Numerical Solution for the Non-linear Dirichlet Problem of a Branching Process

Lupaşcu, Oana; Stanciulescu, Vasile Nicolae

doi:10.1007/s11785-017-0642-z

Cited by 5 publications

(2 citation statements)

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“…According to Proposition 7, we conclude by this computation that the approximate value of E x ( X t 0 ∈ A) (= the probability that the branching process X starting from x lies in the set A at the time moment t 0 ) is h 1 t 0 (x) = 0.9996, which is indeed a value from the error interval (1 − ε, 1]. We finally note that the above Picard iterations have been recently used in [14] to give a probabilistic numerical approach for a nonlinear Dirichlet problem associated with a branching process.…”

Section: )mentioning

confidence: 59%

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

Beznea

Deaconu

Lupaşcu-Stamate

2019

Mathematics and Computers in Simulation

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This paper builds and develops an unifying method for the construction of a continuous time fragmentation-branching processes on the space of all fragmentation sizes, induced either by continuous fragmentation kernels or by discontinuous ones. This construction leads to a stochastic model for the fragmentation phase of an avalanche. We introduce also an approximation scheme for the process which solves the corresponding stochastic differential equations of fragmentation. A new achievement of the paper is to compute the distributions of the branching processes approximating the forthcoming branching-fragmentation process. This numerical approach of the associated branching-fragmentation process, is, to our knowledge, one of the first in this direction. We present also numerical results that confirm the validity of the fractal property which was emphasized by our model for an avalanche.

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Section: )mentioning

confidence: 59%

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

Beznea

Deaconu

Lupaşcu-Stamate

2019

Mathematics and Computers in Simulation

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“…(iv) A nonlinear Dirichlet problem associated to the equation ( 1) is considered and solved in [7] and [8], while a probabilistic numerical approach is developed in [14].…”

Section: Branching Processes On the Finite Configurations Of R Dmentioning

confidence: 99%

From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes

2019

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We study solutions to the evolution equation ut = ∆u − u + k 1 q k u k , t > 0, in R d . Here the coefficients q k 0 verify k 1 q k = 1 < k 1 kq k < ∞. First, we deal with existence, uniqueness, and the asymptotic behavior of the solutions as t → +∞. We then deduce results on the long time behavior of the associated branching process, with state space the set of all finite configurations of R d , under the assumption that k≥1 k 2 q k < ∞. It turns out that the distribution of the branching process behaves when the time tends to infinity like that of the Brownian motion on the set of all finite configurations of R d . However, due to the lack of conservation of the total mass of the initial non linear equation, a deformation with a multiplicative coefficient occurs. Finally, we establish asymptotic properties of the occupation time of this branching process.

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h-transform of Doob and nonlocal branching processes

2020

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Numerical Solution for the Non-linear Dirichlet Problem of a Branching Process

Abstract: We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.

Cited by 5 publications

References 20 publications

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

Numerical approach for stochastic differential equations of fragmentation; application to avalanches

From Gaussian estimates for nonlinear evolution equations to the long time behavior of branching processes

h-transform of Doob and nonlocal branching processes

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