The positive maximum principle on symmetric spaces
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Positivity
The positive maximum principle on symmetric spaces David Applebaum1
· Trang Le Ngan1
Received: 23 July 2019 / Accepted: 22 February 2020 © The Author(s) 2020
Abstract We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller–Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo-differential operators having scalar symbols. Keywords Positive maximum principle · Courrege theorem, symmetric space · Lie group · Lie algebra · Levy kernel · Feller process · Spherical Levy process · Pseudo-differential operator Mathematics Subject Classification 47E20 · 47D07 · 43A85 · 47G30 · 60B15
1 Introduction Consider a linear operator A defined on the space Cc∞ (Rd ) of smooth functions of compact support. If it satisfies the positive maximum principle (PMP), then by a classical theorem of Courrège [8] A has a canonical form as the sum of a second-order elliptic differential operator and a non-local integral operator. Furthermore A may also be written as a pseudo-differential operator whose symbol is of Lévy–Khintchine type (but with variable coefficients). This result is of particular importance for the study of Feller–Markov processes in stochastic analysis. The infinitesimal generator of such a process always satisfies the PMP, and so has the canonical form just indicated. The
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David Applebaum [email protected] Trang Le Ngan [email protected]
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School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, England
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D. Applebaum, T. L. Ngan
use of the symbol to investigate probabilistic properties of the process has been an important theme of much recent work in this area (see e.g. [13], [7] and references therein). In a recent paper [5], the authors generalised the Courrège theorem to linear operators satisfying the PMP on a Lie group G. The key step was to replace the set of first order partial derivatives {∂1 , . . . , ∂d } by a basis {X 1 , . . . , X d } for the Lie algebra g of G. In this case, when G is compact, we find that the operator is a pseudo-differential operator in the sense of Ruzhansky and Turunen [18], with matrix-valued symbols, obtained using Peter–Weyl theory. In the current paper, we extend the Courrège theorem to symmetric spaces M. Since any such space is a homogeneous space G/K , where K is a compact subgroup of the Lie group G, we may conjecture that the required result can be obtained from that of [5] by use of projection techniques; however, this is not the case as a linear operator that satisfies the PMP on Cc∞ (G/K ) may not satisfy it on Cc∞ (G). Nonetheless, a straightforward variation on the proof given in [5] does enable us to derive the required re
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