Diffusions, Superdiffusions and Partial Differential Equations

Branching Brownian motion in a strip: Survival near criticality

“…We point out that the motion of theP R,K -superdiffusion remains unchanged and it is therefore different from the motion of the P R,K -branching diffusion in Definition 4. However, setũ R f (x, t) = λ * (1 − p K (x))(1 − u R f (x, t)), where u R f is the Laplace functional of theP R,K -branching diffusion (Laplace functionals for branching diffusions are defined in a similar fashion to Lemma 6, see for instance Section 4.1.4 [Dyn02]). Then together with the relations (6.5) and (6.6) we can find thatũ R f is the Laplace functional of theP R,K -superdiffusion.…”

An upper bound for positive solutions of the equation Δ𝑢=𝑢^{𝛼}

“…The equivalence of (1.6) and (1.7) is proved, for instance, in [1] (see Theorem 5.1 in Chapter 13). To prove the equivalence of (1.1) and (1.5), we note that ν ∈ M(K) is equal to tµ where t = ν(K) and µ = ν/t ∈ P(K) and…”

Stochastic Solutions of Nonlinear Pde's: McKean Versus Superprocesses

“…[5]. Because of the central role of this result for the construction of superprocesses, a proof is included in the Appendix with the notations used in this paper.…”

“…The …rst stochastic solution for a nonlinear pde was constructed by McKean [4] for the KPP equation. Later on, the exit measures provided by di¤usion plus branching processes [5] [6] as well as the stochastic representations recently constructed for the Navier-Stokes [7] [8] [9] [10] [11], the Vlasov-Poisson [12] [13] [15], the Euler [14] and a fractional version of the KPP equation [16] de…ne solution-independent processes for which the mean values of some functionals are solutions to these equations. Therefore, they are exact stochastic solutions.…”