Potential Kernel Research Articles

Stable processes with drift on the line

The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets.

GROWTH CHARACTERISTICS, YIELD COMPONENTS AND RATE OF GRAIN DEVELOPMENT OF TWO HIGH-YIELDING WHEATS, HY320 AND DT367, COMPARED TO TWO STANDARD CULTIVARS, NEEPAWA AND WAKOOMA

A high-yielding Canada Prairie Spring (CPS cv. HY320) semi-dwarf wheat and a high-yielding durum line (DT367) were compared to standard hard red spring (cv. Neepawa) and durum (cv. Wakooma) wheats to assess the basis for the yield differences and to examine soil moisture use. The study was carried out under dryland (moisture used from seeding to harvest was 287 mm), partial irrigation (moisture used was 374 mm) and full irrigation (moisture used was 547 mm) on a loam soil at Swift Current during 3 yr. Phenological development was delayed as much as 30 d by improved moisture conditions. At ligule of last leaf visible (LLV), HY320 had one or two more green leaves on the main tiller than did Neepawa; Wakooma generally had one more leaf than DT367. However, leaf area index was generally not different among cultivars. Harvest index was greater for the higher yielding cultivars even though cultivars did not differ in total aboveground dry matter. HY320 generally had the highest grain yield, followed by DT367, Wakooma and Neepawa. Neepawa consistently had the most heads per unit area at harvest but, because of high kernel numbers per head, HY320 produced the greatest number of kernels per unit area; as well, HY320 had large kernels; thus, it had the highest yield potential under optimum growing conditions. HY320 had the highest base temperature (Tb) for kernel growth and the highest kernel growth rates immediately after anthesis; consequently, HY320 kernels completed growth with the lowest accumulated degree days above Tb. DT367 had the highest potential kernel weight and Neepawa the lowest. There were no differences among the cultivars in the amount of moisture used between seeding and harvest.Key words: Wheat (Spring), yield components, grain development, moisture use, growing degree days, harvest index

Continuity of excessive harmonic functions for certain diffusions

This paper shows that under certain analytic conditions concerning its potential kernel, a strong Markov process, which is transient and with continuous sample paths, has all of its excessive harmonic functions, which are not identically infinite, continuous, Harnack inequality holds, and a family of its excessive functions is equi-continuous on compacts.

Couplings of Markov chains by randomized stopping times

Summary. We consider a 0-recurrent ergodic Markov chain on (E,N), generated by a kernel P. Again we consider couplings of two chains (~X,), (uX,) starting with the initial distributions v respectively ~t and evolving with P. The coupling consists of two randomized stopping times: T, S, with ~(~xT) = ~(~,Xs). Under additional regularity assumptions we characterize the existence of "short" couplings for two chains (~X~), (uX~) by the property: (v-#,h)=0 for all harmonic functions h fulfilling certain growth conditions. By "short" we mean that the probability to hit CAm before T respectively S decays faster than the analogue quantity for the recurrence times of v and #. Here the -~m are constructed in terms of recurrence times for a certain class of measures (.3, m T E). We will also show that the couplings of the chains (~X~)k~, (~X~')k~ obtained by stopping the original chains, when first hitting C ~]m, converge to a coupling of the original chains, which is also distinguished from other exact couplings by space-time properties. We use these results to characterize the recurrent potential kernel.

Exact singularity cancelling for the potential kernel in the boundary element method

AbstractIt is known that the transformation of a triangular curved element to a local plane square introduces a Jacobian which is effective in dealing with the numerical problems of evaluating the singular integrands arising from the presence of the inverse distance potential kernel in the boundary element method. It is shown that exact cancelling of the singularity occurs for triangular elements with linear, quadratic or cubic shape function representions.

The local structure of the sample paths of asymmetric cauchy processes

Precise information about the local evolution of the sample path is obtained using asymptotic estimates for the potential kernel, for the capacity of small intervals and unions of intervals, and for the hitting distribution of such sets. A new construction of local time is obtained, and it is shown that, except for the completely asymmetric case, the range of the process on a line is a nowhere dense set of positive Lebesgue measure.

Generator coordinate method for the multicluster system

We have introduced a new "hole" formulation by using two rules for deriving the generator coordinate method kernels of the multicluster system with any cluster partitions. The uniform formulae for the overlap kernel, kinetic energy kernel, and potential energy kernel given in this paper can be used easily and are available for computing codes without any further treatment. We have used them to calculate the binding energies of $1p$ shell nuclei and $1p$ shell $\ensuremath{\Lambda}$ hypernuclei.NUCLEAR STRUCTURE Derived formulae of kernels. Calculated binding energies of $1p$ shell nuclei, $\ensuremath{\Lambda}$ hypernuclei.

The Behavior of Asymmetric Cauchy Processes for Large Time

This paper develops precise estimates for the potential kernel, capacities of large intervals, and the probabilities of hitting large intervals for the asymmetric Cauchy processes. These are then applied to study three problems concerning the sample paths: (i) the rate of escape of $|X_t|$ as $t \rightarrow \infty$; (ii) the sizes of the large holes in the range of the process; (iii) the asymptotic behavior of the Lebesgue measure of that part of the range of the process that is in a large interval.

Growth of Spring Wheat Kernels as Influenced by Reduced Kernel Number per Spike and Defoliation1

Field studies with the spring wheat (Triticum aestivum L.) cultivars ‘Era’ and ‘OlaF’ were conducted to assess the effects of reductions in kernel number per spike and defoliation of individual culms on kernel growth characteristics. The importance of stage of kernel growth at which treatments were applied was also examined by treating both at anthesis and anthesis + 14 days, the later date being after the beginning of the kernels linear dry matter accumulation period.The effects of the treatments on final weight per kernel depended on the time of treatment and on the relative weights of kernels from the untreated (control) culms. Reduction in kernel number per spike increased final kernel weight, particularly when control weights were low, whereas defoliation always reduced final weight. Treatment at anthesis usually had a larger effect.Growth rates of kernels during the linear dry matter accumulation period were enhanced by kernel removal and depressed by defoliation. Within each genotype, growth rates were highly correlated with final kernel weight. This association, however, differed between years because of variation in length of the grain filling period. No consistent effect of the treatments on grain filling duration was found.The response to early kernel number reduction, as well as the response in additional experiments to spikelet “trimming” suggested that upper limits, or potentials, for both final weight and growth rate were approached. These potential kernel weights and growth rates appeared to depend on genotype, kernel position in a spikelet, and possibly environment.

Excursions of dual processes

There has been considerable interest in the excursions of a Markov process in recent years. Perhaps the most important general result is that of Maisonneuve [13], in which the existence of an exit system is established. This general result was applied to the study of specific excursions by Getoor in [5]. Then Maisonneuve recast and generalized these results in [14] where he showed how to discuss the excursion straddling a certain class of stopping times T. These included the excursion straddling a fixed time and the first excursion whose length exceeds a given value which were discussed in [5]. The purpose of this paper is to study excursions for processes which satisfy duality conditions. A start in this direction was made in Section 10 of [5] under assumptions that were much too strong. It turns out that the appropriate duality hypotheses are stronger than those of classical duality as described in [I]. We assume the existence of dual transition densities rather than the weaker assumption of dual potential kernel densities. Our precise assumptions are described in Section 2 and especially in Section 3. In particular it is shown in these sections that a systematic use of space-time processes obviates the need for any regularity assumptions on the density over and above the Chapman-Kolmogorov equation. Section 4 summarizes the special properties of additive functionals under our hypotheses that are needed in later sections. In Theorem 4.5 we show that if M is a closed homogeneous optional set, then, under duality assumptions, there exists a Borel set F c E × E such that Me3 ]0, ~[ and {t > 0: (Xt_, Xt) ~ F} are indistinguishable. In Section 5 we begin the discussion of excursions from a closed homogeneous optional set M with R i n f { t > 0 : t C M } . If F ~ E × E corresponds to M as above, R----inf{t: (Xt_,Art) ~ F } almost surely. Especially important is (5.17) which gives the joint distribution of R, X R, and X R . This is obtained by applying a reversal argument to the excursion

Subordinates of the Poisson semigroup

Subordinates of the Poisson semigroup (π t ) t>0 on ℕ0, i.e. measures on ℕ0 of the form $$\int\limits_0^\infty {\pi _t } d\mu (t) = \sum\limits_{n = 0}^\infty {a_n } \varepsilon _n $$ , with an infinitely divisible probability on ℝ+, or a potential kernel on ℝ+, as “mixing” distributionμ, are characterized in terms of recurrence relations for the sequence (a n ) n ≧0

The branching random field

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.

The branching random field

The branching random field is studied under general branching and diffusion laws. Under a renormalization transformation it is shown that at finite fixed time the branching random field converges in law to a generalized Gaussian random field with independent increments. Very mild moment conditions are imposed on the branching process. Under more restrictive conditions on the branching and diffusion processes, the existence of a steady state distribution is proven in the critical case. A central limit theorem is proven for the renormalized steady state, but the limiting Gaussian random field no longer has independent increments. The covariance kernel is now a multiple of the potential kernel of the diffusion process.

Potential Theory of Lévy Processes

Let X be a transient Lévy process in euclidean space. In this paper we consider some aspects of the potential theory of X when the potential kernel is not necessarily absolutely continuous. First we find exactly when the resolvent operators and the semigroup operators are strong Feller operators. Next we extend the Orey–Kanda theorem on the comparison of essentially polar sets and consider some properties of the energy minimizing distributions for non-symmetric processes. Finally we obtain results on the relationships between the classes of exceptional sets. We also give a number of examples illustrating possible kinds of behaviour of Lévy processes.

POLICY IMPROVEMENT IN MARKOV DECISION PROCESSES AND MARKOV POTENTIAL THEORY

A connection between Markov Decision Process (MDP) and Markov potential theory has two sides. One is the potential theoritic development of MDP and the other is the alternative proof of the results in MDP owing to Markov potential theory. Shaufele [12] belongs to the later, but it seems interesting from the standpoint of the mathematical programming to establish the development of MDP by using certain potential notion. Several approaches have been tried. Watanabe [16] interpreted the monotonicity of Howard's iteration [8] in the relation to the a dual problem of Linear Programming. By the property of a potential kernel, Furukawa [6] and Aso and Kimura [1] proved a policy improvement. A formulation of MDP by potential theoretic notion has been tried by Hordijk [7]. In many cases it is restricted to a transient potential theory because its analysis is simpler. In this paper we shall define a new potential in order to serve a general policy improvement. Our aim is to expose theorems which are available to several cases of MDP. By the potential theoretic terms, we can interpret policy improvements of MDP as follows ; The increase of rewards in MDP consists of the potential with a charge of an increment of the policy improvement and a regular function. If it is transient, then the potential is reduced to the ordinary one and the regular function equals zero. Hence this consists with that of Watanabe [16]. The merit of the potential is that it connects the policy improvement with the increment of rewards. We shall consider the following cost criteria of MDP ; (1) discounted case, (2) average case, (3) nearly optimal case and (4) sensitive discounted case. Case (1) and (2) are representitive and discussed by many authors. Especially we list up Howard [7] and Blackwell [2], [3] for (1) and Howard [8] and Derman [4], [5] for (2). Case (3) is due to Blackwell [2]. Extending case (3), case (4) is studied by Miller and Veinott [11] and Veinott [14], [15].

A characterization of potential kernels on the positive half-line

The set P of potential kernels on the half-line [0, τ[ is characterized as the set of positive measures k on [0,τ[ for which the Laplace transform f=ℒк has the property that the sequence $$\left( {S^n ( - 1)^n \frac{{D^n f(s)}}{{n!}}} \right)_{n \geqq 0} $$ is proportional with a renewal sequence for each s>0. This is analogous to a characterization, due to F. Hirsch, of a convex cone H contained in P. The corresponding result for another convex cone S⫅H is also given.

Effect of environment and cultivar on source limitation to grain weight in wheat

The relative increase in final kernel weight (SR, %) with an artificial reduction of about 80% in the number of grains per spike was assumed to estimate the degree of post-anthesis assimilate or source limitation to grain weight in wheat. This assumption was supported by comparisons with other treatments designed to alter the ratio of source to grain number. SR was determined in crops grown under irrigation and high fertility in north-west Mexico over a 5 year period, in order to examine the effect of environment (year, sowing date) and cultivar on source limitation. Some old tall cultivars and many modern short cultivars of wheat (Triticum aestivum L., T. turgidum L.) and triticale (X Tritosecale Wittmack) were studied. For modern cultivars, kernel weight always appeared to be limited by source, SR averaging 21% over all years for eight such cultivars. For a given cultivar, environmental effects were considerable, being partly explained by a positive relationship between SR and mean temperature after anthesis. Each year SR was significantly affected by cultivar and, despite cultivar by year interaction, showed some consistency between years. Old bread wheats and triticales usually showed low source limitation (SR <10%), whereas most modern cultivars, regardless of species, showed higher source limitation (SR up to 50%) and higher grain yields, probably because of higher grain numbers per sq metre. Within this latter group, there was no relationship between SR and grain yield. The kernel weight attained as a result of grain number reduction, defined here as potential kernel weight, was a stable cultivar trait useful in understanding yield variation.

On a construction of a recurrent potential kernel by mean of time change and killing

On a construction of a recurrent potential kernel by mean of time change and killing

Harmonic spaces associated with non-symmetric translation invariant Dirichlet forms

Continuing earlier work on construction of harmonic spaces from translation invariant Dirichlet spaces defined on locally compact abelian groups, it is shown that the potential kernel for a non-symmetric translation invariant Dirichlet form on a locally compact abelian group under the extra assumptions that (i) the potential kernel is absolutely continuous and the “canonical” l.s.c. density is continuous in the complement of the neutral element. (ii) the theory is of local type. (iii) the underlying group is not discrete, can be interpreted as the potential kernel for a translation invariant axiomatic theory of harmonic functions, in which (among other properties) the domination axiom is fulfilled.

The definition of energy in non-symmetric translation invariant Dirichlet spaces

In classical potential theory the set of positive measures of finite energy can be characterized in several ways: by finiteness of the "energy-integral", by continuity of the "energy-function" and by a condition of square-integrability involving the Fourier transformation, see Deny [5]. The purpose of the paper is to study analogous characterizations m the framework of a non-symmetric translation invariant Dirichlet space Y~ on a locally compact abelian group. It turns out that the property of being of finite energy with respect to ~V is equivalent with continuity of the "energy-function" and (for bounded measures) equivalent with square-integrability of the Fourier transform of the measure with respect to the reciprocal of the continuous negative definite function associated with ~V. If the potential kernel of Y" is absolutely continuous then, imitating a construction of It6 for the symmetric case, cf. [7], a lower semicontinuous density function for the potential kernel is constructed. Finiteness of the "energy-integral" with respect to this density is a necessary but not a sufficient condition for a positive measure to be of finite energy. It is shown that finiteness of the "energyintegral" characterizes the set of positive measures of finite energy if and only if the density function satisfies the continuity principle, and this is in particular the case when the Dirichlet space is symmetric.