Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise

Abstract

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation $dX(t)=\mathrm{div}[\frac{\nabla X(t)}{|\nabla X(t)|}]dt+X(t)dW(t)in(0,\infty )\times \mathcal{O},$ where $\mathcal{O}$ is a bounded and open domain in ${\mathbb{R}}^N$, $N\ge 1$, and $W(t)$ is a Wiener process of the form $W(t)={\sum }_{k=1}^{\infty }{\mu }_k{e}_k{\beta }_k(t)$, $e_k \in C^2(\bar\mathcal{O})\cap H^1_0(\mathcal{O}),$ and ${\beta }_k$, $k\in \mathbb{N}$, are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term and one main result established here is that, for all initial conditions in $L^2(\mathcal{O})$, it is well posed in a class of continuous solutions to the corresponding stochastic variational inequality. Thus one obtains a stochastic version of the (minimal) total variation flow. The new approach developed here also allows to prove the finite time extinction of solutions in dimensions $1\le N\le 3$, which is another main result of this work. Keywords: stochastic diffusion equation, Brownian motion, bounded variation, convex functions, bounded variation flow.

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• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
• Convergent numerical approximation of the stochastic total variation flow
• Corrigendum to `Convergence of invariant measures for singular stochastic diffusion equations'
• Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
• A posteriori estimates for the stochastic total variation flow
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• An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
• Well-posedness of SVI solutions to singular-degenerate stochastic porous media equations arising in self-organised criticality
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• Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation
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