Anne de Bouard, Arnaud Debussche
We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.
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On blow up NLS with a multiplicative noise
研究乘性噪声对NLS爆破速度的影响，证明噪声导致爆破的概率较小，并给出大偏差上界。
We review the recent progress on the long-time behavior for a general class of focusing $L^2$-critical nonlinear Schrödinger equations (NLS) with lower order perturbations. Two canonical models are the stochastic NLS driven by linear multiplicative noise and the classical deterministic NLS. We show the construction and uniqueness of the corresponding blow-up solutions and solitons, including the multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons, which provide new examples for the mass quantization conjecture and the soliton resolution conjecture. The refined uniquen…
In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic…
Blowup for the multiplicative stochastic heat equation with superlinear drift
研究了具有超线性漂移的乘法随机热方程在不同区域的爆破条件，证明了有限时间爆破的判定条件及即时爆炸的机制。
Here is investigated the bilinear optimal control problem of quantum mechanical systems with final observation governed by a stochastic nonlinear Schrödinger equation perturbed by a linear multiplicative Wiener process. The existence of an open loop optimal control and first order Lagrange optimality conditions are derived, via Skorohod's representation theorem, Ekeland's variational principle and the existence for the linearized dual backward stochastic equation. Moreover, our approach in particular applies to the deterministic case.
This paper is devoted to the study of noise effects on blow-up solutions to stochastic nonlinear Schrödinger equations. It is a continuation of our recent work \cite{BRZ14}, where the (local) well-posedness is established in $H^1$, also in the non-conservative critical case. Here we prove that in the non-conservative focusing mass-(super)critical case, by adding a large multiplicative Gaussian noise, with high probability one can prevent the blow-up on any given bounded time interval $[0,T]$, $0<T<\9$. Moreover, in the case of spatially independent noise, the explosion even can be prevented wi…
In this short note, we present a construction for the log-log blow up solutions to focusing mass-critical stochastic nonlinear Schröidnger equations with multiplicative noises. The solution is understood in the sense of controlled rough path as in \cite{SZ20}.
No blow-up by nonlinear Itô noise for the Euler equations
本文通过引入具径向结构且超线性增长的乘性Itô噪声，证明二维和三维随机Euler方程存在唯一整体强解，并推广至一类SPDEs，证明此类噪声能有效防止解的爆破。
In this article, we study Stochastic mass critical nonlinear Schrödinger equations with a multiplicative noise in 3D with a slight time decay ($\langle t \rangle^{-ε}$), and prove associated space-time bound and scattering behavior.
Blow-up of stochastic semilinear parabolic equations driven by Lévy noise
针对加性与线性乘性Lévy噪声驱动的随机半线性抛物方程，本文改进随机情形下的凹性方法，证明了有界区域上系统解的爆破现象。
We study a quite general class of stochastic dispersive equations with linear multiplicative noise, including especially the Schrödinger and Airy equations. The pathwise Strichartz and local smoothing estimates are derived here in both the conservative and nonconservative case. In particular, we obtain the P-integrability of constants in these estimates, where P is the underlying probability measure. Several applications are given to nonlinear problems, including local well-posedness of stochastic nonlinear Schrödinger equations with variable coefficients and lower order perturbations, integra…
We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schrödinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate at $K$ distinct points, $1\leq K<\infty$, and behave asymptotically like a sum of pseudo-conformal blow-up solutions in the pseudo-conformal space $Σ$ near the blow-up time. The upper bound of the asymptotic behavior is closely related to the flatness of noise at blow-up points. Moreover, we prove the conditional uniqueness of multi-bubble solutions in the case…
We give decay estimates of the solution to the linear Schrödinger equation in dimension $d \geq 3$ with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé-Soffer-Sogge for decay of deterministic Schrödinger operators.
Stochastic Schrödinger-Korteweg de Vries systems driven by multiplicative noises
本文研究乘性噪声驱动的随机S-KdV系统在$H_x^1\times H_x^1$中的适定性，通过建立Bourgain范数估计和引入物理与频率空间截断的近似方程，证明了系统的整体适定性。
Instantaneous blowup for interacting SDEs with superlinear drift
研究了满足有限Osgood条件的相互作用SDEs系统，证明初始轮廓衰减速度慢于特定指数函数时会出现瞬时爆炸。
We consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (2019) with a sub-quadratic nonlinearity.
Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrodinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. It is white in time and colored in space. The path space considered allows blow-up and is endowed with a topology analogue to a projective limit topology. Thus a large variety of large deviation principle may be deduced by contraction. As a consequence, asymptotics of the tails of the law of the blow-up time when the noise converges to zero are obtained.
The energy-critical stochastic nonlinear Schrödinger equation: well-posedness and blow-up
本文研究了具有随机扰动的聚焦和反聚焦能量临界随机非线性薛定谔方程的局部适定性及破裂准则，针对加性或乘性噪声，提供了初始数据的适定性估计及正能量解的破裂概率分析。
Full Discretization of Stochastic Semilinear Schrödinger equation driven by multiplicative Wiener noise
本文针对乘性Wiener噪声驱动的随机半线性薛定谔方程，提出有限元与随机三角法结合的全离散格式，推导时空强收敛速率，并通过数值实验验证理论结果。
In this paper we focus on the stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise. We first establish two new cancellation properties on pseudo-differential operators, which play a key role in energy estimate. Then, we obtain results on local solution, blow-up criterion and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data are also studied for the multiplicative noise case.
Global well posedness and ergodic results in regular Sobolev spaces for the nonlinear Schrödinger equation with multiplicative noise and arbitrary power of the nonlinearity
研究了 $d$ 维环面上具多项式型非线性的薛定谔方程，证明加入适当随机强迫项后任意正则 Sobolev 初始数据存在唯一整体解，噪声可防止有限时间爆破，并证明了不变测度的存在性与条件唯一性。
For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions on the nonlinear term, the blow-up is delayed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Keller-Segel, 3D Fisher-KPP and 2D Kuramoto-Sivashinsky equations, yielding long-time existence for large initial data with high probability.
Let $ H:=-\tfrac12Δ+V$ be a one-dimensional continuum Schrödinger operator. Consider ${\hat H}:= H+ξ$, where $ξ$ is a translation invariant Gaussian noise. Under some assumptions on $ξ$, we prove that if $V$ is locally integrable, bounded below, and grows faster than $\log$ at infinity, then the semigroup $\mathrm e^{-t {\hat H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb R$, the half line $(0,\infty)$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of p…
We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that we study the influence of noise on the global behavior of solutions. In particular, we show that the noise may induce blow-up, thus, ceasing the global ex…
We prove the global space-time bound for the mass critical nonlinear Schrödinger equation perturbed by a small multiplicative noise in dimension three. The associated scattering behavior are also obtained. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from \cite{fan2018global}, \cite{fan2020long} as well as local smoothing estimates for linear Schrödinger operators.
We study the focusing stochastic nonlinear Schrödinger equation in one spatial dimension with multiplicative noise, driven by a Wiener process white in time and colored in space, in the $L^2$-critical and supercritical cases. The mass ($L^2$-norm) is conserved due to the multiplicative noise defined via the Stratonovich integral, the energy (Hamiltonian) is not preserved. We first investigate how the energy is affected by various spatially correlated random perturbations. We then study the influence of the noise on the global dynamics measuring the probability of blow-up versus scattering beha…
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