Non-Autonomous Kato Classes and Feynman-Kac Propagators

Gulisashvili, Archil; Casteren, Jan A. van

doi:10.1142/5972

Cited by 36 publications

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“…71], [24], and [14,Theorem 19,Ch.VIII] for a selection of results in the perturbation theory of linear operators and semigroups on Banach spaces. For recent developments in Schrödinger perturbations of time-nonhomogeneous transition probabilities we refer the reader to [17], [27], [16]. We point out that our main estimate, Theorem 3 below, is more precise and explicit than those mentioned above.…”

Section: (T − S)mentioning

confidence: 87%

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Time-dependent Schrödinger perturbations of transition densities

2008

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Abstract. We construct the fundamental solution of ∂t − ∆y − q(t, y) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition.The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces.We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator. Main results and overview. Let d be a natural number. The Gausand we let g(s, x, t, y) = 0 if s ≥ t. Here x, y ∈ R d are arbitrary. It is well-known that g is a time-homogeneous transition density with respect to the Lebesgue measure, dz, on R d . In particular, for x, y ∈ R d , R d g(s, x, u, z)g(u, z, t, y) dz = g(s, x, t, y) if s < u < t.2000 Mathematics Subject Classification: 47A55, 60J35, 60J57.

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Section: (T − S)mentioning

confidence: 87%

“…We note that if n is a natural number and s < t < s + nh, then ChapmanKolmogorov, (25), Theorem 3,and (17) imply that, for all x, y ∈ X,…”

Section: Y)mentioning

confidence: 99%

Time-dependent Schrödinger perturbations of transition densities

2008

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“…We discuss the strong continuity of propagators on the space L r , the (L r − L q )-smoothing property, and various versions of the Feller property. More results concerning the similarities in the behaviour of semigroups (propagators) and their perturbations by potentials can be found in [1,2,3,4,5,6,7,14,16,17,18,19,20,21,22,32,34,37,38,39,40,42,43,44,45]. In Section 8 we establish that the Feller property, the Feller-Dynkin property, and the BUC-property are inherited by Feynman-Kac propagators from free propagators under additional restrictions on functions and time-dependent measures generating Feynman-Kac propagators.…”

Section: Introductionmentioning

confidence: 93%

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Gulisashvili

2008

Trans. Amer. Math. Soc.

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Abstract. We study the inheritance of properties of free backward propagators associated with transition probability functions by backward FeynmanKac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space L r , the (L r − L q )-smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.

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“…More to the point, the Markov processes of [36] actually emerge as particular cases of reversible diffusions that belong to the larger class of the so-called reciprocal or Bernstein processes, whose theory was launched many years ago in [2] following Schrödinger's seminal contribution in [27]. The theory of Bernstein processes was subsequently further developed and systematically investigated in [19], and since then has played an important rôle in relating various fields such as the Malliavin calculus and Euclidean quantum mechanics, or Markov bridges with jumps and Lévy processes, to name only a few (see for instance [7], [8], [16], [25], [32] and the references therein for a more complete account).…”

Section: Introduction and Outlinementioning

confidence: 99%

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Vuillermot¹,

Zambrini²

2012

J Theor Probab

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In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial-and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a twodimensional disk.

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Non-Autonomous Kato Classes and Feynman-Kac Propagators

Cited by 36 publications

References 0 publications

Time-dependent Schrödinger perturbations of transition densities

Time-dependent Schrödinger perturbations of transition densities

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

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