A superprocess with dependent spatial motion and interactive immigration is constructed as the pa... more A superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by Poisson processes of one-dimensional excursions.
The objective of these lectures is to serve as an introduction to the theory of measure-valued br... more The objective of these lectures is to serve as an introduction to the theory of measure-valued branching processes or super processes.This class of processes first arose from the study of continuous state branching in the work of Jirina (1958, 1964) and (1968). It was also linked to the study of stochastic evolution equations in (1975). In this introduction we look at two roots of this subject, namely, spatially distributed birth and death particle systems and stochastic partial differential equations with non-negative solutions. In Section 2 we carry out some exploratory calculations concerning the continuous limit of branching particle systems and their relation to stochastic partial differential equations. In addition, we introduce the ideas of local spatial clumping with a set of informal calculations that lead to the prediction that the continuum limit of branching particle systems in dimensions d≥3 will lead to infinitely divisible random measures which are almost surely singular.
The main objective of this paper is to examine in some detail the dynamics and fluctuations in th... more The main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior. The model under consideration is a dynamic model of a collection of anharmonic oscillators in a two-well potential together with an attractive mean-field interaction. The system is studied in the limit as the number of oscillators goes to infinity. The limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established. The main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity. It is established that these fluctuations are non-Gaussian and occur at a time scale slower than the noncritical fluctuations. The method used is based on the perturbation theory for Markov processes developed by Papanicolaou, Stroock, and Varadhan adapted to the context of probability-measure-valued processes.
We consider the questions: how can the long term behavior of large systems of interacting compone... more We consider the questions: how can the long term behavior of large systems of interacting components be described in terms of infinite systems? On what time scale does the infinite system give a qualitatively correct description and what happens at large (resp. critical) time scales? LetY N (t) be a solution (y iN(t))i∈[−N,N]of the system of stochastic differential equations (w i(t) are i.i.d. brownian motions) $$dy_i^N (t) = \left( {\frac{1}{{2N + 1}}\sum\limits_{j = - N}^N {y_j^N (t) - y_i^N (t)} } \right)dt + \sqrt {2g(y_i^N (t))} dw_i (t).$$ In the McKean-Vlasov limit,N→∞, we obtain the infinite independent system $$dy_i^\infty (t) = (E(y_i^\infty (t)) - y_i^\infty (t))dt + \sqrt {2g(y_i^\infty (t))} dw_i (t),i \in Z.$$ This infinite system has a one parameter set of invariant measures $v_\Theta = \mathop \otimes \limits_{x \in Z} \Gamma _\Theta $ with Γθ the equilibrium measure of $dx(t) = (\Theta - x(t))dt + \sqrt {2g(x(t))} dw(t)$ . LetQ s(·,·) be the transition kernel of the diffusion with generator $u_g (x)\left( {\frac{\partial }{{\partial x}}} \right)^2 $ with $u_g (x) = \int {g(y)\Gamma x(dy)} $ . Then one main result is that asN→∞ $$\mathcal{L}((Y^N (s(2N + 1)))) \Rightarrow \int {Q_s (\Theta ',d\Theta )v_\Theta ,\Theta '} = E(y_0 ).$$ This provides a new example of a phenomenon also exhibited by the voter model and branching random walk. In particular we are also able to modify our model by adding the termcN −1(A−y iN (t))dt to obtain the first example in which the analog ofQ s (·,·) converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature. Finally, we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales.
The purpose of this paper is to explore the connection between multiple space-time scale behaviou... more The purpose of this paper is to explore the connection between multiple space-time scale behaviour for block averages and phase transitions, respectively formation of clusters, in infinite systems with locally interacting components. The essential object is the associated Markov chain which describes the joint distribution of the block averages at different time scales. A fixed-point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions.
A model of one-dimensional critical branching (superprocess) is constructed in a random medium fl... more A model of one-dimensional critical branching (superprocess) is constructed in a random medium fluctuating both in time and space. The medium describes a moving system of point catalysts, and branching occurs only in the presence of these catalysts. Although the medium has an infinite overall density, the clumping features of the branching model can be exhibited by rescaling time, space, and mass by an exactly calculated scaling power which is stronger than in the constant medium case. The main technique used is the asymptotic analysis of a generalized diffusion-reaction equation in the space-time random medium, which (given the medium) prescribes the evolution of the Laplace transition functional of the Markov branching process.
A critical branching random walk in a d-dimensional spatial random medium (environment) is consid... more A critical branching random walk in a d-dimensional spatial random medium (environment) is considered. It is said to be “persistent” if there is no loss of particle intensity in the large time limit. A critical dimension d c is shown to exist such that the system is persistent if d>d c and fails to be persistent if d<d c.
A superprocess with dependent spatial motion and interactive immigration is constructed as the pa... more A superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation carried by a stochastic flow and driven by Poisson processes of one-dimensional excursions.
The objective of these lectures is to serve as an introduction to the theory of measure-valued br... more The objective of these lectures is to serve as an introduction to the theory of measure-valued branching processes or super processes.This class of processes first arose from the study of continuous state branching in the work of Jirina (1958, 1964) and (1968). It was also linked to the study of stochastic evolution equations in (1975). In this introduction we look at two roots of this subject, namely, spatially distributed birth and death particle systems and stochastic partial differential equations with non-negative solutions. In Section 2 we carry out some exploratory calculations concerning the continuous limit of branching particle systems and their relation to stochastic partial differential equations. In addition, we introduce the ideas of local spatial clumping with a set of informal calculations that lead to the prediction that the continuum limit of branching particle systems in dimensions d≥3 will lead to infinitely divisible random measures which are almost surely singular.
The main objective of this paper is to examine in some detail the dynamics and fluctuations in th... more The main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior. The model under consideration is a dynamic model of a collection of anharmonic oscillators in a two-well potential together with an attractive mean-field interaction. The system is studied in the limit as the number of oscillators goes to infinity. The limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established. The main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity. It is established that these fluctuations are non-Gaussian and occur at a time scale slower than the noncritical fluctuations. The method used is based on the perturbation theory for Markov processes developed by Papanicolaou, Stroock, and Varadhan adapted to the context of probability-measure-valued processes.
We consider the questions: how can the long term behavior of large systems of interacting compone... more We consider the questions: how can the long term behavior of large systems of interacting components be described in terms of infinite systems? On what time scale does the infinite system give a qualitatively correct description and what happens at large (resp. critical) time scales? LetY N (t) be a solution (y iN(t))i∈[−N,N]of the system of stochastic differential equations (w i(t) are i.i.d. brownian motions) $$dy_i^N (t) = \left( {\frac{1}{{2N + 1}}\sum\limits_{j = - N}^N {y_j^N (t) - y_i^N (t)} } \right)dt + \sqrt {2g(y_i^N (t))} dw_i (t).$$ In the McKean-Vlasov limit,N→∞, we obtain the infinite independent system $$dy_i^\infty (t) = (E(y_i^\infty (t)) - y_i^\infty (t))dt + \sqrt {2g(y_i^\infty (t))} dw_i (t),i \in Z.$$ This infinite system has a one parameter set of invariant measures $v_\Theta = \mathop \otimes \limits_{x \in Z} \Gamma _\Theta $ with Γθ the equilibrium measure of $dx(t) = (\Theta - x(t))dt + \sqrt {2g(x(t))} dw(t)$ . LetQ s(·,·) be the transition kernel of the diffusion with generator $u_g (x)\left( {\frac{\partial }{{\partial x}}} \right)^2 $ with $u_g (x) = \int {g(y)\Gamma x(dy)} $ . Then one main result is that asN→∞ $$\mathcal{L}((Y^N (s(2N + 1)))) \Rightarrow \int {Q_s (\Theta ',d\Theta )v_\Theta ,\Theta '} = E(y_0 ).$$ This provides a new example of a phenomenon also exhibited by the voter model and branching random walk. In particular we are also able to modify our model by adding the termcN −1(A−y iN (t))dt to obtain the first example in which the analog ofQ s (·,·) converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature. Finally, we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales.
The purpose of this paper is to explore the connection between multiple space-time scale behaviou... more The purpose of this paper is to explore the connection between multiple space-time scale behaviour for block averages and phase transitions, respectively formation of clusters, in infinite systems with locally interacting components. The essential object is the associated Markov chain which describes the joint distribution of the block averages at different time scales. A fixed-point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions.
A model of one-dimensional critical branching (superprocess) is constructed in a random medium fl... more A model of one-dimensional critical branching (superprocess) is constructed in a random medium fluctuating both in time and space. The medium describes a moving system of point catalysts, and branching occurs only in the presence of these catalysts. Although the medium has an infinite overall density, the clumping features of the branching model can be exhibited by rescaling time, space, and mass by an exactly calculated scaling power which is stronger than in the constant medium case. The main technique used is the asymptotic analysis of a generalized diffusion-reaction equation in the space-time random medium, which (given the medium) prescribes the evolution of the Laplace transition functional of the Markov branching process.
A critical branching random walk in a d-dimensional spatial random medium (environment) is consid... more A critical branching random walk in a d-dimensional spatial random medium (environment) is considered. It is said to be “persistent” if there is no loss of particle intensity in the large time limit. A critical dimension d c is shown to exist such that the system is persistent if d>d c and fails to be persistent if d<d c.
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Papers by Donald Dawson