arXiv subjects

Marco Rehmeier

Publications and source records attributed to Marco Rehmeier.

17 recordsLinked to original sources

arXiv2026

Brownian motion in Minkowski normed spaces

A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean--Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. Furthermore, we show that solutions to this stochastic differential equation are pathwise unique, and thus probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process \emph{Brownian motion in Minkowski normed spaces.} To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.

math.PR↗

arXiv2026

Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs

We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker--Planck equations and by constructing, for each of these equations, a solution to the associated McKean--Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier--R\"ockner (J.\,Theor.\,Probab. 38, 60 (2025)) are not uniquely determined by their one-dimensional time marginals. This is in sharp contrast to the case of classical Markov processes, which are uniquely determined by their one-dimensional time marginals. We demonstrate our results by constructing a continuum of nonlinear Markov processes with one-dimensional time marginal densities given by the Barenblatt solutions to the porous medium and $p$-Laplace equations, as well as by the fundamental solution to the heat equation. This includes a novel martingale representation for the $p$-Laplace Barenblatt solutions. We also prove that a nonlinear Markov process is uniquely determined by its two-dimensional time marginals. Moreover, for the porous medium equation, we show that the different McKean--Vlasov SDEs we investigate are consistent with corresponding gradient flow interpretations of the equation in the sense of Otto calculus.

math.PR↗

arXiv2025

The Leibenson process

Consider the Leibenson equation \begin{equation*} \partial_t u = \Delta_p u^q, \end{equation*} where $\Delta_p f = div(|\nabla f|^{p-2}\nabla f)$ 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↗

arXiv2024

Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise

For the $3D$ fractional Navier--Stokes equations perturbed by transport noise, we prove the existence of infinitely many H\"older continuous analytically weak, probabilistically strong Leray--Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,\tau]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force $f$ in the equation. In this unforced regime, we prove the first result of Leray--Hopf nonuniqueness for fractional Navier--Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients.

math.AP↗

arXiv2024

$p$-Brownian motion and the $p$-Laplacian

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-) Laplacian.

math.PR↗

arXiv2024

$2D$ vorticity Euler equations: Superposition solutions and nonlinear Markov processes

In this note we contribute two results to the theory of the $2D$ Euler equations in vorticity form on the full plane. First, we establish a generalized Lagrangian representation of weak (in general measure-valued) solutions, which includes and extends classical results on the Lagrangianity of weak solutions. Second, we construct nonlinear Markov processes which are uniquely determined by a selection of weak solutions from initial data in $L^1\cap L^p$, $p \geq 2$, and related spaces such as the classical and uniformly localized Yudovich space. It is well-known that for $p <\infty$ weak solutions are in general not unique, which renders a suitable selection nontrivial.

math.AP↗

arXiv2024

Remarks on regularization by noise, convex integration and spontaneous stochasticity

This note is devoted to a discussion of the potential links and differences between three topics: regularization by noise, convex integration, spontaneous stochasticity. All of them deal with the effect on large scales of a small-scale perturbation of fluid dynamic equations. The effects sometimes have something in common, like convex integration and spontaneous stochasticity, sometimes they look the opposite, as in regularization by noise. We are not aware of rigorous links or precise explanations of the differences, and hope to drive new research with this comparative examination.

math.PR↗

arXiv2024

Average dissipation for stochastic transport equations with L\'evy noise

We show that, in one spatial and arbitrary jump dimension, the averaged solution of a Marcustype SPDE with pure jump L\'evy transport noise satisfies a dissipative deterministic equation involving a fractional Laplace-type operator. To this end, we identify the correct associated L\'evy measure for the driving noise. We consider this a first step in the direction of a non-local version of enhanced dissipation, a phenomenon recently proven to occur for Brownian transport noise and the associated local parabolic PDE by the first author. Moreover, we present numerical simulations, supporting the fact that dissipation occurs for the averaged solution, with a behavior akin to the diffusion due to a fractional Laplacian, but not in a pathwise sense.

math.PR↗

arXiv2023

Weighted $L^1$-semigroup approach for nonlinear Fokker--Planck equations and generalized Ornstein--Uhlenbeck processes

For the nonlinear Fokker--Planck equation $$\partial_tu = \Delta\beta(u)-\nabla \Phi \cdot \nabla \beta(u) - div_{\varrho}\big(D(x)b(u)u\big),\quad (t,x) \in (0,\infty)\times \mathbb{R}^d,$$ where $\varrho = \exp(-\Phi)$ is the density of a finite Borel measure and $\nabla \Phi$ is unbounded, we construct mild solutions with bounded initial data via the Crandall--Liggett semigroup approach in the weighted space $L^1(\mathbb{R}^d,\mathbb{R};\varrho dx)$. By the superposition principle, we lift these solutions to weak solutions to the corresponding McKean--Vlasov SDE, which can be considered a model for generalized nonlinear perturbed Ornstein--Uhlenbeck processes. Finally, for these solutions we prove the nonlinear Markov property in the sense of McKean.

math.AP↗

arXiv2023

Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures

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 flow does not depend on any underlying metric structure such as the Wasserstein distance, but is derived from purely differential geometric considerations. We explicitly identify the associated entropy functions, which are not necessarily convex, and also the corresponding energy functionals, in particular their domains of definition. Furthermore, the latter functions are Lyapunov functions for the solutions of the FPK equations. Moreover, we show uniqueness results for such gradient flows, and we also prove that the gradient of $E$ is a gradient field on $\mathbb{R}^d$, which can be approximated by smooth gradient fields. These results cover classical and generalized porous media equations, where the latter have a generalized diffusivity function and a nonlinear transport-type first-order perturbation.

math.AP↗

arXiv2023

Flow selections for (nonlinear) Fokker-Planck-Kolmogorov equations

We provide a method to select flows of solutions to the Cauchy problem for linear and nonlinear Fokker--Planck--Kolmogorov equations (FPK equations) for measures on Euclidean space. In the linear case, our method improves similar results of a previous work of the author. Our consideration of flow selections for nonlinear equations, including the particularly interesting case of Nemytskii-type coefficients, seems to be new. We also characterize the (restricted) well-posedness of FPK equations by the uniqueness of such (restricted) flows. Moreover, we show that under suitable assumptions in the linear case such flows are Markovian, i.e. they fulfill the Chapman-Kolmogorov equations.

math.PR↗

arXiv2022

On nonlinear Markov processes in the sense of McKean

We study nonlinear time-inhomogeneous Markov processes in the sense of McKean's seminal work [32]. These are given as families of laws $\mathbb{P}_{s,\zeta}$, $s\geq 0$, on path space, where $\zeta$ runs through a set of admissible initial probability measures on $\mathbb{R}^d$. In this paper, we concentrate on the case where every $\mathbb{P}_{s,\zeta}$ is given as the path law of a solution to a McKean-Vlasov SDE, where the latter is allowed to have merely measurable coefficients, which in particular are not necessarily weakly continuous in the measure variable. Our main result is the identification of general and checkable conditions on such general McKean-Vlasov SDEs, which imply that the path laws of their solutions form a nonlinear Markov process. Our notion of nonlinear Markov property is in McKean's spirit, but more general in order to include processes whose one-dimensional time marginal densities solve a nonlinear parabolic PDE, more precisely, a nonlinear Fokker-Planck-Kolmogorov equation, such as Burgers' equation, the porous media equation and variants thereof with transport-type drift, and also the very recently studied $2D$ vorticity Navier-Stokes equation and the $p$-Laplace equation. In all these cases, the associated McKean-Vlasov SDEs are such that both their diffusion and drift coefficients singularly depend (i.e. Nemytskii-type) on the one-dimensional time marginals of their solutions. We stress that for our main result the nonlinear Fokker-Planck-Kolmogorov equations do not have to be well-posed. Thus, we establish a one-to-one correspondence between solution flows of a large class of nonlinear parabolic PDEs and nonlinear Markov processes.

math.PR↗

arXiv2022

Emergence of phase-locked states for a deterministic and stochastic Winfree model with inertia

We study the emergence of phase-locking for Winfree oscillators under the effect of inertia. It is known that in a large coupling regime, oscillators governed by the deterministic second-order Winfree model with inertia converge to a unique equilibrium. In contrast, in this paper we show the asymptotic emergence of non-trivial synchronization in a suitably small coupling regime. Moreover, we study the effect of a new stochastically perturbed Winfree system with multiplicative noise and obtain lower estimates in probability for the pathwise emergence of such a synchronizing pattern, provided the noise is sufficiently small. We also provide numerical simulations which hint at the possibility of more general and stronger analytical results.

math.DS↗

arXiv2022

Linearization and a superposition principle for deterministic and stochastic nonlinear Fokker-Planck-Kolmogorov equations

We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence, we obtain equivalence of existence and uniqueness results for these equations. Moreover, we prove an analogous result for stochastically perturbed Fokker-Planck-Kolmogorov equations. To do so, we particularly show that such stochastic equations for measures are, similarly to the deterministic case, intrinsically related to linearized second-order equations on the space of Borel (sub-)probability measures.

math.PR↗

arXiv2022

Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations

We study the incompressible hypodissipative Navier-Stokes equations with dissipation exponent $0 < α< \frac{1}{2}$ on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.

math.PR↗

arXiv2019

Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness

Let the coefficients $a_{ij}$ and $b_i$, $i,j \leq d$, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) $$\partial_tμ_t = \partial_i\partial_j(a_{ij}μ_t)-\partial_i(b_iμ_t)$$ be Borel measurable, bounded and continuous in space. Assume that for every $s \in [0,T]$ and every Borel probability measure $ν$ on $\mathbb{R}^d$ there is at least one solution $μ= (μ_t)_{t \in [s,T]}$ to the FPK-eq. such that $μ_s = ν$ and $t \mapsto μ_t$ is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution $μ^{s,ν}$ for each pair $(s,ν)$ such that this family of solutions fulfills $$μ^{s,ν}_t = μ^{r,μ^{s,ν}_r}_t \text{ for all }0 \leq s \leq r \leq t \leq T,$$which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unqiue if and only if the FPK-eq. is well-posed.

math.PR↗

arXiv2018

On Cherny's results in infinite dimensions: A theorem dual to Yamada-Watanabe

We prove that joint uniqueness in law and the existence of a strong solution imply pathwise uniqueness for variational solutions to stochastic partial differential equations of the form \begin{align*} \text{d}X_t=b(t,X)\text{d}t+σ(t,X)\text{d}W_t, \,\,\,t\geq 0, \end{align*} and show that for such equations uniqueness in law is equivalent to joint uniqueness in law. Here $W$ is a cylindrical Wiener process in a separable Hilbert space $U$ and the equation is considered in a Gelfand triple $V \subseteq H \subseteq E$, where $H$ is some separable (infinite-dimensional) Hilbert space. This generalizes the corresponding results of A. Cherny for the case of finite-dimensional equations.

math.PR↗