scholarly journals Nonexplosion of a class of semilinear equations via branching particle representations

2008 ◽  
Vol 40 (01) ◽  
pp. 250-272
Author(s):  
Santanu Chakraborty ◽  
Jose Alfredo López-Mimbela
Keyword(s):  
Differential Equation ◽  
Infinitesimal Generator ◽  
Branching Process ◽  
Random Number ◽  
Particle System ◽  
Sufficient Conditions ◽  
Multiplicative Semigroup ◽  
Branching Particle System ◽  
Semilinear Equations ◽  
Semilinear Partial Differential Equation

We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given bypk,k= 2, 3, …. The corresponding branching process is related to the semilinear partial differential equationforx∈ ℝd, whereAis the infinitesimal generator of a multiplicative semigroup and thepks,k= 2, 3, …, are nonnegative functions such thatWe obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

scholarly journals Nonexplosion of a class of semilinear equations via branching particle representations

2008 ◽  
Vol 40 (1) ◽  
pp. 250-272 ◽  
Author(s):  
Santanu Chakraborty ◽  
Jose Alfredo López-Mimbela
Keyword(s):  
Differential Equation ◽  
Infinitesimal Generator ◽  
Branching Process ◽  
Random Number ◽  
Particle System ◽  
Sufficient Conditions ◽  
Multiplicative Semigroup ◽  
Branching Particle System ◽  
Semilinear Equations ◽  
Semilinear Partial Differential Equation

We consider a branching particle system where an individual particle gives birth to a random number of offspring at the place where it dies. The probability distribution of the number of offspring is given by pk, k = 2, 3, …. The corresponding branching process is related to the semilinear partial differential equation for x ∈ ℝd, where A is the infinitesimal generator of a multiplicative semigroup and the pks, k = 2, 3, …, are nonnegative functions such that We obtain sufficient conditions for the existence of global positive solutions to semilinear equations of this form. Our results extend previous work by Nagasawa and Sirao (1969) and others.

PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

2007 ◽  
Vol 10 (03) ◽  
pp. 439-464 ◽  
Author(s):  
ANNA TALARCZYK
Keyword(s):  
Local Time ◽  
Particle System ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Particle Systems ◽  
The Self ◽  
Intersection Local Time ◽  
Branching Particle System ◽  
Independent Particle ◽  
Density Process

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Oscillation and non-oscillation of some neutral differential equations of odd order

1992 ◽  
Vol 15 (3) ◽  
pp. 509-515 ◽  
Author(s):  
B. S. Lalli ◽  
B. G. Zhang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Neutral Differential Equation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Odd Order ◽  
Existence Criterion ◽  
Oscillation Of Solutions

An existence criterion for nonoscillatory solution for an odd order neutral differential equation is provided. Some sufficient conditions are also given for the oscillation of solutions of somenth order equations with nonlinearity in the neutral term.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

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