No Arabic abstract
We establish finite time extinction with probability one for weak solutions of the Cauchy-Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate $O(t^frac{1}{2})$ whereas the support of solutions to the deterministic PME grows only with rate $O(t^{frac{1}{m{+}1}})$. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong-Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.
We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( u Delta X + Delta psi (X) ) dt = sum_{i=1}^N langle b_i, abla X rangle circ dbeta_i$ in $]0,T[ times mathcal{O}$, with $X(0) = x(xi)$ 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.
We study the stochastic cubic complex Ginzburg-Landau equation with complex-valued space-time white noise on the three dimensional torus. This nonlinear equation is so singular that it can only be under- stood in a renormalized sense. In the first half of this paper we prove local well-posedness of this equation in the framework of regularity structure theory. In the latter half we prove local well-posedness in the framework of paracontrolled distribution theory.
In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Levy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls.
Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Levy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.
We prove the existence of nonnegative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the liquid-solid interface. Since their introduction more than 15 years ago, by Davidovitch, Moro, and Stone and by Grun, Mecke, and Rauscher, the existence of solutions to stochastic thin-film equations for cubic mobilities has been an open problem, even in the case of sufficiently regular noise. Our proof of global-in-time solutions relies on a careful combination of entropy and energy estimates in conjunction with a tailor-made approximation procedure to control the formation of shocks caused by the nonlinear stochastic scalar conservation law structure of the noise.