Superprocesses and Partial Differential Equations

Dynkin, E. B.

doi:10.1214/aop/1176989116

Cited by 165 publications

(151 citation statements)

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“…In the As is usual in the analysis of nonlinear phenomena, we use a geometric approach to (5). For a continuous function u define the weighted norm u φ −1 = sup x u(x)φ −1 (x) , where φ is the ground state of L + β − λ c .…”

Section: Definition 6 (Local Extinction)mentioning

confidence: 99%

“…For a continuous function u define the weighted norm u φ −1 = sup x u(x)φ −1 (x) , where φ is the ground state of L + β − λ c . Under certain conditions guaranteed by Theorem 1 or 2 below, we prove in Lemma 20 (Section 3) the existence of a special smooth curve u = ψ(σ ), σ ∈ [0, ∞), in the space of nonnegative functions bounded in the norm · φ −1 , such that ψ(0) = 0 and ψ (0) = φ and that the curve is invariant under the positive time shift u(0) → u(t) defined by (5). Thus, the curve emanates from zero and is tangent at zero to the one-dimensional invariant (with respect to the semigroup {T t } t≥0 ) subspace, spanned by φ.…”

Section: Definition 6 (Local Extinction)mentioning

confidence: 99%

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A scaling limit theorem for a class of superdiffusions

Engländer¹,

Turaev²

2002

Ann. Probab.

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Consider the σ -finite measure-valued diffusion corresponding to the evolution equation u t = Lu + β(x)u − f (x, u), whereand n is a smooth kernel satisfying an integrability condition. We assume that β, α ∈ C η (R d ) with η ∈ (0, 1], and α > 0. Under appropriate spectral theoretical assumptions we prove the existence of the random measure(with respect to the vague topology), where λ c is the generalized principal eigenvalue of L + β on R d and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover, we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator L + β.When β is bounded from above, X is finite measure-valued. In this case, under an additional assumption on L + β, we can actually prove the existence of the previous limit with respect to the weak topology.As a particular case, we show that if L corresponds to a positive recurrent diffusion Y and β is a positive constant, then lim t ↑∞ e −βt X t (dx) exists and equals a nonnegative nondegenerate random multiple of the invariant measure for Y .Taking L = 1 2 on R and replacing β by δ 0 (super-Brownian motion with a single point source), we prove a similar result with λ c replaced by 1 2 and with the deterministic measure e −|x| dx, giving an answer in the affirmative to a problem proposed by Engländer and Fleischmann [Stochastic Process. Appl. 88 (2000) 37-58].The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.

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Section: Definition 6 (Local Extinction)mentioning

confidence: 99%

Section: Definition 6 (Local Extinction)mentioning

confidence: 99%

A scaling limit theorem for a class of superdiffusions

Engländer¹,

Turaev²

2002

Ann. Probab.

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“…They consider (1.4) in the case σ ≡ 1 to study the so-called superdiffusions (see [30], [14], [15]) for 1 < q ≤ 2. Unfortunately, at the moment no probabilistic models seem to be known for (1.2), or (1.4) with q > 2.…”

Section: (2) Conversely (12) Has a Solution With Zero Boundary Valumentioning

confidence: 99%

Nonlinear equations and weighted norm inequalities

Kalton

Verbitsky

1999

Trans. Amer. Math. Soc.

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Abstract. We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem −∆u = v u q + w, u ≥ 0 on Ω,on a regular domain Ω in R n in the "superlinear case" q > 1. The coefficients v, w are arbitrary positive measurable functions (or measures) on Ω. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between v, w, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on v and w; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if v ≡ 1 and Ω is a ball or half-space.The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called 3G-inequality by an elementary "integration by parts" argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.

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“…It is also worth to mention the deep connection of problems like (1.3) with stochastic superprocesses and the so-called Brownian snake, see [15,24] and references therein. Additionally, we remark that the study of large solutions is also addressed in second-order quasilinear problems associated to coercive gradient terms in [23] and in infinity Laplacian problems in [21].…”

Section: 1)mentioning

confidence: 99%

Large solutions for a class of semilinear integro-differential equations with censored jumps

Rossi

Topp

2016

Journal of Differential Equations

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Abstract. We study existence of large solutions, that is, solutions that verify u(x) → +∞ as x → ∂Ω, for equations likewhere Ω is a bounded smooth domain in R N , p > 1 and I is a nonlocal operator of the formwhere α ∈ (0, 2) and :Ω → R is a function whose main particularity is that 0 < (x) ≤ dist(x, ∂Ω). We also obtain uniqueness of the solution in a class of large solutions whose blow-up rate depends on p, α and the rate at which shrinks near the boundary.

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Superprocesses and Partial Differential Equations

Cited by 165 publications

References 0 publications

A scaling limit theorem for a class of superdiffusions

A scaling limit theorem for a class of superdiffusions

Nonlinear equations and weighted norm inequalities

Large solutions for a class of semilinear integro-differential equations with censored jumps

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