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1.
Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process
can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming
principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability
sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for
this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean
curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications
in this direction are also discussed.
Received October 24, 2000 / final version received July 24, 2001?Published online November 27, 2001 相似文献
2.
Summary. The super-Brownian motion X
ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour)
in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical
dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical'
super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction.
Received: 21 November 1996 / In revised form: 31 March 1997 相似文献
3.
Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability
one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundaryΩ of the tree. The random subset Λ of the boundary consisting of all ends of the tree in which the population survives, called
the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric
boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff
dimension of Λ∩Ωμ, where Ωμ is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ωμ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of Λ and Λ∩Ωμ, and it is shown that the log Hausdorff dimension of Λ has critical exponent 1/2 at the phase separation point.
Received: 30 June 1998 / Revised version: 10 March 1999 相似文献
4.
S. Fourati 《Probability Theory and Related Fields》1998,110(1):13-49
Summary. We prove a conjecture of J. Bertoin: a Lévy process has increase times if and only if the integral is finite, where G and H are the distribution functions of the minimum and the maximum of the Lévy process killed at an independent exponential time.
The “if” part of the statement had been obtained before by R. Doney. Our proof uses different techniques, from potential theory
and the general theory of processes, and is self-contained. Our results also show that if P(X
t
<0)≤1/2 for all t small enough, then the process does not have increase times.
Received: 4 May 1995/In revised form: 6 May 1997 相似文献
5.
We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply 相似文献
6.
Summary. A super-Brownian motion in with “hyperbolic” branching rate , is constructed, which symbolically could be described by the formal stochastic equation (with a space-time white noise ). Starting at
this superprocess will never hit the catalytic center: There is an increasing sequence of Brownian stopping times strictly smaller than the hitting time of such that with probability one Dynkin's stopped measures vanish except for finitely many
Received: 27 November 1995 / In revised form: 24 July 1996 相似文献
7.
We study the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. We
show that if the lateral boundary satisfies a capacity density condition and if boundary and initial values are smooth enough,
then quasiminimizers globally belong to a higher Sobolev space than assumed a priori. We derive estimates near the lateral
and the initial boundaries. 相似文献
8.
《偏微分方程通讯》2013,38(1-2):43-100
It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view. In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model). 相似文献
9.
This paper studies, under some natural monotonicity conditions, the theory (existence and uniqueness, a priori estimate,
continuous dependence on a parameter) of forward–backward stochastic differential equations and their connection with quasilinear
parabolic partial differential equations. We use a purely probabilistic approach, and allow the forward equation to be degenerate.
Received: 12 May 1997 / Revised version: 10 January 1999 相似文献
10.
11.
12.
Kenneth S. Alexander 《Probability Theory and Related Fields》1998,110(4):441-471
Summary. For lattice models on ℤ
d
, weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance
from that region. For a wide class of models on ℤ2, including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an
appropriate form of connectivity, and a natural coupling property. In particular, on ℤ2, the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially,
and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied
if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range,
weak mixing implies ratio weak mixing.
Received: 27 August 1996 / In revised form: 15 August 1997 相似文献
13.
We show that in dimensions two or more a sequence of long range contact processes suitably rescaled in space and time converges to a super-Brownian motion with drift. As a consequence of this result we can improve the results of Bramson, Durrett, and Swindle (1989) by replacing their order of magnitude estimates of how close the critical value is to 1 with sharp asymptotics. Received: 2 February 1998 / Revised version: 28 August 1998 相似文献
14.
《偏微分方程通讯》2013,38(7-8):1117-1132
Barrier type boundary conditions are modeled for describing the substance diffusion in a medium when obstacles in the medium are considered. The existence of solutions of parabolic and elliptic differential equations with barrier boundary condition is presented in this paper. 相似文献
15.
《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(3-4):153-164
We prove that all the Dirichlet forms associated with certain diffusions on a d-set are equivalent and that their common domain is an integral Lipschitz space. We also provide an analytic characterisation of the walk dimension dw of a d-set F and show that all fractional diffusions on F share dw as their walk dimension. 相似文献
16.
17.
Ito's rule is established for the diffusion processes on the graphs. We also consider a family of diffusions processes with
small noise on a graph. Large deviation principle is proved for these diffusion processes and their local times at the vertices.
Received: 12 February 1997 / Revised version: 3 March 1999 相似文献
18.
Silvia Cingolani José Luis Gámez 《Calculus of Variations and Partial Differential Equations》2000,11(1):97-117
We study a symmetric semilinear elliptic problem in all and we prove existence of an asymmetric positive solution by using variational arguments. The corresponding problem in dimension N=2, which provides the motivation of this work, arises in Nonlinear Optics from the study of the behaviour of optical cylindrical waveguides. Received September 28, 1999/ Accepted January 14, 2000 / Published online June 28, 2000 相似文献
19.
《随机分析与应用》2013,31(6):1281-1307
The paper is devoted to the generalized stochastic differential equations of the Ito? type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions. 相似文献
20.
We consider a class of multitype particle systems in
d
undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria). 相似文献