We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed by Baudoin, Driver, Gordina, and Melcher previously, these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state.
We prove smoothing properties and optimal Schauder type estimates for a class of nonautonomous evolution equations driven by time dependent Ornstein-Uhlenbeck operators in a separable Hilbert space. They arise as Kolmogorov equations of linear nonautonomous stochastic differential equations with Gaussian noise.
In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are $n$-times continuously Fréchet differentiable, then so are the generalized solutions at every positive time. In addition, a key contribution of this work is to prove suitable enhanced regularity properties for the derivatives of the generalized solutions of the Kolmogorov equations in the sense that the dominating linear operato…
In this paper, a Banach space framework is introduced in order to deal with finite-dimensional path-dependent stochastic differential equations. A version of Kolmogorov backward equation is formulated and solved both in the space of $L^p$ paths and in the space of continuous paths using the associated stochastic differential equation, thus establishing a relation between path-dependent SDEs and PDEs in analogy with the classical case. Finally, it is shown how to establish a connection between such Kolmogorov equation and the analogue finite-dimensional equation that can be formulated in terms …
Infinite-horizon time-inhomogeneous Kolmogorov equation and non-autonomous multi-scale stochastic systems
School of Mathematics and Statistics, Jiangsu Normal University（江苏师范大学数学与统计学院）；School of Mathematics and Big Data, Anhui University of Science and Technology（安徽理工大学数学与大数据学院）；School of Mathematical Sciences, Tiangong University（天津工业大学数学科学学院）
We consider the nonlinear Kolmogorov equation posed in a Hilbert space $H$, not necessarily of finite dimension. This model was recently studied by Cox et al. [24] in the framework of weak convergence rates of stochastic wave models. Here, we propose a complementary approach by providing an infinite-dimensional Deep Learning method to approximate suitable solutions of this model. Based in the work by Hure, Pham and Warin [45] concerning the finite dimensional case, and our previous work [20] dealing with Lévy based processes, we generalize an Euler scheme and consistency results for the Forwar…
We prove a version of the Ambrosio-Figalli-Trevisan superposition principle for a restricted subclass of solutions to the Fokker-Planck-Kolmogorov equation, that is valid on separable infinite-dimensional Hilbert spaces. Furthermore, we transfer this restricted superposition principle into a nonlinear setting.
We study the law of the iterated logarithm (Khinchin (1924), Kolmogorov (1929)) and related strong invariance principles in stochastic geometry. As potential applications, we think of well-known functionals such as functionals defined on the $k$-nearest neighbors graph and important functionals in topological data analysis such as the Euler characteristic and persistent Betti numbers.
Let $S(n)$ be a centered random walk with finite second moment. We consider the integrated random walk $T(n) = S(0)+S(1)+\dots+S(n)$. We prove invariance principles for the meander and for the bridge of this process, under the condition that the integrated random walk remains positive. Furthermore, we prove the functional convergence of its Doob's $h$-transform to the $h$-transform of the Kolmogorov diffusion conditioned to stay positive.
We consider the translation dynamics on the $C^*$-algebra $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ of two sided infinite tensor product of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$ and its restriction to the maximal abelian $C^*$ sub-algebra $\ID^e = \otimes_{n \in \IZ}\!D_e^{(n)}(\IC)$ of $\IM$, where each $\!D_e^{(n)}(\IC)=\!D_d(\IC)$ is the algebra of $d$ dimensional diagonal matrices with respect to an orthonormal basis $e=(e_i)$ of $\IC^d$. We prove that any two translation invariant Kolmogorov pure states of $\IM$ give unitarily equivale…
Moscow Lomonosov State University（莫斯科国立罗蒙诺索夫大学）；National Research University “Higher School of Economics”（国立高等经济大学）；Bielefeld University, Germany（德国比勒费尔德大学）
$L^1$-Uniqueness of Kolmogorov Operators Associated to 2D Stochastic Navier-Stokes Coriolis Equations with Space-Time White Noise
我们考虑Kolmogorov算子$K$与时空白噪声驱动的随机斯托克斯方程在二维环面上的周期边界条件和一个旋转的参考框架，引入虚拟的力量，如科里奥利力。这个方程可以作为地球物理流动的一个简单模型。我们证明了$K$在基于非线性的柱检验函数上由拟能诱导的高斯测度是无穷小不变的，并且$K$是$L^1$ -唯一的. R. t.足够大的粘度的涡度拟能测量。
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…
One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even…
We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortex dynamics. We als…
利用[G. R. Pantsulaia，关于无限维矩形中有限集的递增族的均匀分布序列，Real Anal。交易所36（2）（2010/2011），325--340 ]，本文介绍了一种新的无穷维Monte-Carlo积分方法，并得到了一些无穷维强律型定理的有效性.此外，利用单位区间上均匀分布序列的性质，得到了Kolmogorov强大数律的一个新的证明，它与原来的证明有本质的区别.
For Kolmogorov equations associated to finite dimensional stochastic differential equations (SDEs) in high dimension, a numerical method alternative to Monte Carlo simulations is proposed. The structure of the SDE is inspired by stochastic Partial Differential Equations (SPDE) and thus contains an underlying Gaussian process which is the key of the algorithm. A series development of the solution in terms of iterated integrals of the Gaussian process is given, it is proved to converge - also in the infinite dimensional limit - and it is numerically tested in a number of examples.
We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.
This paper presents a novel method for computing certain particle locations in the multi-species $q$-TAZRP (totally asymmetric zero range process). The method is based on a decomposition of the process into its discrete-time embedded Markov chain, which is described more generally as a monotone process on a graded partially ordered set; and an independent family of exponential random variables. A further ingredient is explicit contour integral formulas for the transition probabilities of the $q$-TAZRP. The main result of this method is a shift invariance for the multi-species $q$-TAZRP on the …
We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution. For a first local approximation we assign a manifold structure to the family and a metric. We then project the vector field of the partial differential equation (PDE) onto the tangent space of the chosen family, thus obtaining an ordinary differential equation for the family parameter. A second global approximation will be based on projecting directly the exac…
We consider a stationary and ergodic random field $\{ω(e) : e \in E_d\}$ that is parameterized by the edge set of the Euclidean lattice $\mathbb{Z}^d$, $d \geq 2$. The random variable $ω(e)$, taking values in $[0, \infty)$ and satisfying certain moment bounds, is thought of as the conductance of the edge $e$. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster $\mathcal{C}_{\infty}(ω)$, we prove a quenched invariance principle for the continuous-time random walk among random conductances under relatively mild conditions on the structure of the infin…
Consider the radial nonlinear wave equation $-\partial_t^2 u + Δu = u^3$, $u :\mathbb{R}_t \times \mathbb{R}_x^3 \to \mathbb{R}$, $u(t,x) = u(t,|x|)$. In this paper, we construct a Gibbs measure for this system and prove its invariance under the flow of the NLW. In particular, we are in the infinite volume setting. For the finite volume analogue, specifically on the unit ball with zero boundary values, an invariant Gibbs measure was constructed by Burq, Tvetkov, and de Suzzoni as a Borel measure on super-critical Sobolev spaces. In this paper, we advocate that the finite volume Gibbs measure b…
Consider an infinite sequence $(U_n)_{n\in\mathbb{N}}$ of independent Cauchy random variables, defined by a sequence $(δ_n)_{n\in\mathbb{N}}$ of location parameters and a sequence $(γ_n)_{n\in\mathbb{N}}$ of scale parameters. Let $(W_n)_{n\in\mathbb{N}}$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $(σ_nγ_n)_{n\in\mathbb{N}}$ of scale parameters, with $σ_n\neq 0$ for all $n\in\mathbb{N}$. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $(U_n)_{n…
Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The gene…
