On Besov regularity of Brownian motions in infinite dimensions

A Probabilistic proof of the breakdown of Besov regularity in $L$-shaped domains

{We provide a probabilistic approach in order to investigate the smoothness of the solution to the Poisson and Dirichlet problems in $L$-shaped domains. In particular, we obtain (probabilistic) integral representations for the solution. We also recover Grisvard's classic result on the angle-dependent breakdown of the regularity of the solution measured in a Besov scale.

On the Besov Regularity of Periodic Lévy Noises

In this paper, we study the Besov regularity of Lévy white noises on the $d$-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for general Lévy white noises. Then, we focus on two subclasses of noises: compound Poisson and symmetric-$α$-stable (including Gaussian), for which we make more precise statements. Before measuring regularity, we show that the question is well-posed; we prove that Besov spaces are in the cylindrical $σ$-field of the space of generalized functions.…

Besov-Orlicz path regularity of non-Gaussian processes

In the article, Besov-Orlicz regularity of sample paths of stochastic processes that are represented by multiple integrals of order $n\in\mathbb{N}$ is treated. We give sufficient conditions for the considered processes to have paths in the exponential Besov-Orlicz space $$B_{\varPhi_{2/n},\infty}^α(0,T)\qquad \mbox{with}\qquad \varPhi_{2/n}(x)=\mathrm{e}^{x^{2/n}}-1.$$ These results provide an extension of what is known for scalar Gaussian stochastic processes to stochastic processes in an arbitrary finite Wiener chaos. As an application, the Besov-Orlicz path regularity of fractionally filte…

一般过程局部时的一致Besov正则性研究

On the uniform Besov regularity of local times of general processes

该研究引入α-局部非确定性条件，研究连续过程局部时的一致Besov正则性，推广Adler定理，得到经典高斯过程与非线性随机热方程组解及其局部时的最优Besov正则性结果。

Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs

We introduce a decoupling method on the Wiener space to define a wide class of an\-iso\-tro\-pic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly …

On Besov regularity and local time of the stochastic heat equation

Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, Hölder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.

关于具有非正正则性 Besov 相互作用核的稳定过程驱动 McKean–Vlasov 随机微分方程的适定性

On well-posedness of stable-driven McKean-Vlasov stochastic differential equations with Besov interaction kernel of negative regularity

本文研究稳定过程驱动的 McKean–Vlasov SDE，借助 Besov 乘积法则、初始数据光滑性及核散度条件，将相互作用核正则性推进至 -1 以下并证明适定性。

布朗运动与多边形部分和过程的Besov-Orlicz模

Besov-Orlicz moduli of Brownian motion and polygonal partial sum processes

研究布朗运动和多边形部分和过程在Besov-Orlicz尺度上的正则性，推导出子高斯Orlicz模的精确极限及收敛速度，改进了Donsker不变原理。

Regularity of Gaussian Processes on Dirichlet spaces

We are interested in the regularity of centered Gaussian processes (Z_x), x in M indexed by compact metric spaces M. It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance K(x,y) = E(Z_xZ_y) under the assumption that (i) there is an underlying Dirichlet structure on M which determines the Besov space regularity, and (ii) the operator K with kernel K(x,y) and the underlying operator A of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processe…

Besov regularity of the uniform empirical process

The paths of Brownian motion have been widely studied in the recent years relatively in Besov spaces $B_{p, \infty}^\a$. The results are the same as to the Brownian bridge. In fact these regularities properties are established in some sequence spaces $S_{p, \infty}^\a$ using an isomorphisim between them and $B_{p, \infty}^\a$. In this note, we are concerned with the regularity of the paths of the continuous version of the uniform empirical process in the space $S_{p, \infty}^\a$ and in one of his separable sub space $S_{p, \infty}^{\a, 0}$ for a suitable choice of $\a$ and $p$.

Multidimensional Lévy White Noises in Weighted Besov Spaces

In this paper, we study the Besov regularity of d-dimensional Lévy white noises. More precisely, we describe new sample paths properties of a given white noise in terms of weighted Besov spaces. In particular, the smoothness and integrability properties of Lévy white noises are characterized using the Blumenthal-Getoor indices. Our techniques rely on wavelet methods and generalized moments estimates for Lévy white noises.

某些高斯过程的Besov-Orlicz路径正则性研究

On the Besov-Orlicz path regularity of some Gaussian processes

本文利用随机过程的加法分解性质与分数布朗运动的正则性，统一证明了广泛过程类的Besov-Orlicz正则性，包括双分数布朗运动、子分数布朗运动及自相似过程。

Wavelet Analysis of the Besov Regularity of Lévy White Noises

We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-alpha-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the L…

Lipschitz 域上随机偏微分方程的空间 Besov 正则性

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

利用 Besov 空间尺度研究有界 Lipschitz 域上线性抛物型随机偏微分方程解的空间正则性，通过加权 Sobolev 估计和小波展开刻画 Besov 空间，确定非线性逼近的收敛阶。

Stochastic Analysis with Modelled Distributions

Using a Besov topology on spaces of modelled distributions in the framework of Hairer's regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type $2$. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness results for mild solutions of semilinear stochastic partial differential equations in these spaces of modelled distributions and for distribution-valued SDEs. Furthermore, we provide a Fubini …

有界Lipschitz域上随机抛物方程的Lq(Lp)-正则性与Besov光滑性

On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

研究有界Lipschitz域上带零Dirichlet边界的线性随机抛物方程在Krylov加权Sobolev空间中的存在唯一性、时间Hölder正则性，并通过嵌入定理得到Besov尺度下的正则性结果。

The reconstruction theorem in Besov spaces

The theory of regularity structures sets up an abstract framework of modelled distributions generalising the usual Hölder functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of "modelled distributions" into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces $\mathcal{B}^γ_{p,q}$ with non-integer regularity indices. We then show that these spaces behave…

On fractional regularity of distributions of functions in Gaussian random variables

We study fractional smoothness of measures on $\mathbb{R}^k$, that are images of a Gaussian measure under mappings from Gaussian Sobolev classes. As a consequence we obtain Nikolskii--Besov fractional regularity of these distributions under some weak nondegeneracy assumption.

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…

On temporal regularity of stochastic convolutions in $2$-smooth Banach spaces

We show that paths of solutions to parabolic stochastic differential equations have the same regularity in time as the Wiener process (as of the current state of art). The temporal regularity is considered in the Besov-Orlicz space $B^{1/2}_{Φ_2,\infty}(0,T;X)$ where $Φ_2(x)=\exp(x^2)-1$ and $X$ is a $2$-smooth Banach space.

三角阵列独立随机变量在若干Besov空间中的弱不变原理

A weak invariance principle for triangular arrays of independent random variables in some Besov spaces

本文在Besov空间中建立Donsker-Prokhorov不变原理，证明三角阵列行内独立随机变量的折线过程弱收敛于布朗运动，扩展了已有结果。

A Littlewood-Paley description of modelled distributions

We exhibit a fundamental link between Hairer's theory of regularity structures and the paracontrolled calculus of Gubinelli, Imkeller and Perkowski. By using paraproducts we provide a Littlewood-Paley description of the spaces of modelled distributions in regularity structures that is similar to the Besov description of classical Hölder spaces.

On the pointwise regularity of the Multifractional Brownian Motion and some extensions

We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process.

用差分刻画带主导混合光滑性的Besov空间

Characterization of Besov spaces with dominating mixed smoothness by differences

该研究针对实直线与环面乘积空间及有界域上的带主导混合光滑性Besov空间，给出其差分形式刻画，并基于相关Kolmogorov准则探讨了在高斯场、随机热方程等随机场中的应用。

Time Reversal of free diffusions I : Reversed Brownian motion, Reversed SDE and first order regularity of conjugate variables

We show that solutions of free stochastic differential equations with regular drifts and diffusion coefficients, when considered backwards in time, still satisfy free SDEs for an explicit free Brownian motion and drift. We also study the stochastic integral part with respect to this reversed free Brownian motion of canonical closed martingales. We deduce that conjugate variables computed along a free Brownian motion, an example of such a reversed martingale appearing in the definition of non-microstates free entropy, are in the $L^2$ domain of corresponding free difference quotients for almost…

BSDEs driven by G-Brownian motion under degenerate case and its application to the regularity of fully nonlinear PDEs

In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.

Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.

Elliptic regularity results: n-regularized Liouville Brownian motion and non-symmetric diffusions associated with degenerate forms

We apply improved elliptic regularity results to a concrete symmetric Dirichlet form and various non-symmetric Dirichlet forms with possibly degenerate symmetric diffusion matrix. Given the (non)-symmetric Dirichlet form, using elliptic regularity results and stochastic calculus we show weak existence of the corresponding singular stochastic differential equation for any starting point in some subset E of R^d. As an application of our approach we can show the existence of n-regularized Liouville Brownian motion only via Dirichlet form theory starting from all points in R^2.

Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.

退化麦肯-瓦拉兹方程在各向异性负贝索夫空间中的漂移

Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces

本文研究了在各向异性负贝索夫空间中具有退化扩散矩阵的麦肯-瓦拉兹型SDE，证明了在弱霍尔曼德条件下的解在分布意义下的存在性和唯一性。

Estimation of the global regularity of a multifractional Brownian motion

This paper presents a new estimator of the global regularity index of a multifractional Brownian motion. Our estimation method is based upon a ratio statistic, which compares the realized global quadratic variation of a multifractional Brownian motion at two different frequencies. We show that a logarithmic transformation of this statistic converges in probability to the minimum of the Hurst function, which is, under weak assumptions, identical to the global regularity index of the path.

On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model

We study the pathwise regularity of the map $$ ϕ\mapsto I(ϕ) = \int_0^T < ϕ(X_t), dX_t>$$ where $ϕ$ is a vector function on $\R^d$ belonging to some Banach space $V$, $X$ is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A \emph{stochastic current} is a continuous version of this map, seen as a random element of the topological dual of $V$. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient condi…

Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet

We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space $\mathbb{R}^{1+1}$ with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.

Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations

\def\R{\mathbb R} We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times \R^d$, which is not invariant by translation. We then prove that the hitting probability of a Borel set $A\subset E$ for the graph of the Brownian snake starting at $(0,0)$ is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass $δ_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for $A^c$) if and only if its capacity is infinite. As a di…

On the regularity of SLE trace

We revisit regularity of SLE trace, for all $κ\neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia--Rodemich--Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + κ/ 8, 2) $, improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.

双曲群上的 Besov 空间与随机游走：Dirichlet 型的边界迹与反射延拓

Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms

本文利用 Dirichlet 型理论证明低共形维数双曲群随机游走的无穷远迹过程存在并关联反射随机游走，同时建立相关 Besov 空间同构并分析调和测度正则性。

SDEs with additive stable noise和singular drift的弱误差：在相同的空间中选择测试函数！

Weak error for SDEs with additive stable noise and singular drift: choose the test function in the same space as the drift!

本文研究了具有加性稳定噪声和奇异漂移的SDE的弱误差，强调在相同空间中选择测试函数以提高收敛率。

Brownian regularity for the KPZ line ensemble

This paper seeks a quantitative comparison between the curves in the KPZ line ensemble [CH16] and a standard Brownian bridge under the $t^{1/3}$ vertical and $t^{2/3}$ horizontal scaling. The estimate we obtained is parallel to the one established in [Ham1], where the Airy line ensemble was studied. Our main tool is the soft Brownian Gibbs property enjoyed by the KPZ line ensemble. In view of the Gibbs property, the KPZ line ensemble differs from the Airy line ensemble mainly due to the intersecting nature of its curves, which results in the main technical difficulty in this paper. We develop …

分数布朗运动自相交局部时的高阶导数的存在性、重整化及正则性性质

Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

利用Wiener混沌展开，重新证明了分数布朗运动自相交局部时高阶导数的存在性，并引入Varadhan型重整化以扩展偶次导数的收敛范围。

Stochastic integral representation and regularity of the density for the Exit measure of super-Brownian motion

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1+β)-stable branching mechanism. It establishes the continuity of the density in dimension d=2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu=u^{1+β}.

KPZ固定点的定量布朗运动正则性：任意初始数据

Quantitative Brownian regularity of the KPZ fixed point with arbitrary initial data

研究KPZ固定点在任意初始数据下的空间增量与速率二布朗运动的强定量比较，证明其在紧集上的一致性，并应用得到空间增量的一侧大偏差不等式及中心KPZ固定点的韦纳密度熵有限性。

随机测度路径的正则性

Regularity of paths of stochastic measures

研究随机测度生成的随机函数，证明其连续路径的 Besov 正则性并给出 Fourier 级数展开，且所需条件弱于既有文献。

随机树Besov先验：数据驱动的正则化参数选择

Random tree Besov priors: Data-driven regularisation parameter selection

本文提出了一种基于随机树Besov先验的数据驱动正则化参数选择算法，用于提高贝叶斯反演的计算效率和表达能力。

带空间齐次有色噪声和从属布朗运动无穷小生成元的随机反应-扩散-平流方程的Sobolev正则性理论

Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions

该研究针对带空间齐次有色噪声与从属布朗运动无穷小生成元的非线性随机反应-扩散-平流方程，提出强强化Dalang条件，在混合范数空间下证明了解的存在唯一性并建立了时空Hölder型正则性。

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in $\mathbb{R}^d$ and a one-dimensional Brownian motion in $\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood-Paley functions which are obtained by the $α$-harmonic extension of an $L^p(\mathbb{R}^d)$ function.

On temporal regularity for strong solutions to stochastic $p$-Laplace systems

In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$ time differentiability in an expontential Besov-Orlicz space for the solution process $u$. Furthermore, we prove $1/2$ time differentiability of the nonlinear gradient $|\nabla u|^{\frac{p-2}{2}} \nabla u$ in a Nikolskii space.

球面上的各向同性 Q-分数布朗运动：正则性与快速模拟

Isotropic Q-fractional Brownian motion on the sphere: regularity and fast simulation

本文提出球面上各向同性 Q-分数布朗运动，证明其时空 Hölder 正则性，用谱方法近似并证明收敛，通过循环嵌入或条件化随机中点位移快速模拟，数值验证了时间精度与复杂度。

Time regularity of the densities for the Navier--Stokes equations with noise

We prove that the density of the law of any finite dimensional projection of solutions of the Navier--Stokes equations with noise in dimension $3$ is Hölder continuous in time with values in the natural space $L^1$. When considered with values in Besov spaces, Hölder continuity still holds. The Hölder exponents correspond, up to arbitrarily small corrections, to the expected diffusive scaling.

Multilevel Monte Carlo FEM for Elliptic PDEs with Besov Random Tree Priors

We develop a multilevel Monte Carlo (MLMC)-FEM algorithm for linear, elliptic diffusion problems in polytopal domain $\mathcal D\subset \mathbb R^d$, with Besov-tree random coefficients. This is to say that the logarithms of the diffusion coefficients are sampled from so-called Besov-tree priors, which have recently been proposed to model data for fractal phenomena in science and engineering. Numerical analysis of the fully discrete FEM for the elliptic PDE includes quadrature approximation and must account for a) nonuniform pathwise upper and lower coefficient bounds, and for b) low path-regu…