On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. (English) Zbl 1105.60043
The authors consider the initial problem for the stochastic nonlinear equation
\[
(\partial_t - S)u(t,x)+ \partial_x (q(t,x,u(t,x)))= f(t,x,u(t,x))+ g(t,x,u(t,x))F_{t,x},
\]
on the given domain \([0,\infty) \times \mathbb R\) with \(L^2(\mathbb R)\) initial condition, where \(A\) is an integro-differential operator associated with a symmetric, nonlocal, regular Dirichlet form, and \(F_{t,x}\) stands for a Lévy space-time white noise. The initial problem for this equation is interpreted as a jump type stochastic integral equation involving the transition density kernels associated with the symmetric integro-differential operators as the convolution kernels. Then, they prove the existence and uniqueness of a local mild solution to the initial problem for this equation.
Reviewer: Carles Rovira (Barcelona)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
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