International audienceWe consider stochastic equations of the prototype $du(t,x) =\left( \Delta u(t,x)+u(t,x)^{1+\beta}\right)dt+\kappa u(t,x)\,dW_{t}$ on a smooth domain $D\subset \mathord{\rm I\mkern-3.6mu R\:\!\!}^d$, with Dirichlet boundary condition, where $\beta$, $\kappa$ are positive constants and $\{W_t $, $t\ge0\}$ is a one-dimensional standard Wiener process. We estimate the probability of finite time blowup of positive solutions, as well as the probability of existence of non-trivial positive global solutions

We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions. By extending the method of moving planes to fractional powers of the Laplacian we prove that all positive steady states of the corresponding equation in a finite ball are radially symmetric.

Abstract. We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂w/∂t = −(−∆) α/2 w + w 1+β in the critical dimension d = α/β. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as t → ∞. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of α-Laplacians with possibly different parameters α.

The existence of the multitype measure branching process is established as a small particle limit of a system of particles of several types in Rd with immigration undergoing migration, branching and mutation. The process is characterized as a solution of a martingale problem. The single-type case was studied by Dawson (1975), (1977) and Watanabe (1968).

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