The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise

A Practical Underwater Information Sensing System Based on Intermittent Chaos Under the Background of Lévy Noise

“…The Lévy process was proposed by the French mathematician Paul Lévy to study of the generalized central limit theorem. It is a random process with independent and fixed increments, indicating that the movement of a point and its continuous displacement are random [30]. The difference between two disjoint time intervals displacement is independent.…”

The stochastic nonlinear Schrödinger equations driven by pure jump noise

“…De Bouard and Hausenblas investigated the existence of martingale solutions of the nonlinear Schrödinger equation with a Lévy noise with infinite activity in [5] and pathwise uniqueness was studied in a separate paper [6] with Ondrejat. Recently, Brzeźniak et al established a new version of the stochastic Strichartz estimate for the stochastic convolution driven by a jump noise in [8]. By applying the stochastic Strichartz estimates in a fixed point argument, they proved the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation with a Marcus-type jump noise in L 2 (R d ) with either focusing or defocusing nonlinearity in the full subcritical range of exponents.…”

“…The purpose of this paper is to prove the existence and uniqueness of mild solutions for the stochastic nonlinear Schrödinger equation (1) with the additive jump noise. By means of the deterministic and stochastic Strichartz's estimates due to Brzeźniak et al from [8], we apply the classical truncation procedure of the nonlinearities and use the well-known fixed point argument to construct a local solution. One main ingredient of establishing global solution in [8] is the mass conservation for the stochastic nonlinear Schrödinger equaiton.…”

“…By means of the deterministic and stochastic Strichartz's estimates due to Brzeźniak et al from [8], we apply the classical truncation procedure of the nonlinearities and use the well-known fixed point argument to construct a local solution. One main ingredient of establishing global solution in [8] is the mass conservation for the stochastic nonlinear Schrödinger equaiton. Unlike the L 2 (R d )-norm conservation of solutions in the literature [8], we establish some uniform L 2 (R d )-norm estimate of the local solution and selects some specific stopping times that we believe are critical to the proof.…”

“…One main ingredient of establishing global solution in [8] is the mass conservation for the stochastic nonlinear Schrödinger equaiton. Unlike the L 2 (R d )-norm conservation of solutions in the literature [8], we establish some uniform L 2 (R d )-norm estimate of the local solution and selects some specific stopping times that we believe are critical to the proof. Let us formulate our main result of this paper.…”