Nonlinear PDEs and measure-valued branching type processes
Let \(E\) be a topological space homeomorphic to a universally measurable set in a metrizable compact, \(E^{(m)}\) be \(E^{m}\) factorized by the action of the permutation group \(S_{m}\). Let \(S=\oplus_{m\geq 0}E^{(m)}\). \(S\) appears as the set of all finite sums of Dirac measures \(\delta_{x}\) on \(E\), this defines an addition for elements in \(S\) and a convolution for measures on \(S\). A multiplicative function on \(S\) is a \(\varphi^{\wedge}(\mu )=e^{\int \log\varphi d\mu}\), where \(\varphi :E\rightarrow (0,1]\). A kernel \(N\) on \(S\) is called branching if \(N_{\mu +\nu}=N_{\mu}\ast N_{\nu}\), where \(N_{\mu}\) is \(\mu N\). The branching kernels are in a bijective correspondence with the submarkovian ones \(B\) taking functions on \(S\) to ones on \(E\) by \(B^{\wedge}(\varphi^{\wedge})=((B\varphi )^{\wedge})^{\wedge}\). Every kernel \(P\) on \(E\) defines a \(P\otimes^{m}\) on \(E^{m}\) and, by restriction, one \(P^{(m)}\) on \(E^{(m)}\). Then, if \(P\) is submarkovian, a kernel \(P^{\sim}\) on \(S\) follows. The family of transition kernels of a right Markov process \(X\) with state space \(E\) defines in this way such a process on \(S\), with the corresponding resolvent. Suppose that \(E\) is Lusin and let \((U_{\alpha})\) be the resolvent of \(X\). Let \(A\) be a continuous additive functional of \(X\) with bounded \((U_{A}^{q}f)(x)=E^{x}\int e^{- qt}f(X_{t})dA_{t}\), \(A_{t}^{m}(\omega_{1},\dots ,\omega_{n})= \sum_{i=1}^{n}A_{t}(\omega_{i})\), \(A_{t}^{'m}=A_{t}^{m}/m\). Consider the resolvents \((K_{\alpha}^{q,m\sim})_{\alpha >0}\) and \((K_{\alpha}^{'q,m\sim})_{\alpha >0}\) with initial kernels the restrictions, to \(E^{(m)}\), of \(U_{A^{m}}^{q}\), \(U_{A'{}^{m}}{}^{q}\) respectively, \[ K_{1}^{q\sim}=\oplus_{m\geq 1}K_{1}^{q,m\sim}, \quad K_{1}^{'q\sim}=\oplus_{m\geq 1}K_{1}^{'q,m\sim}. \] Let \((W_{\alpha}^{\sim})\), \((W_{\alpha}^{'\sim})\) be the (subordinate) resolvents with initial kernels \[ U_{q}^{\sim}-K_{1}^{q\sim}U_{q}^{\sim}, \quad U_{q}^{\sim}-K_{1}^{'q\sim}U_{q}^{\sim}. \] Consider a submarkovian kernel \(B\) taking functions on \(S\) to ones on \(E\) and \[ P^{\sim}=K_{1}^{'q\sim}B^{\wedge},\quad P_{\alpha}^{\sim}=P^{\sim}-\alpha W_{\alpha}^{\sim}P^{\sim},\quad P_{\alpha}^{'\sim}=P^{\sim}-\alpha W_{\alpha}^{'\sim}P^{\sim}, \] \[ V_{\alpha}^{\sim}=\sum_{k\geq 0}(P_{\alpha}^{\sim})^{k}W_{\alpha}^{\sim},\quad V_{\alpha}^{'\sim}=\sum_{k\geq 0}(P_{\alpha}^{'\sim})^{k}W_{\alpha}^{'\sim}. \] Then \((V_{\alpha}^{\sim})\), \((V_{\alpha}^{'\sim})\) are resolvents of right Markov processes with state space \(S\). If \(Uf\) is bounded for some \(f>0\), then the result holds also for \(q=0\), the \((V_{\alpha}^{\sim})\) process is obtained by killing \(X^{\sim}\) with \(K_{1}^{0\sim}\) and its semigroup is a branching one. Let \({\mathcal B}_{1}\) be the set of measurable \(\varphi \leq 1\) on \(E\), \(c\) be bounded on \(E\) and \((T_{t})\) be the transition function with infinitesimal generator \({\mathcal L}-c\), where \({\mathcal L}\) is that of \(X\). Then, for every \(\varphi \in {\mathcal B}_{1}\), \(w_{t}(x)=T_{t}\varphi (x)+\int_{_{0}}^{^{t}}T_{t}(cBw_{t-s}^{\wedge})(x)ds\) has a unique solution \(w\in {\mathcal B}_{1}\), denoted by \(H_{t}\varphi\), and there exists a branching semigroup \({\mathcal H}_{t}\) on \(S\), with \({\mathcal H}_{t}\varphi^{\wedge}=(H_{t}\varphi )^{\wedge}\). If \(u\in {\mathcal B}_{1}\) and \({\mathcal H}_{t}^{\wedge}u^{\wedge}=u^{\wedge}\), then \(u\in \mathrm{dom}\,{\mathcal L}\) and \(({\mathcal L}-c)u+cBu^{\wedge}=0\). An example is \(Bg= \sum_{k\geq 1}q_kg|_{E^{(k)}}\), \(\sum_{k\geq 1}q_k\leq 1\). The authors show how \({\mathcal H}_{t}^{\wedge}\) solves the Dirichlet problem when \(E\) is the closure of an open \(D\). The paper contains detailed proofs of some results leading to the principal ones, together with examples and comments.
scientific article; zbMATH DE number 5947106
scientific article