Stochastic wave equation in a plane driven by spatial stable noise
Анотація
The main object of this paper is the planar wave equation \[ \bigg(\frac{{\partial }^{2}}{\partial {t}^{2}}-{a}^{2}\varDelta \bigg)U(x,t)=f(x,t),\hspace{1em}t\ge 0,\hspace{2.5pt}x\in {\mathbb{R}}^{2},\] with random source f. The latter is, in certain sense, a symmetric α-stable spatial white noise multiplied by some regular function σ. We define a candidate solution U to the equation via Poisson’s formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point $(x,t)$. We further show that U is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where σ does not vanish. Finally, we prove that U is a generalized solution to the equation.
Класифікація
Ідентифікатори
Рецензії (0)
Написати рецензіюРецензій ще немає. Будьте першим!
Схожі роботи
Numerical approximation and dynamics of periodic solution in distribution of stochastic differential equations
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods
Invariant Sets of Systems of Stochastic Differential Equations with Jumps
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods
Approximation of solution of the cable equation driven by a stochastic measure
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods
Existence and uniqueness of solutions of stochastic differential equations with non-Lipschitz diffusion and Poisson measure
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods
On the Asymptotics of Solutions of Stochastic Differential Equations with Jumps
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods
Comparison theorem for solutions of parabolic stochastic equations with an absorber
Схоже за: Advanced Mathematical Modeling in Engineering · Stochastic processes and financial applications · Differential Equations and Numerical Methods