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Jonas M. Tölle

Publications and source records attributed to Jonas M. Tölle.

28 records · Page 2Linked to original sources
arXiv2018

A gradient flow formulation for the stochastic Amari neural field model

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.

math.AP↗
arXiv2016

Ergodicity and local limits for stochastic local and nonlocal p-Laplace equations

Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, Tölle; J. Math. Pures Appl., 2014], [Liu, Tölle; Electron. Commun. Probab., 2011], [Liu; J. Evol. Equations, 2009]. In particular, the results include the multivalued case of the stochastic (nonlocal) total variation flow, which solves an open problem raised in [Barbu, Da Prato, Röckner; SIAM J. Math. Anal., 2009]. Moreover, under appropriate rescaling, the convergence of the unique invariant measure for the nonlocal stochastic $p$-Laplace equation to the unique invariant measure of the local stochastic $p$-Laplace equation is proven.

math.PR↗
arXiv2016

Nonlinear stochastic partial differential equations with singular diffusivity and gradient Stratonovich noise

We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich gradient-type noise. We derive a commutator relation for the unbounded noise coefficients in terms of a geometric Killing vector condition. The drift term is given by the total variation flow, respectively, by a singular $p$-Laplace-type operator. We impose nonlinear zero Neumann boundary conditions and precisely investigate their connection with the coefficient fields of the noise. This solves an open problem posed in [Barbu, Brzeźniak, Hausenblas, Tubaro; Stoch. Proc. Appl., 123 (2013)] and [Barbu, Röckner; J. Eur. Math. Soc., 17 (2015)].

math.AP↗
arXiv2015

Stability of solutions to stochastic partial differential equations

We provide a general framework for the stability of solutions to stochastic partial differential equations with respect to perturbations of the drift. More precisely, we consider stochastic partial differential equations with drift given as the subdifferential of a convex function and prove continuous dependence of the solutions with regard to random Mosco convergence of the convex potentials. In particular, we identify the concept of stochastic variational inequalities (SVI) as a well-suited framework to study such stability properties. The generality of the developed framework is then laid out by deducing Trotter type and homogenization results for stochastic fast diffusion and stochastic singular p-Laplace equations. In addition, we provide an SVI treatment for stochastic nonlocal p-Laplace equations and prove their convergence to the respective local models.

math.AP↗
arXiv2013

Multi-valued, singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-* mean ergodic invariant measure.

math.PR↗
arXiv2012

Uniqueness of weighted Sobolev spaces with weakly differentiable weights

We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where $d\in\N$ and $p\in [1,\infty)$. If $w$ admits a (weak) logarithmic gradient $\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we propose an alternative definition of the weighted $p$-Sobolev space based on an integration by parts formula involving $\nabla w/w$. We prove that weights of the form $\exp(-β|\cdot|^q-W-V)$ are $p$-admissible, in particular, satisfy a Poincaré inequality, where $β\in (0,\infty)$, $W$, $V$ are convex and bounded below such that $|\nabla W|$ satisfies a growth condition (depending on $β$ and $q$) and $V$ is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.

math.FA↗
arXiv2012

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic $p$-Laplace equation, $p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter $r\in (0,1)$ on a bounded open domain $Λ\subset\R^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters $p$ and $r$ respectively (in the Hilbert spaces $L^2(Λ)$, $H^{-1}(Λ)$ respectively). The highly singular limit case $p=1$ is treated with the help of stochastic evolution variational inequalities, where $\mathbbm{P}$-a.s. convergence, uniformly in time, is established. It is shown that the associated unique invariant measures of the ergodic semigroups converge in the weak sense (of probability measures).

math.PR↗
arXiv2011

Convergence of solutions to the $p$-Laplace evolution equation as $p$ goes to 1

We prove that the set of solutions to the parabolic singular $p$-Laplace equation with Dirichlet boundary conditions on a bounded Lipschitz domain $Ω$ for all space dimensions is continuous in the parameter $p\in [1,+\infty)$ and the initial data. The highly singular limit case p=1 is included. In particular, we show that the solutions $u_p$ converge strongly in $L^2(Ω)$, uniformly in time, to the solution $u_1$ of the parabolic 1-Laplace equation as $p\to 1$.

math.AP↗
arXiv2011

Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].

math.PR↗