Stochastic partial differential equation driven by stable noise
The aim of the article is to construct a process \(Y\) which is a weak solution for the stochastic partial differential equation \[ \partial Y_t(x)/\partial t =\Delta Y_t(x)/2 + Y_{s-}^\beta (x)\dot{L},\qquad t\geq 0,\quad x\in R^d, \] with \(Y_0\geq 0\). Here \(\dot{L}\) is an \(\alpha\)-stable white noise on \(R_+\times R^d\) without negative jumps and \(\Delta\) is the \(d\)-dimensional Laplacian. Existence of a weak solution is established for parameters \(0<\alpha\beta<(2/d)+1\) and \(1<\alpha<\min(2,(2/d)+1)\). However, for \(\beta<1\) the function \(f(u)=u^\beta\) is not a Lipschitz function. The author adopts \textit{N. Konno} and \textit{T. Shiga's} martingale problem approach [Probab. Theory Relat. Fields 79, No. 2, 201-225 (1988; Zbl 0631.60058)] to show that any solution of a corresponding martingale problem is a weak solution of the spde above. This reduces the problem to the construction of an approximating sequence of processes \((Y^n)\) which is tight and whose weak limit points solve the martingale problem. An appropriate integrability condition on \(Y^n\) to prove the tightness of the sequence was derived through introducing auxiliary evolution equations. This method may be applied to other stochastic partial differential equations in the absence of high moments. The martingale problem approach allows also a connection to super Brownian motion. Let \(1<\alpha<\min(2,(2/d)+1)\) and \(Y\) be a super Brownian motion with \((\alpha-1)\)-stable branching mechanism. Then there exists an \(\alpha\)-stable white noise without negative jumps such that \(Y\) is a weakly unique and weak solution of the spde above for \(\alpha\beta=1\). Uniqueness follows from the theory of superprocesses.
stochastic partial differential equations
scientific article; zbMATH DE number 1836959