Stochastic Variational Inequalities and Applications to the Total Variation Flow Perturbed by Linear Multiplicative Noise

By Viorel Barbu, Michael Röckner. 2012

64 citations recorded by the source corpus.

In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation $${{\rm d}X(t) = {\rm div} \left[\frac{\nabla X(t)}{|\nabla X(t)|}\right]{\rm d}t + X(t){\rm d}W(t) {\rm in} (0, \infty) \times \mathcal{O},}$$ where $${\mathcal{O}}$$ is a bounded and open domain in $${\mathbb{R}^N, N \geqq 1}$$ and W(t) is a Wiener process of the form $${W(t) = \sum^{\infty}_{k = 1}\mu_{k}e_{k}\beta_{k}(t), e_{k} \in C^{2}(\overline{\mathcal{O}}) \cap H^{1}_{0}(\mathcal{O}),}$$ and $${\beta_{k}, k \in \mathbb{N}}$$ are independent Brownian motions. This is a stochastic diffusion equation with a highly singular diffusivity term. 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 us to prove the finite time extinction of solutions in dimensions $${1\leqq N \leqq3}$$, which is another main result of this work.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e45bf1bc5d8f59855c9f4e449d29ceb2f85d9e5d

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In Applied Mathematics

  1. Viorel Barbu
  2. Michael Röckner
  3. Feng-Yu Wang
  4. Wei Liu
  5. Xicheng Zhang
  6. Giuseppe Da Prato
  7. Franco Flandoli
  8. Tusheng Zhang

In Finance

  1. Viorel Barbu
  2. Michael Röckner
  3. Wei Liu
  4. Tusheng Zhang
  5. Feng-Yu Wang
  6. Benjamin Gess
  7. Giuseppe Da Prato
  8. Wei David Liu

In Statistics and Probability

  1. Viorel Barbu
  2. Michael Röckner

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