Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Journal of Theoretical Probability - We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $$Af=g$$ associated with...     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.     

Measure-valued diffusion

We consider the class of continuous measure-valued processes {μ _t } on a finite-dimensional Euclidean space X for which ∫fd μ _t is a semimartingale with absolutely continuous characteristics with respect to t for all f:X→R smooth enough. It is shown that, under some general condition, the Markov process with this property can be obtained as a weak limit for systems of randomly interacting particles that are moving in X along the trajectories of a diffusion process in X as the number of particles increases to infinity.     

Measure-valued almost periodic functions

We consider Stepanov almost periodic functions μ ∈ ranging in the metric space of Borel probability measures on a complete separable metric space is equipped with the Prokhorov metric). The main result is as follows: a function , belongs to if and only if for each bounded continuous function , the function is Stepanov almost periodic (of order 1) and

Measure-valued solutions for a hierarchically size-structured population

We present a hierarchically size-structured population model with growth, mortality and reproduction rates which depend on a function of the population density (environment). We present an example to show that if the growth rate is not always a decreasing function of the environment (e.g., a growth which exhibits the Allee effect) the emergence of a singular solution which contains a Dirac delta mass component is possible, even if the vital rates of the individual and the initial data are smooth functions. Therefore, we study the existence of measure-valued solutions. Our approach is based on the vanishing viscosity method.     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

Measure-valued Markov branching processes conditioned on non-extinction

We consider a particular class of measure-valued Markov branching processes that are constructed as “superprocesses” over some underlying Markov process. Such a processX dies out almost surely, so we introduce various conditioning schemes which keepX alive at large times. Under suitable hypotheses, which include the convergence of the semigroup for the underlying process to some limiting probability measureν, we show that the conditional distribution oft ⁻¹ X _t converges to that ofZν ast → ∞, whereZ is some strictly positive, real random variable. Research supported in part by NSF grant DMS 8701212. Research supported in part by an NSERC operating grant.     

One Dimensional Stochastic Differential Equations with Distributional Drifts

In this paper we study the existence, pathwise uniqueness and homeomorphism flow of strong solutions to a class of one dimensional SDEs driven by infinitely many Brownian motions, and with Yamada- Watanabe diffusion coefficients and distributional drift coefficients.     

Brownian Particles with Rank-Dependent Drifts: Out-of-Equilibrium Behavior

We study the long-range asymptotic behavior for an out-of-equilibrium, countable, one-dimensional system of Brownian particles interacting through their rank-dependent drifts. Focusing on the semi-infinite case, where only the leftmost particle gets a constant drift to the right, we derive and solve the corresponding one-sided Stefan (free-boundary) equations. Via this solution we explicitly determine the limiting particle-density profile as well as the asymptotic trajectory of the leftmost particle. While doing so we further establish stochastic domination and convergence to equilibrium results for the vector of relative spacings among the leading particles. © 2019 Wiley Periodicals, Inc.     

Measure-valued immigration diffusions and generalized ornstein-uhlenbeck diffusions

1.IntroductionLetAbeaboundedlinearoperatoronS(R')whichadmitsanon-positivedefiniteself-adjoillteXtensiononL'(R').AssumethatAgeneratesastronglycontinuoussellilgroupicofboundedlinearoperatorsonS(R').LetBbeaboundedlinearoperatoron).aamtheresultsof[l],wek...     

On Feller processes with sample paths in Besov spaces

Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real-valued Feller processes in terms of Besov spaces . Our result extends previous papers on the paths of Gaussian, symmetric -stable [6], [20], and Lévy processes [11]. Received: 31 May 1996 / Revised version: 10 December 1996     

Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.     

A Note on Feller Semigroups

We present an integral representation of the operators which form a Feller semigroup using a result of P. Courrège.     

关于Feller的一个未解决问题（英）

In Feller＇s well-known book, Volume Ⅱ, second edition [1], there is a very interesting problem：No.29 on p.288 with a large Hint, as follows. If a probability distribution F has the property：for some α 〉 0,     

Elliptic problems with nonlocal boundary conditions and Feller semigroups

This monograph is devoted to the following interrelated problems: the solvability and smoothness of elliptic linear equations with nonlocal boundary conditions and the existence of Feller semigroups that appear in the theory of multidimensional diffusion processes.     

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首先, 当$Q$是一个拟单调的q矩阵的时候, 我们找出最小的$Q$函数是一个Feller的转移函数的准则. 然后我们把这个结论应用于生成分支q矩阵并得到相应的生成分支过程的Feller准则. 特别地, 设$\theta$是分支q矩阵中的非线性数, 总是存在一个分点$\theta_0$满足$1\leq\theta_0\leq2$或$\theta_0<+\infty$使得 生成分支过程是否是Feller的要依据$\theta<\theta_0$或者$\theta>\theta_0$.