u(0)≡1, then we prove that, under mild decay condition on f, process x↦u(t,x) stationary ergodic at all times t>0. It has been argued when coupled moment estimates, ergodicity u teaches us about intermittent nature solution to...

Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous non random, $η$ centered Gaussian noise that white in time colored space, with possibly-signed homogeneous spatial correlation function $f$. If, addition, $u(0)\equiv1$, then we prove that, under mild decay condition on $f$, process $x\mapsto u(t\,,x)$ stationary ergodic at all times $t&gt;0$. It has...

In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis partial-differential Subscript t Superscript beta Baseline plus StartFraction nu Over 2 EndFraction left-parenthesis negative normal upper Delta right-parenthesis alpha slash u comma x equals I gamma...

Keywords: nonlinear stochastic heat equation ; wave parabolic Anderson model hyperbolic rough initial data Holder continuity Lyapunov exponents growth indices These Ecole polytechnique federale de Lausanne EPFL, n° 5712 (2013)Programme doctoral MathematiquesFaculte des sciences baseInstitut mathematiques d'analyse et applicationsChaire probabilitesJury: J. Krieger (president), D. Khoshnevisan, T. Mountford, R. Tribe Public defense: 2013-4-19 Reference doi:10.5075/epfl-thesis-5712Print copy...

Dans cet article, nous étudions l'équation des ondes stochastique en dimensions d≤3, dirigée par un bruit gaussien W˙ qui ne dépend pas du temps. On suppose que soit le est blanc, la fonction de covariance satisfait une propriété d'échelle similaire au noyau Riesz. La solution interprétée dans sens Skorohod utilisant calcul Malliavin. obtient comportement asymptotique exact p-ième moment lorsque temps grand, p grand. Pour cas critique, i.e. si d=3 et on transition pour deuxième fini.