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A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $φ$, \begin{eqnarray*}X_t(φ)=X_0(φ)+\frac{1}{2}\int _0^tX_s(Δφ)\,ds+θ\int_0^tX_s(φ)\,ds\\{}-\int_0^tX_s(L_sφ)\,ds+M_t(φ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(φ)$ is a martingale with quadratic variation $[M(φ)]_t=\int_0^tX_s(φ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $θ_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $θ>θ_c(d)$, then the solution survives forever with positive probability, but if $θ<θ_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $θ$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $θ$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.