Existence and Uniqueness of Nonnegative Solutions to the Stochastic Porous Media Equation

A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation

The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $ω$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.

Probabilistic representation for solutions of an irregular porous media type equation

We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.

Stochastic Generalized Porous Media and Fast Diffusion Equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for $σ$-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called $N$-functions in the theory of Orlicz spaces.

The obstacle problem for stochastic porous media equations

We prove the existence and uniqueness of non-negative entropy solutions of the obstacle problem for stochastic porous media equations. The core of the method is to combine the entropy formulation with the penalization method.

Ergodicity for singular-degenerate porous media equations

The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising in self-organised criticality. Based on the so-called lower-bound method, the existence and uniqueness of an invariant measure is proved.

Stochastic Porous Media Equation and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states.

Stochastic Porous Media Equation on General Measure Spaces with Increasing Lipschitz Nonlinearties

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E, \mathscr{B}(E), μ)$, and the Laplacian replaced by a self-adjoint operator $L$. In the case of Lipschitz nonlinearities $Ψ$, we in particular generalize previous results for open $E\subset \mathbb{R}^d$ and $L\!\!=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed i…

The stochastic porous media equation in $\R^d$

Existence and uniqueness of solutions to the stochastic porous media equation $dX-\Dψ(X) dt=XdW$ in $\rr^d$ are studied. Here, $W$ is a Wiener process, $ψ$ is a maximal monotone graph in $\rr\times\rr$ such that $ψ(r)\le C|r|^m$, $\ff r\in\rr$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space $\calh=\{φ\in\mathcal{S}'(\rr^d);\ |ξ|(\calfφ)(ξ)\in L^2(\rr^d)\}$, for $d\le2$. When $ψ$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sob…

Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality

We consider a class of generalised stochastic porous media equations with multiplicative Lipschitz continuous noise. These equations can be related to physical models exhibiting self-organised criticality. We show that these SPDEs have unique SVI solutions which depend continuously on the initial value. In order to formulate this notion of solution and to prove uniqueness in the case of a slowly growing nonlinearity, the arising energy functional is analysed in detail.

Existence and extinction in finite time for Stratonovich gradient noise porous media equations

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( νΔX + Δψ(X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ dβ_i$ in $]0,T[ \times \mathcal{O}$, with $X(0) = x(ξ)$ in $\mathcal{O}$ and $X = 0$ on $]0,T[ \times \partial \mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions

In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.

非负卦限上Robin边界问题的一种分解-齐次化方法

A decomposition-homogenization method for Robin boundary problems on the nonnegative orthant

本文研究了非负卦限上Robin边界问题经典解的存在性与唯一性，提出一种基于概率表示的分解-齐次化方法，导出两个辅助问题，并利用随机流分析证明了原始问题唯一经典解的存在性。

An Identity for Two Integral Transforms Applied to the Uniqueness of a Distribution via its Laplace-Stieltjes Transform

It is well known that the Laplace-Stieltjes transform of a nonnegative random variable (or random vector) uniquely determines its distribution function. We extend this uniqueness theorem by using the Muntz-Szasz Theorem and the identity for the Laplace-Stieltjes and Laplace-Carson transforms of a distribution function. The latter appears for the first time to the best of our knowledge. In particular, if X and Y are two nonnegative random variables with joint distribution H, then H can be characterized by a suitable set of countably many values of its bivariate Laplace-Stieltjes transform. The …

Capacities, removable sets and $L^p$-uniqueness on Wiener spaces

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set $Σ$ of zero Gaussian measure. To prove the equivalence we show the $W^{r,p}(B,μ)$-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We also give connections to Gaussian Hausdorff measures. Roughly speaking, if $L^p$-uniqueness holds then the 'removed' se…