Michael Röckner
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Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space Open
In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures $\{μ^i_0\}_{1\le…
The Leibenson process Open
Consider the Leibenson equation \begin{equation*} \partial_t u = Δ_p u^q, \end{equation*} where $Δ_p f = div(|\nabla f|^{p-2}\nabla f)$ for $p>1$ and $q>0$, which is a simultaneous generalization of the porous media and the $p$-Laplace equ…
On Nonlinear Markov Processes in the Sense of McKean Open
We study nonlinear time-inhomogeneous Markov processes in the sense of McKean’s (Proc Natl Acad Sci USA 56(6):1907–1911, 1966) seminal work. These are given as families of laws $$\mathbb {P}_{s,\zeta }$$ , $$s\ge 0$$ , on path …
Stochastic intrinsic gradient flows on the Wasserstein space Open
We construct stochastic gradient flows on the $2$-Wasserstein space $\mathcal P_2$ over $\mathbb R^d$ for energy functionals of the type $W_F(ρd x)=\int_{\mathbb R^d}F(x,ρ(x))d x$. The functions $F$ and $\partial_2 F$ are assumed to be loc…
Uniqueness of distributional solutions to the 2D vorticity Navier–Stokes equation and its associated nonlinear Markov process Open
In this work, we prove uniqueness of distributional solutions to 2D Navier–Stokes equations in vorticity form u_{t}-\nu\Delta u+\mathrm{div}(K(u)u)=0 on { (0,\infty)\times\mathbb{R}^{2} } with Radon measures as initial data, where K is the…
On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ Open
UDC 517.9 We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suit…
View article: McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels
McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels Open
We prove the existence and conditional uniqueness in the Krylov class for SDEs with singular divergence-free drifts in the endpoint critical Lorentz space $L^{\infty}(0,T; L^{d,\infty}(\mathbb{R}^d))$, $d \geqslant 2$, which particularly i…
Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations Open
We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport equatio…
Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows Open
Under suitable assumptions on $β:\mathbb{R}\!\to\!\mathbb{R}, \,D:\mathbb{R}^d\!\to\!\mathbb{R}^d$ and $b:\mathbb{R}^d\!\to\!\mathbb{R}$, the nonlinear Fokker-Planck equation $u_t-Δβ(u)+{\rm div}(Db(u)u)=0$, in $(0,\infty)\times\mathbb{R}^…
Recent Developments in Dirichlet Form Theory and Related Fields Open
Theory of Dirichlet forms is one of the main achievements in modern probability theory. It has numerous interactions with other areas of mathematics and sciences. The recent notable developments are its role in the study of Liouville Brown…
The stochastic evolution of an infinite population with logistic-type interaction Open
An infinite population of point entities dwelling in the habitat $X=\mathds{R}^d$ is studied. Its members arrive at and depart from $X$ at random. The departure rate has a term corresponding to a logistic-type interaction between the entit…
The energy-critical stochastic Zakharov system Open
This work is devoted to the stochastic Zakharov system in dimension four, which is the energy-critical dimension. First, we prove local well-posedness in the energy space $H^1\times L^2$ up to the maximal existence time and a blow-up alter…
Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups Open
In this paper we deal with the problem of characterizing those generalized Mehler semigroups that do correspond to càdlàg Markov processes, which is highly non-trivial and has remained open for more than a decade. Our approach is to recons…
$p$-Brownian motion and the $p$-Laplacian Open
In this paper we construct a stochastic process, more precisely, a (nonlinear) Markov process, which is related to the parabolic $p$-Laplace equation in the same way as Brownian motion is to the classical heat equation given by the (2-) La…
Operator semigroups in the mixed topology and the infinitesimal description of Markov processes Open
We define a class of not necessarily linear C 0 -semigroups (P t ) t≥0 on C b (E) (more generally, on C κ (E) := 1 κ C b (E), for some bounded function κ, which is the pointwise limit of a decreasing sequence of continuous functions) equip…
Variational inequalities and smooth-fit principle for singular stochastic control problems in Hilbert spaces Open
We consider a class of infinite-dimensional singular stochastic control problems. These can be thought of as spatial monotone follower problems and find applications in spatial models of production and climate transition. Let $(D,\mathcal{…
View article: Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations
Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations Open
Motivated by the probabilistic representation for solutions of the Navier-Stokes equations, we introduce a novel class of stochastic differential equations that depend on the entire flow of its time marginals. We establish the existence an…
View article: Propagation of chaos for moderately interacting particle systems related to singular kinetic Mckean-Vlasov SDEs
Propagation of chaos for moderately interacting particle systems related to singular kinetic Mckean-Vlasov SDEs Open
We study the propagation of chaos in a class of moderately interacting particle systems for the approximation of singular kinetic McKean-Vlasov SDEs driven by alpha-stable processes. Diffusion parts include Brownian (alpha=2) and pure-jump…
Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise Open
This work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator $$(- \Delta )^s$$ for $$s\in \left( \frac{1}{2},1\right) $$ . The uniqueness of Schwartz di…
SVI solutions to stochastic nonlinear diffusion equations on general measure spaces Open
We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator $L$ being the generator of a transient Diric…
Cameron--Martin Type Theorem for a Class of non-Gaussian Measures Open
In this paper, we study the quasi-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron--Martin space and derive the explicit Rad…
Homogenization of diffusions on the lattice ${\mathbf Z}^d$ with periodic drift coefficients; Application of logarithmic Sobolev inequality Open
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…
Uniqueness of distributional solutions to the 2D vorticity Navier-Stokes equation and its associated nonlinear Markov process Open
In this work we prove uniqueness of distributional solutions to $2D$ Navier-Stokes equations in vorticity form $u_t-νΔu+ div (K(u)u)=0$ on $(0,\infty)\times\mathbb{R}^2$ with Radon measures as initial data, where $K$ is the Biot-Savart ope…
Averaging principle and normal deviation for multi-scale SDEs with polynomial nonlinearity Open
We investigate three types of averaging principles and the normal deviation for multi-scale stochastic differential equations (in short, SDEs) with polynomial nonlinearity. More specifically, we first demonstrate the strong convergence of …
Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems Open
This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $Ψ(-Δ)$, where $Ψ$ is a Bernstein function. As applications, the existe…
Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures Open
We propose a general method to identify nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) as gradient flows on the space of probability measures on $\mathbb{R}^d$ with a natural differential geometry. Our notion of gradient fl…
On the restriction of a right process outside a negligible set Open
The objective of this paper is to examine the restriction of a right process on a Radon topological space, excluding a negligible set, and investigate whether the restricted object can induce a Markov process with desirable properties. We …