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Michael Rockner

Publications and source records attributed to Michael Rockner.

12 recordsLinked to original sources
arXiv2023

Homogenization of diffusions on the lattice ${\mathbf Z}^d$ with periodic drift coefficients; Application of logarithmic Sobolev inequality

A homogenization problem of infinite dimensional diffusion processes indexed by ${\mathbf Z}^d$ having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an homogenization property of the processes starting from an almost every arbitrary point in the state space with respect to an invariant measure is proved. This result is also interpreted as solution to a homogenization problem of infinite dimensional diffusions with random coefficients, which is essentially analogous to the known ones in finite dimensions.

math.PR↗
arXiv2021

Nonlinear Fokker-Planck equations with time-dependent coefficients

An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in $(0,\infty)\times \mathbb{R}$ $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space $H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and $b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$ It is proved also that, if $u_0$ is a density of a probability measure, so is $u(t,\cdot)$ for all $t\ge0$. Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that $u(t)$ is the density of its time marginal law. MSC: 60H15, 47H05, 47J05. Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential equation, Sobolev space, periodic solution.

math.AP↗
arXiv2019

Superposition principle for non-local Fokker-Planck operators

We prove the superposition principle for probability measure-valued solutions to non-local Fokker-Planck equations, which in turn yields the equivalence between martingale problems for SDEs with jumps and such non-local PDEs with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.

math.PR↗
arXiv2019

Linearization of Nonlinear Fokker-Planck Equations and Applications

We associate a coupled nonlinear Fokker-Planck equation on $\R^d$, i.e. with solution paths in $\scr P$, to a linear Fokker-Planck equation for probability measures on the product space $\R^d\times \scr P$, i.e. with solution paths in $\scr P(\R^d\times\scr P)$. We explicitly determine the corresponding linear Kolmogorov operator $\tilde{\mathbf{L}}_t$ using the natural tangent bundle over $\scr P$ with corresponding gradient operator $\nabla^\scr P$. Then it is proved that the diffusion process generated by $\tilde{\mathbf{L}}_t$ on $\R^d\times\scr P$ is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by $\tilde{\mathbf{L}}_t$ in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurebility conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of "Nemytskii-type". As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schr\"odinger type PDEs on $\R^d\times\scr P_2$, through the Feynman-Kac formula for the corresponding diffusion processes.

math.PR↗
arXiv2018

Conservative stochastic 2-dimensional Cahn-Hilliard equation

We consider the stochastic 2-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution $Y$ to the shifted equation (see (1.4) below), then $X:=Y+{Z}$ is the unique solution to stochastic Cahn-Hilliard equaiton, where ${Z}$ is the corresponding O-U process. Moreover, we use Dirichlet form approach in \cite{Albeverio:1991hk} to construct the probabilistically weak solution the the original equation (1.1) below. By clarifying the precise relation between the solutions obtained by the Dirichlet forms aprroach and $X$, we can also get the restricted Markov uniquness of the generator and the uniqueness of martingale solutions to the equation (1.1).

math.PR↗
arXiv2017

Stochastic Heat Equations with Values in a Riemannian Manifold

The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.

math.PR↗
arXiv2016

Ergodicity for the stochastic quantization problems on the 2D-torus

In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the $Φ^4_2$ quantum field on the torus in terms of its density under translation. We also deduce that the $Φ^4_2$ quantum field on the torus is an extreme point in the set of all $L$-symmetrizing measures, where $L$ is the corresponding generator.

math.PR↗
arXiv2016

The total variation flow perturbed by gradient linear multiplicative noise

We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show extinction in finite time with probability one. These kind of equations arise, e.g. in the use for simulation of image restoring techniques or for modeling turbulence.

math.PR↗
arXiv2014

Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions

In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.

math.PR↗
arXiv2013

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction-diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier-Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.

math.PR↗
arXiv2011

General Extinction Results for Stochastic Partial Differential Equations and Applications

Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a $\si$-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(μ)$. By using an Itô type inequality for the $H$-norm and an integrability condition for the hyperbound of the semigroup $P_t:=\e^{-Lt}$, general extinction results are derived for a class of continuous adapted processes on $H$. Main applications include stochastic and deterministic fast diffusion equations with fractional Laplacians. Furthermore, we prove exponential integrability of the extinction time for all space dimensions in the singular diffusion version of the well-known Zhang-model for self-organized criticality, provided the noise is small enough. Thus we obtain that the system goes to the critical state in finite time in the deterministic and with probability one in finite time in the stochastic case.

math.PR↗