Exponential Decay of $L^2$-Solutions to Stochastic Nonlinear Schrödinger Equations Driven by Continuous Martingales
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
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2026
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| author | Dôku, Isamu Hashimoto, Shunya Machihara, Shuji |
| author_facet | Dôku, Isamu Hashimoto, Shunya Machihara, Shuji |
| contents | We investigate the global well-posedness and asymptotic behavior of $L^2$-solutions to stochastic nonlinear Schrödinger equations with multiplicative noise driven by continuous square integrable martingales with density. Our approach relies on a rescaling transformation that converts the stochastic system into a random nonlinear Schrödinger equation with a potential acting as a damping term. Unlike the standard Brownian motion case, this induced potential plays a critical role in the dynamics. We establish the global existence of solutions and prove the pathwise exponential decay of the $L^2$-norm. Crucially, the strict positivity of the decay rate is intrinsically induced by the density of the martingale\rq{}s quadratic variation. This result generalizes the stabilization known for standard Brownian motion, thereby characterizing the stabilizing effect of the martingale noise. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10309 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exponential Decay of $L^2$-Solutions to Stochastic Nonlinear Schrödinger Equations Driven by Continuous Martingales Dôku, Isamu Hashimoto, Shunya Machihara, Shuji Probability Analysis of PDEs 60H15, 35B65, 35J10 We investigate the global well-posedness and asymptotic behavior of $L^2$-solutions to stochastic nonlinear Schrödinger equations with multiplicative noise driven by continuous square integrable martingales with density. Our approach relies on a rescaling transformation that converts the stochastic system into a random nonlinear Schrödinger equation with a potential acting as a damping term. Unlike the standard Brownian motion case, this induced potential plays a critical role in the dynamics. We establish the global existence of solutions and prove the pathwise exponential decay of the $L^2$-norm. Crucially, the strict positivity of the decay rate is intrinsically induced by the density of the martingale\rq{}s quadratic variation. This result generalizes the stabilization known for standard Brownian motion, thereby characterizing the stabilizing effect of the martingale noise. |
| title | Exponential Decay of $L^2$-Solutions to Stochastic Nonlinear Schrödinger Equations Driven by Continuous Martingales |
| topic | Probability Analysis of PDEs 60H15, 35B65, 35J10 |
| url | https://arxiv.org/abs/2605.10309 |