On killing fields preserving minimal foliations of polynomial growth at most 2

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

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A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽

10.1007/s11083-021-09570-7 ◽

2021 ◽

Author(s):

Péter Vrana

Keyword(s):

Compact Hausdorff Space ◽

Polynomial Growth ◽

Hausdorff Space ◽

Continuous Functions ◽

Equivalent Condition ◽

Real Numbers ◽

Asymptotic Spectrum ◽

Ring Of Continuous Functions ◽

Archimedean Property ◽

Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

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The structure of the space of solutions of Einstein's equations. II. Several killing fields and the Einstein-Yang-Mills equations

Annals of Physics ◽

10.1016/0003-4916(82)90224-x ◽

1982 ◽

Vol 143 (1) ◽

pp. 240

Keyword(s):

Einstein’S Equations ◽

Einstein's Equations ◽

Yang Mills ◽

Killing Fields

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Soliton solutions in spacetimes with two spacelike killing fields

Physics Reports ◽

10.1016/0370-1573(93)90139-5 ◽

1993 ◽

Vol 229 (1-2) ◽

pp. 1-80 ◽

Cited By ~ 66

Author(s):

Enric Verdaguer

Keyword(s):

Soliton Solutions ◽

Killing Fields

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Heat equation with singular potential and singular data

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210505000442 ◽

2005 ◽

Vol 135 (4) ◽

pp. 863-886 ◽

Cited By ~ 1

Author(s):

M. Nedeljkov ◽

S. Pilipović ◽

D. Rajter-Ćirić

Keyword(s):

Polynomial Growth ◽

Singular Potential ◽

Function Algebra ◽

Energy Estimates ◽

Generalized Function ◽

Type Condition ◽

Classical Energy ◽

The Cauchy Problem ◽

White Noise Calculus ◽

Singular Data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

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