Nonlinear Feynman--Kac formulas for Stochastic Partial Differential Equations with Space-Time Noise

2019 ◽  
Vol 51 (2) ◽  
pp. 955-990
Author(s):  
Jian Song ◽  
Xiaoming Song ◽  
Qi Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Space Time ◽  
Partial Differential

scholarly journals Stochastic partial differential equations driven by space-time fractional noises

2019 ◽  
Vol 19 (02) ◽  
pp. 1950012
Author(s):  
Ying Hu ◽  
Yiming Jiang ◽  
Zhongmin Qian
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Space Time ◽  
Partial Differential ◽  
The Family ◽  
Non Local ◽  
Fractional Noises

In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying at first the non-local SPDEs and thereafter showing the convergence of the family of these equations. The limit gives the solution of the SPDE.

Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise

2008 ◽  
Vol 465 (2102) ◽  
pp. 649-667 ◽  
Author(s):  
Arnulf Jentzen ◽  
Peter E Kloeden
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Numerical Approximation ◽  
Time Discretization ◽  
Space Time ◽  
Euler Scheme ◽  
Numerical Schemes ◽  
Partial Differential ◽  
The Laplace Operator

We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

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