Constructing the super-Brownian process by using SPDEs and Skorokhod's method (Q2726266)

The relationship between the one-dimensional super-Brownian process and the theory of stochastic partial differential equations (SPDEs) is well-known. For example, it is proved that the one-dimensional super-Brownian process at any time \(t>0\) has a density \(u=u(t,x)\) which satisfies the following SPDE NEWLINE\[NEWLINEdu=u_{xx}dt+\sqrt{u}\sum_{i=1}^{\infty}\varphi_{i} dw^{i}_{t},NEWLINE\]NEWLINE where \(\varphi_{i}\) is an orthogonal basis in \(L_{2}(R)\) and \(w^{i}\) are independent Wiener processes. It turns out that the super-Brownian process \(\mu_{t}\) in \(R^{d}\) can be viewed as a solution of the equation NEWLINE\[NEWLINEd\mu_{t}=\Delta\mu_{t}dt+ \sum_{i=1}^{\infty}\varphi_{i}(\mu_{t},\cdot)\mu_{t} dw^{i}_{t},NEWLINE\]NEWLINE where \(\varphi_{i}(\mu,x)\) is the so-called frame function defined for any finite measure \(\mu.\) The fact that one can write down SPDEs for super-Brownian processes suggests that one can prove the existence of super-Brownian processes by showing that the equations are solvable. One of the ways of proving the existence of super-Brownian processes is to construct superdiffusions as Markov processes defining their transition functions in the space of measures and then using the general theory of Markov processes to get a process corresponding to this transition function. The natural question arises: Can one use the theory of SPDEs to construct superdiffusions, say the super-Brownian process? The aim of this article is to show that the answer to this question is positive. Any \(d\geq 1\) is taken but instead of the second equation above the construction is based on approximations of a \(d\)-dimensional analog of the first equation above. Therefore, a way of constructing super-Brownian processes through solving SPDEs is presented.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00022].