Viorel Barbu, Vladimir I. Bogachev, Giuseppe Da Prato, Michael Röckner
A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,ν)$ where $ν$ is an infinitesimally excessive measure. Then a weak solution is constructed.
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We obtain Calder{ó}n-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous coefficients. We then derive the well-posedness of the martingale problem associated to related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.
We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution.
In this paper we show how to derive regularity for the solution of Kolmogorov PIDEs driven by a vector field which is a second order integro differential operator of affine type. These results are valuable in applications, in particular for the analysis of the convergence rate of weak approximation schemes for affine processes.
The Porous Medium Equation: Large Deviations and Gradient Flow with Degenerate and Unbounded Diffusion
本文研究多孔介质方程的热力学梯度流结构推导问题，通过带退化无界扩散的缩放零程过程的大偏差分析，借助路径离散正则性估计得到超指数估计，证明该方程是熵在热力学度量下的梯度流。
The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising in self-organised criticality. Based on the so-called lower-bound method, the existence and uniqueness of an invariant measure is proved.
Doubling variables and uniqueness of probability solutions to degenerate stationary Kolmogorov equations
针对具有退化扩散矩阵的平稳Kolmogorov方程，通过直接应用随机分析中的变量加倍方法，得到其概率解唯一性的充分条件。
Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Even…
We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.
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.
We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins (From Probability to Geometry, vol. in honor of J.M Bismut (2009)) to prove uniqueness of the martingale problem in the framework of non degenerated elliptic operators and the Mc Kean and Singer (Journal of Diff. Geometry, 1967) parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi, Journal of Funct. Analysis,…
We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD 2) + Az, D , where A, B are N x N matrices, z $\in$ R N , N $\ge$ 1, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of L, namely for L + Tr(S(t)D 2) where S(t) is a non-negative N x N matrix depending continuously…
The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.
In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The proof uses the theory of viscosity solutions applied to the associated backward Kolmogorov equations. Under appropriate conditions on drift and diffusion coefficients, we will establish a comparison principle and a one-one correspondence between Feller solutions to the martingale problem and continuous viscosity solutions of the associated Kolmogorov equatio…
We provide here global Schauder-type estimates for a chain of integro-partial differential equations (IPDE) driven by a degenerate stable Ornstein-Uhlenbeck operator possibly perturbed by a deterministic drift, when the coefficients lie in some suitable anisotropic H{ö}lder spaces. Our approach mainly relies on a perturbative method based on forward parametrix expansions and, due to the low regularizing properties on the degenerate variables and to some integrability constraints linked to the stability index, it also exploits duality results between appropriate Besov Spaces. In particular, our…
About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations
本文研究含时可测二阶扩散扰动对退化 Ornstein-Uhlenbeck 型算子的影响，证明 Schauder 与 Sobolev 估计保持不变，并将已有局部结果推广到对称稳定型非局部扩散。
The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.
We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [Gonçalves-Landim-Toninelli '09] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from $0$ and $1$. In this paper we consider the more general…
In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previous…
We study the sensitivity of the densities of some Kolmogorov like degenerate diffusion processes with respect to a perturbation of the coefficients of the non-degenerate component. Under suitable (quite sharp) assumptions we quantify how the pertubation of the SDE affects the density. Natural applications of these results appear in various fields from mathematical finance to kinetic models.
The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.
We prove a priori estimates in $L_\infty$ for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constant $\varepsilon$ and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation.
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are prese…
Hypocoercivity for infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise
本文研究无穷维非线性退化随机微分方程，建立生成算子的耗散性并构造解过程，应用次协调性方法证明指数收敛，推广至反应-扩散及Cahn-Hilliard型方程。
In this book we establish under suitable assumptions the uniqueness and existence of viscosity solutions of Kolmogorov backward equations for stochastic partial differential equations (SPDEs). In addition, we show that this solution is the semigroup of the corresponding SPDE. This generalizes the Feynman-Kac formula to SPDEs and establishes a link between solutions of Kolmogorov equations and solutions of the corresponding SPDEs. In contrast to the literature we only assume that the nonlinear part of the drift is Lipschitz continuous on bounded sets (and not globally Lipschitz continuous) and …
Non-degenerate U-empirical Kolmogorov-Smirnov tests are studied and their large deviation asymptotics under the null-hypothesis is described. Several examples of such statistics used for testing goodness-of-fit and symmetry are considered. It is shown how to calculate their local Bahadur efficiency.
In this article we develop a new abstract strategy for proving ergodicity with explicit computable rate of convergence for diffusions associated with a degenerate Kolmogorov operator L. A crucial point is that the evolution operator L may have singular and nonsmooth coefficients. This allows the application of the method e.g. to degenerate and singular particle systems arising in Mathematical Physics. As far as we know in such singular cases the relaxation to equilibrium can't be discussed with the help of existing approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type te…
We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach.
We develop a general technique to prove uniqueness of solutions for Fokker--Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker--Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part.
We analyze a class of partial differential equations that arise as "backwards Kolmogorov operators" in infinite population limits of the Wright-Fisher models in population genetics and in mathematical finance. These are degenerate elliptic operators defined on manifolds with corners. The classical example is the Kimura diffusion operator, which acts on functions defined on the simplex in R^n. We introduce anisotropic Holder spaces, and prove existence, uniqueness and regularity results for the heat and resolvent equations defined by this class of operators. This suffices to prove that the C^0-…
Let $z = (x,y) \in {\mathbb R}^d \times {\mathbb R}^{N-d}$, with $1 \le d < N$. We prove a priori estimates of the following type :$$\|Δ\_{x}^{\frac α2} v \|\_{L^p({\mathbb R}^N)} \lec\_p\Big \| L\_{x } v + \sum\_{i,j=1}^{N}a\_{ij}z\_{i}\partial\_{z\_{j}} v \Big \|\_{L^p({\mathbb R}^N)}, \;\; 1<p<\infty,$$for $v \in C\_0^{\infty}({\mathbb R}^N)$,where $L\_x$ is a non-local operator comparable with the ${\mathbb R}^d $-fractional Laplacian $Δ\_{x}^{\frac α2}$ in terms of symbols, $α\in (0,2)$. We require that when $L\_x$ is replaced by the classical ${\mathbb R}^d$-Laplacian $Δ\_{x}$, i.e., in …
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。
Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations
本文研究非线性无穷维退化随机微分方程，用预解式方法构造无穷维Langevin动力学弱解，结合亚耗散性方法与算子本质m-耗散性结果，证明L²指数遍历性，并应用于显式退化扩散方程。
We show the existence and uniqueness of strong solutions for stochastic differential equation driven by partial $α$-stable noise and partial Brownian noise with singular coefficients. The proof is based on the regularity of degenerate mixed type Kolmogorov equation.
Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures
本文提出一种不依赖度量结构的微分几何方法，将非线性Fokker--Planck--Kolmogorov方程识别为概率测度空间上的梯度流，明确熵函数与能量泛函，证明唯一性及梯度场逼近，涵盖经典与广义多孔介质方程。
The essential m-dissipativity for degenerate infinite dimensional stochastic Hamiltonian systems and applications
本文在$L^2(μ^Φ)$上建立了带乘性噪声的退化无限维随机哈密顿系统对应Kolmogorov算子的基本m-耗散性，并利用亚强制性方法构造$μ^Φ$-不变Hunt过程，进而证明指数遍历性，最终应用于退化随机反应扩散方程。
We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the $L^2$-convergence rate by using weak Poincaré inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and…
In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then…
Strong solutions to McKean-Vlasov SDEs associated to a class of degenerate Fokker-Planck equations with coefficients of Nemytskii-type
本文研究具有 Nemytskii 型系数的退化非线性 Fokker-Planck 方程，证明对应 McKean-Vlasov SDE 存在唯一强解，并揭示弱解可表示为驱动 Brown 运动的泛函。
We consider degenerate diffusion equations of the form $\partial_tp_t = Δf(p_t)$ on a bounded domain and subject to no-flux boundary conditions, for a class of nonlinearities $f$ that includes the porous medium equation. We derive for them a trajectorial analogue of the entropy dissipation identity, which describes the rate of entropy dissipation along every path of the diffusion. Our approach is based on applying stochastic calculus to the underlying probabilistic representations, which in our context are stochastic differential equations with normal reflection on the boundary. This trajector…
We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic ge…
Weak regularization by degenerate Lévy noise and its applications
本论文研究退化Lévy噪声的弱正则化，通过前向和后向parametrix方法建立Kolmogorov方程的Schauder估计和SDE的弱适定性，并探讨其在带摩擦动力学中的应用。
We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical b…
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to …
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