Global gradient bounds for dissipative diffusion operators

Bogachev VI, Da Prato G, Röckner M, Sobol Z (2004)
Comptes Rendus Mathematique 339(4): 277-282.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Let L be a second order elliptic operator on R-d with a constant diffusion matrix and a dissipative (in a weak sense) drift b is an element of L-loc(p) with some p > d. We assume that L possesses a Lyapunov function, but no local boundedness of b is assumed. It is known that then there exists a unique probability measurelt satisfying the equation L*mu = 0 and that the closure of L in L-1 (mu) generates a Markov semigroup {T-t}(tgreater than or equal to0) with the resolvent {G(lambda)}(lambda>0). We prove that, for any Lipschitzian function f is an element of L-1 (mu) and all t, lambda > 0, the functions T-t f and G(lambda) f are Lipschitzian and sup(x,t) \delT(t) f(x)\ less than or equal to sup(x) \del f (x)\ and sup(x) \delG(lambda)f (x)\ less than or equal to (1)/(lambda) sup(x) \del f (x)\. In addition, we show that for every bounded Lipschitzian function g, the function G(lambda) g is the unique bounded solution of the equation lambdaf - Lf = g in the Sobolev class H-loc(2,2)(R-d).
Erscheinungsjahr
Zeitschriftentitel
Comptes Rendus Mathematique
Band
339
Ausgabe
4
Seite(n)
277-282
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Bogachev VI, Da Prato G, Röckner M, Sobol Z. Global gradient bounds for dissipative diffusion operators. Comptes Rendus Mathematique. 2004;339(4):277-282.
Bogachev, V. I., Da Prato, G., Röckner, M., & Sobol, Z. (2004). Global gradient bounds for dissipative diffusion operators. Comptes Rendus Mathematique, 339(4), 277-282. doi:10.1016/j.crma.2004.05.016
Bogachev, V. I., Da Prato, G., Röckner, M., and Sobol, Z. (2004). Global gradient bounds for dissipative diffusion operators. Comptes Rendus Mathematique 339, 277-282.
Bogachev, V.I., et al., 2004. Global gradient bounds for dissipative diffusion operators. Comptes Rendus Mathematique, 339(4), p 277-282.
V.I. Bogachev, et al., “Global gradient bounds for dissipative diffusion operators”, Comptes Rendus Mathematique, vol. 339, 2004, pp. 277-282.
Bogachev, V.I., Da Prato, G., Röckner, M., Sobol, Z.: Global gradient bounds for dissipative diffusion operators. Comptes Rendus Mathematique. 339, 277-282 (2004).
Bogachev, Vladimir I., Da Prato, Giuseppe, Röckner, Michael, and Sobol, Zeev. “Global gradient bounds for dissipative diffusion operators”. Comptes Rendus Mathematique 339.4 (2004): 277-282.