Abstract
It is well known that for a large class of Markov process the associated semi-group T(t)f(x)=∫f(y)P(t,x;dy) satisfies the Kolmogorov backward differential equation, that is, if u(t,x)=T(t)f(x) then \(\frac{{\partial U}}{{\partial t}} = \frac{1}{2}a(x)\frac{{\partial ^2 U}}{{\partial x^2 }} + b(x)\frac{{\partial U}}{{\partial x}}\) and \(\mathop {\lim }\limits_{t \downarrow 0} {\text{ }}U(t,x) = U(0,x) = f(x)\).
In this paper we are considering the opposite problem: given the diffusion and drift coefficients we study the differentiability preserving properties of the semigroup T(t) having as infinitesimal generator \(A = \frac{1}{2}a(x)\frac{{\partial ^2 }}{{\partial x^2 }} + b(x)\frac{\partial }{{\partial x}}\).
More specifically, for a large class of functions a(x) and b(x), we will prove for k=0, ..., 3 the existence of T(t) such that T(t): C k (I)→ C k (I) and the existence of a constant Μ k such that |T(t)f| k ≦|f| k exp (Μ k t) for fεC k (I). Moreover an explicit expression of Μ k in terms of the coefficients a(x) and b(x) is obtained. As a side result we obtain the necessity of the boundary conditions imposed.
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This paper is a revised version of the author's Ph. D. dissertation at University of Massachusetts under W. Rosenkrantz
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Dorea, C.C.Y. Differentiability preserving properties of a class of semigroups. Z. Wahrscheinlichkeitstheorie verw Gebiete 36, 13–26 (1976). https://doi.org/10.1007/BF00533206
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DOI: https://doi.org/10.1007/BF00533206