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Analytic semigroups and degenerate elliptic operators with unbounded coefficients: A probabilistic approach. (English) Zbl 0957.35053
The paper deals with second order degenerate elliptic operators \[ A_0u(x)= \textstyle {1\over 2}\sum^d_{i,j=1} a_{ij}(x)D_{ij} u(x)+\sum^d_{i=1} b_i(x)D_i u(x),\quad x\in\mathbb{R}^d, \] where the coefficients \(a_{ij}(x)\) and \(b_i(x)\) can be undefined. The matrix \(a(x)=\{a_{ij}(x)\}\) is positive semi-definite and has quadratic growth, and the vector \(b(x)\) has linear growth and is of class \(C^2\). \(A_0\) is the diffusion operator associated with the stochastic differential problem \[ d\xi(t)= b\bigl(\xi(t) \bigr)dt+\sqrt {a\bigl(\xi(t) \bigr)} dw(t),\;\xi(0)=x, \] where \(w(t)=(w_1(t), \dots,w_d(t))\) is a standard dimensional Brownian motion. Based on this fact and using the technique of investigating stochastic differential equations, the author proves that the operator \(A_0\) generates an analytic semigroup \[ P_t\varphi(x) =E\varphi \bigl( \xi(t,x)\bigr),\;t\geq 0,\;x\in\mathbb{R}^D \] in the space \(C_b(\mathbb{R}^d)\) in the sense that if \(\varphi\in C_b(\mathbb{R}^d)\), then \(P_t\varphi\in D(A_0)\). The mapping of \(P_t\varphi(x)\) is differentiable and one has \[ \sup_{x\in \mathbb{R}^d}\left |{d \over dt}(P_t\varphi) (x)\right|= \bigl\|A_0(P_t \varphi)\bigr \|_0 \leq c(t\wedge 1)^{-1} \|\varphi \|_0. \]

35J70 Degenerate elliptic equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
47D03 Groups and semigroups of linear operators
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[1] Aronson, D.G.; Besala, P., Parabolic equations with unbounded coefficients, J. differential equations, 3, 1-14, (1967) · Zbl 0149.06804
[2] Baouendi, M.S.; Goulaouic, C., Regularité analitique et itérés d’operateurs elliptiques Dégénérés, applications, J. functional anal., 9, 208-248, (1972) · Zbl 0243.35044
[3] Besala, P., On the existence of a foundamental solution for a parabolic equation with unbounded coefficients, Ann. polon. math., 29, 403-409, (1975) · Zbl 0305.35047
[4] Bismut, J.M., Martingales, the Malliavin calculus and hypoellipticity general Hörmander’s conditions, Z. wahrscheinlichkeitstheorie gebiete, 56, 469-505, (1981) · Zbl 0445.60049
[5] Brezis, H.; Rosenkrantz, W.; Singer, B., On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. pure appl. math., 24, 395-416, (1971) · Zbl 0206.11203
[6] Campiti, M.; Metafune, G., Ventcel’s boundary conditions and analytic semigroups, Arch. math., 70, 377-390, (1998) · Zbl 0909.34051
[7] M. Campiti, G. Metafune, and, D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, to appear. · Zbl 0915.47029
[8] Cannarsa, P.; Vespri, V., Generation of 1nalytic semigroups by elliptic operators with unbounded coefficients, SIAM J. math. anal., 18, 857-872, (1987) · Zbl 0623.47039
[9] Cannarsa, P.; Vespri, V., Generation of analytic semigroups in the L^{p} topology by elliptic operators in \(R\)n, Israel J. math., 61, 235-255, (1988) · Zbl 0669.35026
[10] Cerrai, S., A Hille Yosida theorem for weakly continuous semigroups, Semigroup forum, 49, 349-367, (1994) · Zbl 0817.47048
[11] Clément, Ph.; Timmermans, C.A., On C0-semigroups generated by differential operators satisfying Ventcel’s boundary conditions, Indag. math., 89, 379-387, (1986) · Zbl 0618.47035
[12] Da Prato, G.; Lunardi, A., On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. functional anal., 131, 94-114, (1995) · Zbl 0846.47004
[13] Devinatz, A., Self-adjointness of second order degenerate elliptic operators, Indiana univ. math. J., 27, 255-266, (1978) · Zbl 0384.35022
[14] Dynkin, E.B., Markov processes, (1968), Academic Press/Springer-Verlag New York · Zbl 0169.49102
[15] Elworthy, K.D.; Li, X.M., Formulae for the derivatives of heat semigroups, J. functional anal., 125, 252-286, (1994) · Zbl 0813.60049
[16] Ethier, S.N.; Kurtz, Th.G., Markov processes, characterization and convergence, Wiley series in probability and mathematical statistics, (1986), Wiley New York
[17] Favini, A.; Goldstein, J.A.; Romanelli, S., An analytic semigroup associated to a degenerate evolution equation, (), 85-100 · Zbl 0889.35039
[18] Feller, W., Two singular diffusion problems, Ann. of math., 54, 173-182, (1951) · Zbl 0045.04901
[19] Feller, W., The parabolic differential equations and the associated semigroups of transformations, Ann. of math., 55, 468-519, (1952) · Zbl 0047.09303
[20] Freidlin, M., Functional integration and partial differential equations, Annals of mathematics studies, (1985), Princeton University Press Princeton · Zbl 0568.60057
[21] Friedman, A., Stochastic differential equations and applications, (1975), Academic Press New York
[22] F. Gozzi, R. Monte, and, V. Vespri, Generation of analytic semigroups for degenerate elliptic operators arising in financial mathematics, preprint, 1997. · Zbl 1033.47028
[23] Krylov, N.V., Introduction to the theory of diffusions processes, (1995), American Mathematical Society Providence · Zbl 0844.60050
[24] Liskevich, V.A.; Perelmuter, M.A., Analyticity of sub-Markovian semigroups, Proc. amer. math. soc., 123, 1097-1104, (1995) · Zbl 0826.47030
[25] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, (1995), Birkhäuser Basel · Zbl 0816.35001
[26] Stein, E.M., Topics in harmonic analysis (related to littlewood – paley theory), Annals of mathematics studies, (1970), Princeton University Press Princeton · Zbl 0193.10502
[27] Stewart, B., Generation of analytic semigroups by strongly elliptic operators, Trans. amer. math. soc., 199, 141-162, (1974) · Zbl 0264.35043
[28] Stewart, B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. amer. math. soc., 259, 299-310, (1980) · Zbl 0451.35033
[29] Stroock, D.W.; Varadhan, S.R.S., Multidimensional diffusion processes, (1979), Springer-Verlag Berlin
[30] Vespri, V., Analytic semigroups, degenerate elliptic operators and applications to non-linear Cauchy problems, Ann. mat. pura appl., 155, 353-388, (1989) · Zbl 0709.35065
[31] Yosida, K., Functional analysis, (1980), Springer-Verlag Berlin/Heidelberg · Zbl 0152.32102
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