Self-repelling diffusions via an infinite dimensional approach

In the present work we study self-interacting diffusions following an infinite dimensional approach. First we prove existence and uniqueness of a solution with Markov property. Then we study the corresponding transition semigroup and, more precisely, we prove that it has Feller property and we give an explicit form of an invariant probability of the system.Comment: Version 2: Typos are corrected. Section 6 is reorganised in order to make it more transparent; the results are unchanged. The presentation of the proof of Proposition 3 is improved. Statement of Lemma 5 is rephrased. Version 3: Acknowledgement of financial support is added. Accepted for publication in "Stochastic Partial Differential Equations: Analysis and Computations

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 $\Lambda\subset\R^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters $p$ and $r$ respectively (in the Hilbert spaces $L^2(\Lambda)$, $H^{-1}(\Lambda)$ 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).Comment: to appear in Stoch. Proc. Appl. (in press), 18 p

Existence and uniqueness of the solution for stochastic super-fast diffusion equations with multiplicative noise

International audienceIn this paper we prove an existence and uniqueness result for the stochastic porous media equation with very singular diffusion and multiplicative noise, by using monotonicity techniques. The multiplicative Gaussian noise is essential in the proof of existence

Stochastic porous media equations with divergence Itô noise

International audienceWe study the existence and uniqueness of solution to stochastic porous media equations with divergence Itô noise in infinite dimensions. The first result prove existence of a stochastic strong solution and it is essentially based on the non-local character of the noise. The second result proves existence of at least one martingale solution for the critical case corresponding to the Dirac distribution

Nonlinear Fokker-Planck equation with reflecting boundary conditions

We study the existence and uniqueness of a mild solution to a nonlinear Fokker-Planck equation with reflecting boundary conditions, by using a monotonicity approach. Then we prove that the mild solution is also a distributional one