# zbMATH — the first resource for mathematics

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 $\sup_{x\in \mathbb{R}^d}\left |{d \over dt}(P_t\varphi) (x)\right|= \bigl\|A_0(P_t \varphi)\bigr \|_0 \leq c(t\wedge 1)^{-1} \|\varphi \|_0.$

##### MSC:
 35J70 Degenerate elliptic equations 60H30 Applications of stochastic analysis (to PDEs, etc.) 47D03 Groups and semigroups of linear operators
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.