arXiv subjects
Publications and source records attributed to Michael Röckner.
One proves existence and uniqueness of strong solutions to stochastic porous media equations under minimal monotonicity conditions on the nonlinearity. In particular, we do not assume continuity of the drift or any growth condition at infinity.
In this paper, we prove the existence and uniqueness of a smooth solution to a tamed 3D Navier-Stokes equation in the whole space. In particular, if there exists a bounded smooth solution to the classical 3D Navier-Stokes equation, then this solution satisfies our tamed equation. Moreover, using this renormalized equation we can give a new construction for a suitable weak solution of the classical 3D Navier-Stokes equation introduced in [Scheffer: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys., 1977] and [Caffarelli, Kohn, Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 1982].
We develop a new method to uniquely solve a large class of heat equations, so-called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of Stroock--Varadhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic Navier--Stokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.
We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for $σ$-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called $N$-functions in the theory of Orlicz spaces.
We prove that for any semi-Dirichlet form $(ε, D(ε))$ on a measurable Lusin space $E$ there exists a Lusin topology with the given $σ$-algebra as the Borel $σ$-algebra so that $(ε, D(ε))$ becomes quasi-regular. However one has to enlarge $E$ by a zero set. More generally a corresponding result for arbitrary $L^p$-resolvents is proven.
We prove a priori estimates and, as sequel, existence of Euclidean Gibbs states for quantum lattice systems. For this purpose we develop a new analytical approach, the main tools of which are: first, a characterization of the Gibbs states in terms of their Radon-Nikodym derivatives under shift transformations as well as in terms of their logarithmic derivatives through integration by parts formulae, and second, the choice of appropriate Lyapunov functionals describing stabilization effects in the system. The latter technique becomes applicable since on the basis of the integration by parts formulae the Gibbs states are characterized as solutions of an infinite system of partial differential equations. Our existence result generalize essentially all previous ones. In particular, superquadratic growth of the interaction potentials is allowed and $N$-particle interactions for $N\in \mathbb{N}\cup \{\infty \}$ are included. We also develop abstract frames both for the necessary single spin space analysis and for the lattice analysis apart from their applications to our concrete models. Both types of general results obtained in these two frames should be also of their own interest in infinite dimensional analysis.
The large deviation principle is established for the distributions of a class of generalized stochastic porous media equations for both small noise and short time.
We provide an $N/V$-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on $\mathbb R^d$, $d \ge 1$. Starting point is an $N$-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset $Λ\subset {\mathbb R}^d$ with finite volume (Lebesgue measure) $V = |Λ| < \infty$. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above $N$-particle dynamic in $Λ$ as $N \to \infty$ and $V \to \infty$ such that $N/V \to ρ$, where $ρ$ is the particle density.
Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal decay for the solution of the classical (deterministic) porous medium equation.
Given a second order parabolic operator $$ Lu(t,x) :=\frac{\partial u(t,x)}{\partial t} + a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x) + b^i(t,x)\partial_{x_i}u(t,x), $$ we consider the weak parabolic equation $L^{*}μ=0$ for Borel probability measures on $(0,1)\times\mathbb{R}^d$. The equation is understood as the equality $$ \int_{(0,1)\times\mathbb{R}^d} Lu dμ=0 $$ for all smooth functions $u$ with compact support in~$(0,1)\times\mathbb{R}^d$. This equation is satisfied for the transition probabilities of the diffusion process associated with~$L$. We show that under broad assumptions $μ$ has the form $μ=\varrho(t,x) dt dx$, where the function $x\mapsto \varrho(t,x)$ is Sobolev, $|\nabla_x \varrho(x,t)|^2/\varrho(t,x)$ is Lebesgue integrable over $[0,τ]\times\mathbb{R}^d$, and $\varrho\in L^p([0,τ]\times\mathbb{R}^d)$ for all $p\in [1,+\infty)$ and $τ<1$. Moreover, a sufficient condition for the uniform boundedness of $\varrho$ on $[0,τ]\times\mathbb{R}^d$ is given.
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.
Various Poincare-Sobolev type inequalities are studied for a reaction-diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction-diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces $E^n (n \geq 1)$ which determine the motion of particles, and the reaction part induced by a $Q$-process on $\mathbb Z_+$ and a sequence of reference probability measures, where the $Q$-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincare and weak Poincare inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding $Q$-process. But under a mild condition, stronger inequalities rely on both parts: the reaction-diffusion Dirichlet form satisfies a super Poincare inequality (e.g. the log-Sobolev inequality) if and only if so do both the corresponding $Q$-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results.
Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of $b$ is assumed. It is known that then there exists a unique probability measure $μ$ satisfying the equation $L^*μ=0$ and that the closure of $L$ in $L^1(μ)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the resolvent $\{G_λ\}_{λ> 0}$. We prove that, for any Lipschitzian function $f\in L^1(μ)$ and all $t,λ>0$, the functions $T_tf$ and $G_λf$ are Lipschitzian and |\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_λf(x)| \leq \frac{1}λ G_λ|\nabla f|(x). An analogous result is proved in the parabolic case.
We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(Γ_X,π)$, where $Γ_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $π$ is a Poisson measure on $Γ_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels $(P_t)_{t\ge 0}$ and, using it, we prove the main results of the paper: the Feller property of the semigroup $(P_t)_{t\ge 0}$ with respect to the vague topology on the configuration space $Γ_X$, and the ergodic property of $(P_t)_{t\ge 0}$. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every $γ\inΓ_X$, will never leave $Γ_X$ and has cadlag sample paths in $Γ_X$.
A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $Γ$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $μ$ as an invariant measure. We assume that $μ$ corresponds to a symmetric pair potential $ϕ(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_μ^Γ$ on $L^2(Γ;μ)$, and under general conditions on the potential $ϕ$, prove its closability. For a potential $ϕ$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_μ^Γ$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_μ^Γ$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.
The purpose of this paper is to provide a both comprehensive and summarizing account on recent results about analysis and geometry on configuration spaces $Γ_X$ over Riemannian manifolds $X$. Particular emphasis is given to a complete description of the so--called ``lifting--procedure'', Markov resp. strong resp. $L^1$--uniqueness results, the non--conservative case, the interpretation of the constructed diffusions as solutions of the respective classical ``heuristic'' stochastic differential equations, and a self--contained presentation of a general closability result for the corresponding pre--Dirichlet forms. The latter is presented in the general case of arbitrary (not necessarily pair) potentials describing the singular interactions. A support property for the diffusions, the intrinsic metric, and a Rademacher theorem on $Γ_X$, recently proved, are also discussed.
We consider an $L^2$-Wasserstein type distance $ρ$ on the configuration space $Γ_X$ over a Riemannian manifold $X$, and we prove that $ρ$-Lipschitz functions are contained in a Dirichlet space associated with a measure on $Γ_X$ satisfying some general assumptions. These assumptions are in particular fulfilled by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of $ρ$ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that $ρ$ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.
Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on a--priori estimates is used. The extension of the problem to the $L^p$-setting is discussed. As a direct application essential self--adjointness and strong uniqueness in $L^p$ is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume $Λ\subset \R^2$.