Solutions of semilinear wave equation via stochastic cascades

@article{Bakhtin2009SolutionsOS,
  title={Solutions of semilinear wave equation via stochastic cascades},
  author={Yuri Bakhtin and Carl Mueller},
  journal={arXiv: Probability},
  year={2009}
}
We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution. 

Figures from this paper

Stochastic solutions of the wave equation
Unlike the heat equation or the Laplace equation, solutions of the wave equation on general domains have no known stochastic representation. This short note gives a simple solution to this well known
Branching diffusion representation for nonlinear Cauchy problems and Monte Carlo approximation
We provide a probabilistic representations of the solution of some semilinear hyperbolicand high-order PDEs based on branching diffusions. These representations pave theway for a Monte-Carlo
STOCHASTIC ANALYSIS IN THE ACOUSTICS OF DAMPED SOUNDS
A stochastic model of sound propagation in damping medium is proposed. It consists of: (1) Ito’s stochastic differential equation describing the sound propagation, (2) a potential which models the

References

SHOWING 1-10 OF 12 REFERENCES
Intermittency properties in a hyperbolic Anderson problem
Keywords: Stochastic wave equation ; Stochastic partial differential equations ; Moment Lyapunov exponents ; Intermittency ; Stochastic heat equation ; Stochastic Wave-Equation ; Model ; Dimensions
Stochastic cascades and 3-dimensional Navier–Stokes equations
Summary. In this article, we study the incompressible Navier–Stokes equations in ℝ3. The non linear integral equation satisfied by the Fourier transform of the Laplacian of the velocity field can be
A FEYNMAN-KAC-TYPE FORMULA FOR THE DETERMINISTIC AND STOCHASTIC WAVE EQUATIONS AND OTHER
We establish a probabilistic representation for a wide class of linear deterministic p.d.e.’s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies
A stochastic model related to the telegrapher's equation
Some stochastic problems in physics and mathematics. Magnolia Petroleum Co
  • Lectures in Pure and Applied Science
  • 1956
A Feynman - Kac - type formula for deterministic and stochastic wave equations and other p . d . e . ’ s
  • Trans . Amer . Math . Soc .
  • 2008
Some stochastic problems in physics and mathematics
  • Magnolia Petroleum Co. Lectures in Pure and Applied Science
  • 1956
Some stochastic problems in physics and mathematics , Magnolia Petroleum Co
  • Lectures in Pure and Applied Science
  • 1956
Some stochastic problems in physics and mathematics, Magnolia
  • Petroleum Co. Lectures in Pure and Applied Science
  • 1956
...
...