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53
Interior a priori estimates for solutions of fully nonlinear equations
 Ann. of Math
, 1989
"... ABSTRACT. We derive an a prioriC 2,α estimate for solutions of the fully nonlinear elliptic equationF(D 2 u) = 0, provided the level setΣ={MF(M)=0} satisfies: (a)Σ∩{MTrM= t} is strictly convex for all constants t; (b) the angle between the identity matrixI and the normalFij toΣis strictly positi ..."
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Cited by 59 (6 self)
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ABSTRACT. We derive an a prioriC 2,α estimate for solutions of the fully nonlinear elliptic equationF(D 2 u) = 0, provided the level setΣ={MF(M)=0} satisfies: (a)Σ∩{MTrM= t} is strictly convex for all constants t; (b) the angle between the identity matrixI and the normalFij toΣis strictly positive on the nonconvex part of Σ. Moreover, we do not need any convexity assumption onF in the course of the proof for the two dimensional case, as the classical result indicates. 1.
On viscosity solutions of fully nonlinear equations with measurable ingredients
 Comm. Pure Appl. Math
, 1996
"... In this paper we study Hölder regularity for the first and second derivatives of continuous viscosity solutions of fully nonlinear equations of the form (1.1) F(D 2 u) = 0. It is well known that viscosity solutions of (1.1) are C 1,α for some 0 < α < 1, and ..."
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Cited by 34 (5 self)
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In this paper we study Hölder regularity for the first and second derivatives of continuous viscosity solutions of fully nonlinear equations of the form (1.1) F(D 2 u) = 0. It is well known that viscosity solutions of (1.1) are C 1,α for some 0 < α < 1, and
Nonlocal Dirichlet forms and symmetric jump processes
 Transactions of the American Mathematical Society
, 1999
"... We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x ..."
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Cited by 30 (16 self)
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We consider the symmetric nonlocal Dirichlet form (E, F) given by E(f, f) = (f(y) − f(x)) 2 J(x, y)dxdy Rd Rd with F the closure of the set of C 1 functions on R d with compact support with respect to E1, where E1(f, f): = E(f, f) + ∫ R d f(x) 2 dx, and where the jump kernel J satisfies κ1y − x  −d−α ≤ J(x, y) ≤ κ2y − x  −d−β for 0 < α < β < 2, x − y  < 1. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E, F). We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.
On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 27 (3 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
On Polynomial Mixing Bounds For Stochastic Differential Equations
 Appl
, 1997
"... : Polynomial bounds for the coefficient of fimixing are established for stochastic dynamical systems under weak recurrency assumptions. The method is based on direct evaluations for certain functionals of hittingtimes of the system under consideration. Keywords: mixing coefficients, stochastic d ..."
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Cited by 26 (8 self)
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: Polynomial bounds for the coefficient of fimixing are established for stochastic dynamical systems under weak recurrency assumptions. The method is based on direct evaluations for certain functionals of hittingtimes of the system under consideration. Keywords: mixing coefficients, stochastic differential equations, stability of Markov chains (R'esum'e : tsvp) 19 Bolshoy Karetnii (Ermolovoy str.), Institute of Information Transmission Problems, Moscow, Russia This paper was supported by INTAS grants # 930894, and # 931585 CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientiøque Institut National de Recherche en Informatique (URA 227) Université de Rennes 1 Insa de Rennes et en Automatique unité de recherche de Rennes Sur les bornes polynomiales des coefficients de m'elange pour des 'equations diff'erentielles stochastiques R'esum'e : Sous la condition de recurrence faible on etablit des bornes polynomiales pour le coefficient de fimelange...
Traveling waves in a onedimensional heterogeneous medium
, 2008
"... We consider solutions of a scalar reactiondiffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling wa ..."
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Cited by 13 (3 self)
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We consider solutions of a scalar reactiondiffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.
Behavior Near The Boundary Of Positive Solutions Of Second Order Parabolic Equations II
 in Proceedings of El Escorial 96
"... A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in nondivergence form in a bounded cylinder Q =\Omega \Theta (0; T ) which vanish on @xQ = @\Omega \Theta (0; T ) ; where\Omega is a bounded Lipschitz domain in R n . This inequality i ..."
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Cited by 8 (3 self)
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A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in nondivergence form in a bounded cylinder Q =\Omega \Theta (0; T ) which vanish on @xQ = @\Omega \Theta (0; T ) ; where\Omega is a bounded Lipschitz domain in R n . This inequality is applied to the proof of the Holder continuity of the quotient of two positive solutions vanishing on a portion of @xQ: 1.
QUASISTATIONARY DISTRIBUTIONS AND DIFFUSION MODELS IN POPULATION DYNAMICS
, 2009
"... In this paper, we study quasistationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to − ∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller d ..."
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Cited by 5 (3 self)
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In this paper, we study quasistationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to − ∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth–death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near 0 and near + ∞ for the existence of quasistationary distributions, as well as rate of convergence in the Yaglom limit and existence of the Qprocess. We also show that under these conditions, there is exactly one quasistationary distribution, and it attracts all initial distributions under the conditional evolution, if and only if + ∞ is an entrance boundary. In particular this gives a sufficient condition for the uniqueness of quasistationary distributions. In the proofs spectral theory plays an important role on L 2 of the reference measure for the killed process.
Fully nonlinear parabolic equations in two space variables
"... Abstract. Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions. 1. ..."
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Cited by 4 (0 self)
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Abstract. Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions. 1.
Polynomial Growth Solutions Of Uniformly Elliptic Operators Of NonDivergence Form
 Proc. AMS 129
, 2001
"... this article, we will study polynomial growth solutions to a uniformly elliptic operator of nondivergence form defined on R ..."
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Cited by 4 (1 self)
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this article, we will study polynomial growth solutions to a uniformly elliptic operator of nondivergence form defined on R