本文扩展 Otto 和 Weber 的粗糙路径方法，通过可确定性处理的初始边界层修正周期解，为奇异拟线性 SPDE 建立满足 $C^α$ 周期初值的解理论，并证明唯一性与数据扰动稳定性。
摘要：
在本文中，我们针对受初始条件约束的奇异拟线性随机偏微分方程建立了一套解理论。我们的解理论是通过对 Otto 和 Weber（2019）为处理时空周期问题而发展的粗糙路径方法进行扰动而得到的。与其工作一样，我们假设外力属于 $C^{α-2}$ 类，其中 $α\in(\frac{2}{3},1)$，且具有时空周期性；此外还假设初始条件属于 $C^α$ 类并具有周期性。我们对该理论的分析方面作出了贡献。具体而言，我们证明可以通过用一个初始边界层修正先前得到的时空周期解来强制满足初始条件，而该初始边界层可以以完全确定性的方式处理。在以这种初始边界层进行修正的解类中，我们得到了唯一性。此外，还建立了解关于数据扰动的稳定性。
英文摘要：
In this contribution we develop a solution theory for singular quasilinear stochastic partial differential equations subject to an initial condition. We obtain our solution theory as a perturbation of the rough path approach developed to handle the space-time periodic problem by Otto and Weber (2019). As in their work, we assume that the forcing is of class $C^{α-2}$ for $α\in (\frac{2}{3},1)$ and space-time periodic and, additionally, that the initial condition is of class $C^α$ and periodic. We contribute to the analytic aspects of the theory. Indeed, we show that we can enforce the initial condition via correcting the previously obtained space-time periodic solution with an initial boundary layer which may be handled in a completely deterministic manner. Uniqueness is obtained in the class of solutions which are corrected in this way by an initial boundary layer. Moreover, stability of the solutions with respect to perturbations of the data is established.
We give meaning to linear and semi-linear (possibly degenerate) parabolic partial differential equations with (affine) linear rough path noise and establish stability in a rough path metric. In the case of enhanced Brownian motion (Brownian motion with its Lévy area) as rough path noise the solution coincides with the standard variational solution of the SPDE.
We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is Hölder-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to …
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander's bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris's le…
We prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift in divergence form involving maximal monotone graphs defined everywhere, on which neither growth nor continuity assumptions are imposed.
In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough paths theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem, which is stable under a large class of perturbations, includ…
We study the dependence of mild solutions to linear stochastic evolution equations on Hilbert space driven by Wiener noise, with drift having linear part of the type $A+\varepsilon G$, on the parameter $\varepsilon$. In particular, we study the limit and the asymptotic expansions in powers of $\varepsilon$ of these solutions, as well as of functionals thereof, as $\varepsilon \to 0$, with good control on the remainder. These convergence and series expansion results are then applied to a parabolic perturbation of the Musiela SPDE of mathematical finance modeling the dynamics of forward rates.
We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation, based on which we are able to obtain existence, uniqueness and stability results. The case of weak "rough" forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play …
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques …
We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $μ$, solvability of the associated Kolmogorov equation in $L^1(μ)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variati…
We give a construction allowing to construct local renormalised solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [Hai14,BHZ16,CH16] applies, including in particular one-dimensional systems with KPZ-type nonlinearities driven by space-time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renor…
In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +$\infty$ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.
We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} γ(\nabla \cdot)$, where $γ$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $γ$ is single-valued, a solution with better integrability and regularity prop…
We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - σ(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $ C^{α-2}$ for $α> \frac{2}{3}$ on the parabolic Hölder scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, …
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent. Math. 198, 2014) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theorie…
The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to SPDE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamical systems it is proven that the attractor consists of a single random point. In case of real, linear multiplicative noise finite time extinction is obtained. Applications include stochastic generalized fast diffusion equations and stocha…
In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds,…
In this paper, we study weighted $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of singular integrals with homogeneous convolution kernel $K(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$, {under the assumption that $K_0$, the restriction of $K$ to the unit annulus, is mean zero and $L^{q}$ integrable for some $q_{0}<q\leq \infty$,} where $q_{0}$ is a fixed constant depending on $w$. We obtain a quantitative weighted bound, which is consistent with the one obtained by Hytönen--Roncal--Tapiola in the Euclidean setting, for this operator o…
We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.
This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth, and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Si…
We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced---with Brownian-time-type kernel formulations---by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regul…
This article focuses on parabolic equations with rough diffusion coefficients which are ill-posed in the classical sense of distributions due to the presence of a singular forcing. Inspired by the philosophy of rough paths and regularity structures, we introduce a notion of modelled distribution which is suitable in this context. We prove two general tools for reconstruction and integration, as well as a product lemma which is tailor made for the reconstruction of the rough diffusion operator. This yields a partially automated deterministic theory, which we apply to obtain an existence and uni…
The magnetization ripple is a microstructure formed by the magnetization in a thin-film ferromagnet. It is triggered by the random orientation of the grains in the poly-crystalline material. In an approximation of the micromagnetic model, which is sketched in this paper, this leads to a nonlocal (and strongly anisotropic) elliptic equation in two dimensions with white noise as a right hand side. However, like in singular Stochastic PDE, this right hand side is too rough for the non-linearity in the equation. In order to develop a small-date well-posedness theory, we take inspiration from the r…
