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Michael Röckner

Publications and source records attributed to Michael Röckner.

At least 19 recordsLinked to original sources
arXiv2026

Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)β(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $β$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv δ_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

math.PR↗
arXiv2026

On Solutions to Graphon McKean-Vlasov SDEs of Nemytskii-type

We study an uncountable system of McKean-Vlasov SDEs with coefficients of Nemytskii-type which are driven by a family of essentially pairwise independent Wiener processes. These SDEs interact through a Graphon kernel by means of their one-dimensional time marginal law densities evaluated in the spatial coordinate. We prove the existence and uniqueness of probabilistically weak solutions to such SDEs under mild conditions on the drift coefficient and the Graphon kernel. Furthermore, we prove that these solutions are, in fact, probabilistically strong by essentially proving a (restricted) Yamada--Watanabe theorem on Fubini extension spaces. For the associated system of nonlinear Fokker-Planck equations, we prove a new uniqueness result where we can allow for density dependent diffusion coefficients of Nemytskii-type.

math.PR↗
arXiv2026

Regularization of the superposition principle: Potential theory meets Fokker-Planck equations

For a solution to a (possibly nonlinear) Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this work is to go a substantial further step and, by exploiting the superposition principle, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process, which is a particularly useful class of Markov processes, enjoying among other regularity properties the strong Markov property, which is fundamental for the analysis of the underlying FPE as a (nonlinear) parabolic PDE by probabilistic tools. As two main applications we construct fundamental flow solutions for the FPE and we prove a well-posedeness result for the parabolic Dirichlet problem through probabilistic means for more general coefficients than could be treated in the existing literature. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. The validity of the strong Markov property in the context of the superposition principle was an open problem even in the linear case. In this paper we solve this also in the nonlinear case, i.e. for path laws of solutions to McKean-Vlasov SDEs with Nemytskii type coefficients. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE.

math.PR↗
arXiv2026

The stochastic evolution of an infinite population with logistic-type interaction

An infinite population of point entities dwelling in the habitat $X=\mathds{R}^d$ is studied. Its members arrive in and depart from $X$ at random. The departure rate has a term corresponding to a logistic-type interaction between the entities. Thereby, the corresponding Kolmogorov operator $L$ has an additive quadratic term, which usually produces essential difficulties in its study. The population pure states are locally finite counting measures defined on $X$. The set of such states $Γ$ is equipped with the vague topology, which allows one to use probability measures defined thereon. The population evolution is described at two levels. At the first level, one deals with the Fokker-Planck equation for $(L,\mathcal{F},μ_0)$ where $\mathcal{F}$ is an appropriate set of bounded test functions $F:Γ\to \mathds{R}$ (domain of $L$) and $μ_0$ is an initial state, which is supposed to belong to the set $\mathcal{P}_{\rm exp}$ of sub-Poissonian probability measures on $Γ$. We prove that the Fokker-Planck equation has a unique solution $t\mapstoμ_t$, which belongs to $\mathcal{P}_{\rm exp}$. Some of the properties of this solution are also described. The second-level description yields a Markov process with cadlag paths such that its one-dimensional marginals coincide with the mentioned states $μ_t$. The process is obtained as the unique solution of the corresponding martingale problem. The results obtained are discussed and compared with those known for similar models with logistic-type interactions.

math.PR↗
arXiv2026

Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems

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 existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs.

math.AP↗
arXiv2026

Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations

We prove existence of a probability solution to the nonlinear stationary Fokker-Planck-Kolmogorov equation on an infinite dimensional space with a centered Gaussian measure $γ$ with a unit diffusion operator and a drift of the form $-x+v(p,x)$, where $v$ is a bounded mapping with values in the Cameron-Martin space $H$ of $γ$ and $v$ is defined on the space $E\times X$, where is $E$ is the subset of $L^2(γ)$ consisting of probability densities. The equation has the form $L_{b(p,\bullet)} ^*(p\cdot γ)=0$ with $L_{b(p,\bullet)}φ=Δ_H φ+ (b(p,\bullet) , D_{_H}φ)_{_H}$, so that the drift coefficient depends on the unknown solution, which makes the equation nonlinear. This dependence is assumed to satisfy a suitable continuity condition. This result is applied to drifts of Vlasov type defined by means of the convolution of a vector field with the solution. In addition, we consider a more general situation where only the components of $v$ are uniformly bounded and prove the existence of a probability solution under some stronger continuity condition on the drift.

math.AP↗
arXiv2026

Probabilistic representation of solutions to the parabolic $p$-Laplace equation

This work is concerned with the probabilistic representation of solutions to the $p$-Laplace evolution equation $\frac{\partial u}{\partial t}={\rm div}(|\nabla u|^{p-2}\nabla u)$ in $(0,\infty)\times\mathbb{R}^d$, $u(0,x)=u_0(x),$ $x\in\mathbb{R}^d$. One proves that, if $p\geq 4$, and if $u_0$ is a probability density with compact support and $u_0\in L^2$, $|\nabla u_0|\in L^\infty$, then $u$ can be represented as $u(t,x)dx=\mathscr L_{X(t)}(dx)$, where $\mathscr L_{X(t)}$ denotes the time marginal law of $X$ at time $t$ with $X$ being a probabilistically weak solution to a corresponding McKean-Vlasov stochastic differential equation. This result is based on a new second order global regularity result for the weak solutions to the parabolic $p$-Laplace equation.

math.AP↗
arXiv2026

Stochastic intrinsic gradient flows on the Wasserstein space

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 locally Lipschitz on $\mathbb R^d\times (0,\infty)$. This includes the relevant examples of $W_F$ as the entropy functional or more generally the Lyapunov function of generalized porous media equations. First we define a class of Gaussian-based measures $Λ$ on $\mathcal P_2$ together with a corresponding class of symmetric Markov processes ${(R_t)}_{t\geq 0}$. Next, using Dirichlet form techniques we perform stochastic quantization for the perturbations of these objects which result from multiplying such a measure $Λ$ by a density proportional to $e^{-W_F}$. Finally we show that the intrinsic gradient $DW_F(μ)$ is defined for $Λ$-a.e. $μ$ and that the Gaussian-based reference measure $Λ$ can be chosen in such way that the distorted process ${(μ_t)}_{t\geq 0}$ is a martingale solution for the equation $dμ_t=-DW_F(μ_t) d t+d R_t$, $t\geq 0$.

math.PR↗
arXiv2026

Variational inequalities and smooth-fit principle for singular stochastic control problems in Hilbert spaces

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 $(D,\mathcal{M},μ)$ be a finite measure space and consider the Hilbert space $H:=L^2(D,\mathcal{M},μ; \mathbb{R})$. Let then $X$ be an $H$-valued stochastic process on a suitable complete probability space, whose evolution is determined through an SPDE driven by a self-adjoint linear operator $\mathcal{A}$ and affected by a cylindrical Brownian motion. The evolution of $X$ is controlled linearly via an $H$-valued control consisting of the direction and the intensity of action, a real-valued nondecreasing right-continuous stochastic process, adapted to the underlying filtration. The goal is to minimize a discounted convex cost-functional over an infinite time-horizon. By combining properties of semiconcave functions and techniques from viscosity theory, we first show that the value function of the problem $V$ is a {$C^{1,\mathrm{Lip}}(H)$}-viscosity solution to the corresponding dynamic programming equation, which here takes the form of a variational inequality with gradient constraint. Then, by allowing the decision maker to choose only the intensity of the control and requiring that the given control direction $\hat{n}$ is an eigenvector of the linear operator $\mathcal{A}$, we establish that the directional derivative $V_{\hat{n}}$ is of class $C^1(H)$, hence a second-order smooth-fit principle in the controlled direction holds for $V$. This result is obtained by exploiting a connection to optimal stopping and combining results and techniques from convex analysis and viscosity theory.

math.OC↗
arXiv2025

McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular Lorentz kernels

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 includes the $2$D Biot-Savart law. The uniqueness result is shown to be optimal in dimensions $d \geqslant 3$, by constructing different martingale solutions in the case of supercritical Lorentz drifts. As a consequence, the well-posedness of McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular kernels is derived. In particular, this yields the uniqueness of the $2$D vorticity Navier-Stokes equations even in certain supercritical-scaling spaces. Furthermore, we prove that the path laws of solutions to McKean-Vlasov equations with critical singular kernel form a nonlinear Markov process in the sense of McKean.

math.PR↗
arXiv2025

Multi-solitons to focusing mass-supercritical stochastic nonlinear Schrödinger equations

We consider the stochastic nonlinear Schrödinger equation driven by linear multiplicative noise in the mass-supercritical case. Given arbitrary $K$ solitary waves with distinct speeds, we construct stochastic multi-solitons pathwisely in the sense of controlled rough path, which behave asymptotically as the sum of the $K$ prescribed solitons as time tends to infinity. In contrast to the mass-(sub)critical case in \cite{RSZ23}, the linearized Schrödinger operator around the ground state has more unstable directions in the supercritical case. Our pathwise construction utilizes the rescaling approach and the modulation method in \cite{CMM11}. We derive the quantitative decay rates dictated by the noise for the unstable directions, as well as the modulation parameters and remainder in the geometrical decomposition. They are important to close the key bootstrap estimates and to implement topological arguments to control the unstable directions. As a result, the temporal convergence rate of stochastic multi-solitons, which can be of either exponential or polynomial type, is related closely to the spatial decay rate of the noise and reflects the noise impact on soliton dynamics.

math.PR↗
arXiv2025

Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

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 $\{μ^i_0\}_{1\leq i\leq M}$ on $\mathbb{R}^d$ and $\frac{d}{p}+\frac{1}{r}>1$, we construct a divergence-free drift field $v\in L_t^rL^p\cap C_tL^{d-}$ such that the associated SDE admits at least two distinct weak solutions originating from each initial measure $μ^i_0$. This result is sharp in view of the well-known uniqueness of strong solutions for drifts in $C_tL^{d+}$, as established in \cite{KR05}. As a corollary, there exists a measurable set $A\subset\mathbb{R}^d$ with positive Lebesgue measure such that for any $x\in A$, the SDE with drift $v$ admits at least two weak solutions when with start in $x\in A$. The proof proceeds by constructing two distinct probability solutions to the associated Fokker-Planck equation via a convex integration method adapted to all of $\mathbb{R}^d$ (instead of merely the torus), together with refined heat kernel estimate.

math.PR↗
arXiv2025

The Leibenson process

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 equation. In this paper we identify the Leibenson equation as a nonlinear Fokker--Planck equation and prove that it has a nonlinear Markov process in the sense of McKean as its probabilistic counterpart. More precisely, we obtain a probabilistic representation of its Barenblatt solutions as the one-dimensional marginal density curve of the unique solutions to the associated McKean--Vlasov SDE. The latter is of novel type, since its coefficients depend pointwise both on its solution's time marginal densities and also on their first and second order derivatives. Moreover, we show that these solutions constitute the aforementioned nonlinear Markov process, which we call the Leibenson process. A further main result of this work is to prove that despite the strong degeneracy of the diffusion and the irregularity of the drift coefficient (which is merely of bounded variation) of the McKean--Vlasov SDE these solutions are probabilistically strong, i.e., measurable functionals of the driving Brownian motion and the initial condition.

math.PR↗
arXiv2025

Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{align*} are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process. Such SPDEs include many interesting models in applied fields like fluid dynamics etc. In this paper, we establish the well-posedness of the above SPDEs under fully local monotonicity condition solving a longstanding open problem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowed to depend on both the $H$-norm and $V$-norm. In the case of classical SPDEs, this means that $B(\cdot,\cdot)$ could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.

math.PR↗
arXiv2025

SDEs with critical time dependent drifts: strong solutions

Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the unique strong solvability of time-inhomogeneous stochastic differential equations with drift coefficients in critical Lebesgue spaces, which gives an affirmative answer to a longstanding open problem. As an application, we also prove a regularity criterion for solutions of a stochastic system proposed by Constantin and Iyer (Comm. Pure. Appl. Math. 61(3): 330-345, 2008), which is closely related to the Navier-Stokes equations.

math.PR↗
arXiv2025

Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

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 equations advected by weak solutions of the 3D Euler equations that exhibit kinetic energy dissipation with $C_{t,x}^{1/3-}$ regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension $d\geq2$, for any $\frac{1}{p}+\frac{1}{r}>1$ or $p\in(1,2),r=\infty$, we construct solutions to the Euler or Navier-Stokes equations in the space $L_t^rL^p\cap L_t^1W^{1,1}$, demonstrating that the associated (stochastic) Lagrangian trajectories are not unique. Our result is sharp in 2D in the sense that: (1) in the stochastic case, for any vector field $v\in C_tL^p$ with $p>2$, the associated stochastic Lagrangian trajectory associated with $v$ is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution $v\in C_tL^p$ with $p>2$ to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension $d\geq2$ in the sense that for any divergence-free vector field $v\in L_t^1W^{1,s}$ with $s>d$, the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

math.AP↗
arXiv2025

Nonlinear Fokker-Planck equations as smooth Hilbertian gradient flows

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 $(0,\infty)\times\mathbb{R}^d$ where $D=-\nablaΦ$, can be identified as a smooth gradient flow $\frac{d^+}{dt}\,u(t)+\nabla E_{u(t)}=0$, $\forall t>0$. Here, $E:\mathcal{P}^*\cap L^\infty(\mathbb{R}^d)\to\mathbb{R}$ is the energy function associated to the equation, where $\mathcal{P}^*$ is a certain convex subset of the space of probability densities. $\mathcal{P}^*$ is invariant under the flow and $\nabla E_u$ is the gradient of $E$, that is, the tangent vector field to $\mathcal{P}$ at $u$ defined by $\left<\nabla E_u,z_u\right>_u={\rm diff}\,E_u\cdot z_u$ for all vector fields $z_u$ on $\mathcal{P}^*$, where $\left<\cdot,\cdot\right>_u$ is a scalar product on a suitable tangent space $\mathcal{T}_u(\mathcal{P}^*)\subset\mathcal{D}'(\mathbb{R}^d)$.

math.PR↗