Martin Hairer, Jonathan Mattingly
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Φ^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
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Strong Feller property, irreducibility, and uniqueness of the invariant measure for stochastic PDEs with degenerate multiplicative noise
本文研究了退化乘性噪声下随机PDEs的强Feller性质和不可约性，并证明了在无强耗散假设下不变测度的唯一性，通过测量解向势垒的积累，赋予退化噪声系数的逆意义。
In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.
In this paper, we consider the (upper) semigroup envelope, i.e. the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, under suitable assumptions, it yields viscosity solutions to abstract Hamilton-Jacobi-Bellman-type partial differential equations related to stochastic optimal control problems arising in the field of Robust Finance. We further derive conditions for the existence of a Markov process under a nonlinear expectation related to the semigroup envelope for the case where the state space is locally compact. T…
Feller generators with singular drifts in the critical range
本文研究了具有临界阶奇异漂移的扩散算子，通过在足够大的 $L^p$ 空间中应用 De Giorgi 方法，在更宽的奇性幅度范围内构造了强连续 Feller 半群，并在临界边界值下于特定的 Orlicz 空间中构造了强连续半群。
Well-posedness and Feller property for functional stochastic Hamiltonian systems with singular coefficients and state-dependent switching
本文研究带奇异系数和状态依赖切换的泛函随机哈密顿系统，通过Girsanov变换、鞅方法和向量值椭圆方程证明了系统的适定性与Feller性质。
We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.
We construct Skorokhod decompositions for diffusions with singular drift and reflecting boundary behavior on open subsets of $\mathbb R^d$ with $C^2$-smooth boundary except for a sufficiently small set. This decomposition holds almost surely under the path measures of the process for every starting point from an explicitly known set. This set is characterized by the boundary smoothness and the singularities of the drift term. We apply modern methods of Dirichlet form theory and $\mathcal L^p$-strong Feller processes. These tools have been approved as useful for the pointwise analysis of stocha…
Semigroups on generalized Sobolev spaces associated with Laplacians with applications Stochastic PDEs with singular boundary conditions
该研究考察带奇异边界条件区域对应的拉普拉斯算子及其生成的广义Sobolev空间半群，并给出其在带奇异边界条件的随机偏微分方程中的应用。
We show that the weak solutions of parabolic equation $\partial_t u - Δu + b(t,x) \cdot \nabla u=0$, $(t,x) \in (0,\infty) \times \mathbb R^d$, $d \geqslant 3$, for $b(t,x)$ in a wide class of time-dependent vector fields capturing critical order singularities, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses an a priori estimate on the $L^p$-norm of the gradient of solution in terms of the $L^q$-norm of the gradient of initial function, and an iterative procedure that moves the problem of convergence in $L^\infty$ to $L^p$.
We construct a strong Feller process associated with $-Δ+ σ\cdot \nabla$, with drift $σ$ in a wide class of measures (weakly form-bounded measures, e.g. combining weak $L^d$ and Kato class measure singularities), by exploiting a quantitative dependence of the smoothness of the domain of an operator realization of $-Δ+ σ\cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the form-bound of $σ$. Our method admits extension to other types of perturbations of $-Δ$ or $(-Δ)^{\fracα{2}}$, e.g. to yield new $L^p$-regularity results for Schrödinger ope…
The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $σ$-finite measure on the space of $\{0,1\}\times\{0,1\}$-valued arrays. In discrete-time, this characterization amounts to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous-time, the behavior is enriched by a Lévy--Itô-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Furthermore, every such process almost surely projects to a Feller proces…
Renormalization of singular elliptic stochastic PDEs using flow equation
针对含分数阶拉普拉斯算子、加性白噪声和三次非线性项的奇异椭圆随机偏微分方程，基于Wilson重正化群理论和Polchinski流方程建立解理论，覆盖整个亚临界区域。
Stochastic PDEs with Generalized Coercivity: Global Well-Posedness and Finite Time Extinction
本文研究了在广义 coercivity 条件下随机偏微分方程的全局存在性、唯一性和 Feller 性，尤其在确定性方程仅具有局部解的情况下。发现适当乘性噪声可防止爆炸并导致随机动力学的有限时间灭绝。
Lecture notes on the flow equation approach to singular stochastic PDEs
本文介绍流方程方法，一种适用于整个亚临界区域中包括分数阶拉普拉斯算子在内的广泛奇异随机偏微分方程的鲁棒框架，通过粗粒化过程和流方程边界条件归纳解决重整化问题。
Locality for singular stochastic PDEs
本文研究了由非平移不变微分算子驱动的奇异随机偏微分方程，提出了一种可描述极大一类时空相关重正化方程的方法，其核心贡献在于避开了带扩展装饰的装饰树的使用。
Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating and studying the properties of a few examples of such equations. We focus mainly on the dynamical Phi4 equation, KPZ equation and Parabolic Anderson Model, as well as touch on a few other equations which arise mainly in physics.
We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients …
In this paper, we prove that there exists a unique weak solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.
We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hormander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator $M_t$ can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection $Π$ on a subspace of sufficiently regular functions. Then the eigenfunctions of $M_t$ with small eigenvalues have only a very sma…
Mean field singular stochastic PDEs
本文研究了由均场类型奇异随机偏微分方程驱动的相互作用场系统，建立了稳健的研究框架，证明了解的存在性与唯一性及大数定律传播结果。
Pathwise uniqueness by noise for singular stochastic PDEs
本文通过噪声正则化方法，解决了奇异随机偏微分方程中具有微分形式漂移项的路径唯一性问题，为流体动力学和相分离模型等提供了新的唯一性结果。
In $\mathbb R^d$, $d \geq 3$, consider the divergence and the non-divergence form operators \begin{equation} \tag{$i$} - \nabla \cdot a \cdot \nabla + b \cdot \nabla, \end{equation} \begin{equation} \tag{$ii$} - a \cdot \nabla^2 + b \cdot \nabla, \end{equation} where $a=I+c \mathsf{f} \otimes \mathsf{f}$, the vector fields $\nabla_i \mathsf{f}$ ($i=1,2,\dots,d$) and $b$ are form-bounded (this includes the sub-critical class $[L^d + L^\infty]^d$ as well as vector fields having critical-order singularities). We characterize quantitative dependence on $c$ and the values of the form-bounds of the …
We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli' study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we i…
A limit theorem for certain Feller semigroups, and the distribution of the local time for Brownian motion when it exits an interval
本文从半群理论视角研究一维 Laplace 算子的局部奇异摄动，证明其收敛至零点具 Robin 型边界条件的热半群，并借此提出计算 Brownian motion 离开区间时局部时分布的方法。
These notes have been prepared for a series of lectures given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture notes and reviews on the subject, such as (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandra and Weber 2017) on singular stochastic PDEs and regularity structures. The present notes have two main specificities. The first one is that they focus on a particular example, the Allen-Cahn equation, which allows to introduce several of the difficulties of the theory in a gradual way, …
Existence and uniqueness of invariant measures for non-Feller Markov semigroups
本文研究非Feller马尔可夫半群不变测度的存在性和唯一性，通过测度论方法证明在自然假设下唯一性成立。
We define multiple stochastic integrals with respect to càdlàg martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3$-dimensional stochastic quantization equation and the KPZ equation.
We provide in this work a semigroup approach to the study of singular PDEs, in the line of the paracontrolled approach developed recently by Gubinelli, Imkeller and Perkowski. Starting from a heat semigroup, we develop a functional calculus and introduce a paraproduct based on the semigroup, for which commutator estimates and Schauder estimates are proved, together with their paracontrolled extensions. This machinery allows us to investigate singular PDEs in potentially unbounded Riemannian manifolds under mild geometric conditions. As an illustration, we study the generalized parabolic Anders…
We prove unique weak solvability and Feller property for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the critical Ladyzhenskaya-Prodi-Serrin class, the weak $L^d$ class as well as some vector fields that are not even in $L^{2+\varepsilon}_{\rm loc}$, $\varepsilon>0$.
Existence and uniqueness is established for a large class of backward stochastic differential equations which contain singular terms of the form $\pm|z|^2/y$. The results are applied to investigate singular partial differential equations (PDEs) and to decision theory problems that cannot be studied using classical regular BSDEs. The application to PDEs concerns the existence of viscosity solutions to PDEs containing a singular term of the form $\pm|\nabla v|^2/v$ with rather weak assumptions on the regularity of the coefficients. Such PDEs with singularity in the value process appear in severa…
Extended decorations on naturally decorated trees were introduced in the work of Bruned, Hairer and Zambotti on algebraic renormalization of regularity structures to provide a convenient framework for the renormalization of systems of singular stochastic PDEs within that setting. This non-dynamical feature of the trees complicated the analysis of the dynamical counterpart of the renormalization process. We provide a new proof of the renormalized system by-passing the use of extended decorations and working for a large class of renormalization maps, with the BPHZ renormalization as a special ca…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the…
We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This problem can be addressed within the framework of the recently developed theory of "regularity structures", which allows to describe candidate solutions locally by a "jet", but where the usual Taylor polynomials are replaced by a sequence of custom-built objects. In order to illustrate the theory, we focus on the particular example of the Kardar-Parisi-Zhang …
Weak uniqueness by noise for singular stochastic PDEs
该研究针对希尔伯特空间上的奇异SPDEs，提出仅需漂移项定义在索伯列夫型子空间、无需霍尔德连续性的框架，借助有色或更粗糙噪声证明温和解弱唯一性，可推广至多类具体算例。
Feller's test for explosions of stochastic Volterra equations
本文针对一维卷积型连续随机Volterra过程的爆炸问题提出Feller检验，推导非奇异核下过程区间内存续的充要条件，通过三类过程验证结果，并讨论Volterra CIR过程的分数核指数和近似方法。
In this paper, we prove a convergence theorem for singular perturbations problems for a class of fully nonlinear parabolic partial differential equations with ergodic structures. The limit function is represented as the viscosity solution to a fully nonlinear degenerate PDEs. Our approach is mainly based on G-stochastic analysis argument. As a byproduct, we also establish the averaging principle for stochastic differential equations driven by G-Brownian motion with two time-scales. The results extend Khasminskii's averaging principle to nonlinear case.
Parametrization of renormalized models for singular stochastic PDEs
本文提出了一种对奇异随机偏微分方程正规化模型进行参数化的线性方法，并探讨了重整化方案对参数化空间的作用。
We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations. These are expanded notes of the 20th Takagi lectu…
These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non--linear PDEs with stochastic and irregular terms. We explain how it is possible to handle them using two main techniques: the notion of energy solutions and that of paracontrolled distributions. In order to maintain a link with physical intuitions, we motivate such singular SPDEs via a homogenisation result for a diffusion in a random potential.
We obtain a generalisation of the Stroock-Varadhan support theorem for a large class of systems of subcritical singular stochastic PDEs driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup $\mathcal{H}$ of the renormalisation group such that any renormalisation procedure determines a unique coset $g\circ\mathcal{H}…
Non-local parabolic equations with singular (Morrey) time-inhomogeneous drift
该研究针对带局部无界奇异漂移项的非局部抛物方程，借助Adams和Krylov的思路证明抛物算子范数不等式，得到解的Sobolev正则性估计，同时给出对应Feller稳定过程的Krylov界与McKean-Vlasov方程解的先验正则性估计。
We use some tools from nonlinear analysis to study two examples of singular stochastic elliptic PDEs that cannot be solved by the contraction principle or the Schauder fixed point theorem. Let $ξ$ stand for a spatial white noise on a closed Riemannian surface $S$. We prove the existence of a solution to the equation $$ (-Δ+ a)u = f(u) + ξu $$ with a potential $a\in L^p(S)$ and $p>1$, and $f$ subject to growth conditions. Under an additional parity condition on $f$ -- met for instance when $f(u) = u\vert u\vert^\ell$, with $\ell$ an even innteger, we further prove that this equation has infinit…
Flow equation approach to singular stochastic PDEs
该研究利用 Wilson 重整化群与 Polchinski 流方程建立奇异半线性抛物型 SPDE 的新解理论，证明弱非线性区域宏观标度极限的普适性，并覆盖四维三次非线性的整个次临界区域。
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric $α$-stable process with a locally unbounded drift $b:\mathbb R^d \rightarrow \mathbb R^d$, $d \geq 3$, $1<α<2$. In this paper, $b$ belongs to the class of weakly form-bounded vector fields. The latter arises as the class providing the $L^2$ theory of the non-local operator behind the SDE, i.e.\,$(-Δ)^{\fracα{2}} + b \cdot \nabla$, and contains as proper sub-classes the other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, weak $L…
Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian switching. Then, the case that the Markov chain is involved in a large state space is considered. Following the classical approach, a hierarchical approach is adopted to reduce the complexity and a singularly perturbed Markov chain is involved. We will study the asymptotic property of BSDE with the singularly perturbed Markov chain. At last, as an application…
Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strict positivity of the unique continuous version of the density to its invariant measure. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically has an unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of \cite…
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimat…
In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=Δ^{\fracα{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $Δ^{\fracα{2}}=-(-Δ)^{\fracα{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump Lévy processes with Lévy measure of $ν^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\fracα{q_0}$-order d…
Large field problem in coercive singular PDEs
本文研究了严酷偏微分方程中的大领域问题，通过粗糙路径理论和正则性结构推导了先验估计，并利用缩放将局部 coercivity 转换为全局性质。
We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the…
Remarks on parabolic Kolmogorov operator
本文研究含奇异漂移的抛物型 Kolmogorov 方程，通过建立解的梯度估计构造 Feller 演化族，进而得到对应随机微分方程的唯一弱解。
The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state-and control-dependent white noise is formulated and studied. Both finite-and infinite-time horizons are considered. The multi-plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular …
A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular Lévy measures
提出一种蒙特卡洛格式求解完全非线性非局部抛物型偏微分方程，通过截断无限Lévy测度避免困难，并证明格式收敛性及凹（凸）非线性情形的收敛速率。
We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at $1$ and $-1$ and a constraint of conservation of the space average. The equation, driven by a trace-class space-time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this approximated sequence of solutions is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic st…
We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant…
In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift $b$ and the weak gradient of Sobolev diffusion $σ$ are supposed to satisfy $||b(x){1}_{|x|\le R}||_{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})$ and $||\nabla σ(x)1_{|x|\le R} ||_{p_1} \le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$ respectively. The main tools for these results are the decomposition of global two-point motions, Krylov's estimate, Kh…
We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation (PDE) with singular-degenerate diffusion. In addition, the deterministic BTW model is shown to converge to a singular-degenerate PDE. Alternatively, the proof of the scaling limit can be understood as a convergence proof of a finite-difference discretization for singular-degenerate stochastic PDEs. This extends recent work on finite difference approximation of (deterministic)…
Spectral Theory of Krein-Feller Type Operators and Applications in Stochastic Fractional Elliptic and Parabolic Equations
本文研究Krein-Feller型算子的谱理论，推导其特征值的渐近下界和收敛指数上界，并应用于随机分数椭圆和抛物方程的存在性问题。
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