Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces

Martcheva, Maia; Thieme, Horst R.; Dhirasakdanon, Thanate

doi:10.1007/s00285-006-0002-5

Cited by 9 publications

(11 citation statements)

References 28 publications

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“…Proposition IV.2.10 in ). In particular, inputting

M \equiv 0

, we obtain

w_{ess} (E) \leq w_{0} (E) .

Proposition We have that

w_{0} (B) \leq 0

. Proof Again, since the ODEs associated with B satisfy Assumptions 1,2 in , by Theorem 4 in , we infer that

w_{0} (B) \leq α^{⋄} \lor \underset{k \to \infty}{\lim \sup} k^{- 1} \sum_{j = 1}^{\infty} j α_{j, k} = 0,

since

α^{⋄} = 0

and by (SUB)

\underset{k ↑ \infty}{\lim \sup} k^{- 1} \sum_{j = 1}^{\infty} j α_{j, k} = \lim_{k ↑ \infty} q_{0} \frac{w (k + 1)}{k} = 0

.…”

Section: Proof Of Theorem 21mentioning

confidence: 84%

“…We remark the full statement of (SUB) is not used in this verification or in the proof of the following proposition, only that

\sup_{k} w (k) / k < \infty

. Proposition Both A and B generate strongly continuous semigroups

P_{t}, P_{t}^{B} : Ω \to Ω

with bounds

| | P_{t} | | \leq 2 e^{(2 q_{0} W + 2 | | K | |) t}

and

| | P_{t}^{B} | | \leq 2 e^{2 q_{0} W t}

for

t \geq 0

respectively . Proof By Theorem 2 in , there is a strongly continuous semigroup

P_{t}^{B} : Ω \to Ω

, generated by B restricted to domain

D (B) \cap {x : x_{k} \in ℝ}

where

D (B) : = {x \in Ω : \sum_{k \geq 1} w (k) | x_{k} | < \infty, B x \in Ω}

, with bound

| | P_{t}^{B} | | \leq e^{ω t}

. Extending to complex

x \in D (B)

, where operators act linearly on the real and imaginary parts of x , we have

| | P_{t}^{B} | | \leq 2 e^{ω t}

.…”

Section: Proof Of Theorem 21mentioning

confidence: 97%

“…Next, we construct a strongly continuous semigroup P t corresponding to (), and estimate its growth (Proposition ). Although the transformed ODEs () do not fit into the general theoretical framework of linear ‘Kolmogorov’ differential systems, recently considered in certain host patch/parasite models , nevertheless one can use this framework to prove Proposition .…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…Now, we note the ODEs associated to B ,

\dot{ζ} = B ζ

, fall into the framework of the ‘Kolmogorov’ differential equations considered in . Indeed, given

\sup_{k} w (k) / k \leq W

, in the notation of , with

α_{k, k} = - q_{0} w (k + 1), α_{k + 1, k} = q_{0} w (k + 1)

for

k \geq 0, α_{j, k} = 0

otherwise,

α^{⋄} = \sup_{k} \sum_{j = 0}^{\infty} α_{j, k} = 0, c_{0} = 2 q_{0} W, c_{1} = 2 q_{0} W

, ∊ = 1 and

ω = c_{1} \lor (α^{⋄} + c_{0}) = c_{0}

, one inspects

\sum_{j = 1}^{\infty} j α_{j, k} = q_{0} w (k + 1)

, and the ODEs associated to B satisfy Assumptions 1, 2 in . We remark the full statement of (SUB) is not used in this verification or in the proof of the following proposition, only that

\sup_{k} w (k) / k < \infty

.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…By a change of variables, the ODEs can be written in terms of linear ‘Kolmogorov’ differential equations (cf. ), which can be treated by C 0 ‐semigroup/dynamical systems arguments. In particular, perhaps of interest itself, we show (Theorem ) the ODEs admit a unique nonnegative solution.…”

Section: Introductionmentioning

confidence: 99%

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A scaling limit for the degree distribution in sublinear preferential attachment schemes

Choi

Sethuraman

Venkataramani

2015

Random Struct. Alg.

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ABSTRACT:We consider a general class of preferential attachment schemes evolving by a reinforcement rule with respect to certain sublinear weights. In these schemes, which grow a random network, the sequence of degree distributions is an object of interest which sheds light on the evolving structures.In this article, we use a fluid limit approach to prove a functional law of large numbers for the degree structure in this class, starting from a variety of initial conditions. The method appears robust and applies in particular to 'non-tree' evolutions where cycles may develop in the network.A main part of the argument is to show that there is a unique nonnegative solution to an infinite system of coupled ODEs, corresponding to a rate formulation of the law of large numbers limit, through C 0 -semigroup/dynamical systems methods. These results also resolve a question in Chung, Handjani and Jungreis (2003).

show abstract

“…Proposition IV.2.10 in ). In particular, inputting

M \equiv 0

, we obtain

w_{ess} (E) \leq w_{0} (E) .

Proposition We have that

w_{0} (B) \leq 0

. Proof Again, since the ODEs associated with B satisfy Assumptions 1,2 in , by Theorem 4 in , we infer that

w_{0} (B) \leq α^{⋄} \lor \underset{k \to \infty}{\lim \sup} k^{- 1} \sum_{j = 1}^{\infty} j α_{j, k} = 0,

since

α^{⋄} = 0

and by (SUB)

\underset{k ↑ \infty}{\lim \sup} k^{- 1} \sum_{j = 1}^{\infty} j α_{j, k} = \lim_{k ↑ \infty} q_{0} \frac{w (k + 1)}{k} = 0

.…”

Section: Proof Of Theorem 21mentioning

confidence: 84%

“…We remark the full statement of (SUB) is not used in this verification or in the proof of the following proposition, only that

\sup_{k} w (k) / k < \infty

. Proposition Both A and B generate strongly continuous semigroups

P_{t}, P_{t}^{B} : Ω \to Ω

with bounds

| | P_{t} | | \leq 2 e^{(2 q_{0} W + 2 | | K | |) t}

and

| | P_{t}^{B} | | \leq 2 e^{2 q_{0} W t}

for

t \geq 0

respectively . Proof By Theorem 2 in , there is a strongly continuous semigroup

P_{t}^{B} : Ω \to Ω

, generated by B restricted to domain

D (B) \cap {x : x_{k} \in ℝ}

where

D (B) : = {x \in Ω : \sum_{k \geq 1} w (k) | x_{k} | < \infty, B x \in Ω}

, with bound

| | P_{t}^{B} | | \leq e^{ω t}

. Extending to complex

x \in D (B)

, where operators act linearly on the real and imaginary parts of x , we have

| | P_{t}^{B} | | \leq 2 e^{ω t}

.…”

Section: Proof Of Theorem 21mentioning

confidence: 97%

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…Now, we note the ODEs associated to B ,

\dot{ζ} = B ζ

, fall into the framework of the ‘Kolmogorov’ differential equations considered in . Indeed, given

\sup_{k} w (k) / k \leq W

, in the notation of , with

α_{k, k} = - q_{0} w (k + 1), α_{k + 1, k} = q_{0} w (k + 1)

for

k \geq 0, α_{j, k} = 0

otherwise,

α^{⋄} = \sup_{k} \sum_{j = 0}^{\infty} α_{j, k} = 0, c_{0} = 2 q_{0} W, c_{1} = 2 q_{0} W

, ∊ = 1 and

ω = c_{1} \lor (α^{⋄} + c_{0}) = c_{0}

, one inspects

\sum_{j = 1}^{\infty} j α_{j, k} = q_{0} w (k + 1)

, and the ODEs associated to B satisfy Assumptions 1, 2 in . We remark the full statement of (SUB) is not used in this verification or in the proof of the following proposition, only that

\sup_{k} w (k) / k < \infty

.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A scaling limit for the degree distribution in sublinear preferential attachment schemes

Choi

Sethuraman

Venkataramani

2015

Random Struct. Alg.

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show abstract

Well-Mixed Stochastic Reaction Kinetics

Winkelmann

Schütte

2020

Stochastic Dynamics in Computational Biology

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Moment Closure—A Brief Review

Kuehn

2016

Understanding Complex Systems

100

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Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.

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Kolmogorov’s Differential Equations and Positive Semigroups on First Moment Sequence Spaces

Cited by 9 publications

References 28 publications

A scaling limit for the degree distribution in sublinear preferential attachment schemes

A scaling limit for the degree distribution in sublinear preferential attachment schemes

Well-Mixed Stochastic Reaction Kinetics

Moment Closure—A Brief Review

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