Positive Densities of Transition Probabilities of Diffusion Processes

Bogachev VI, Röckner M, Shaposhnikov SV (2009)
Theory of Probability & Its Applications 53(2): 194-215.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Bogachev, Vladimir I.; Röckner, MichaelUniBi; Shaposhnikov, S. V.
Abstract / Bemerkung
For diffusion processes in R-d with locally unbounded drift coefficients we obtain a sufficient condition for the strict positivity of transition probabilities. To this end, we consider parabolic equations of the form L* mu = 0 with respect to measures on R-d x ( 0, 1) with the operator Lu := partial derivative(t)u+partial derivative(xi) (a(ij)partial derivative(xj)u) + b(i)partial derivative(xi)u. It is shown that if the diffusion coefficient A = (a(ij)) is sufficiently regular and the drift coefficient b = (b(i)) satisfies the condition exp(kappa vertical bar b vertical bar(2)) is an element of L-loc(1)(mu), where the measure mu is nonnegative, then mu has a continuous density rho(x, t) which is strictly positive for t > tau provided that it is not identically zero for t <= tau. Applications are obtained to finite-dimensional projections of stationary distributions and transition probabilities of infinite-dimensional diffusions.
Stichworte
stationary distribution; density of transition probability; parabolic; equation; infinite-dimensional diffusion
Erscheinungsjahr
2009
Zeitschriftentitel
Theory of Probability & Its Applications
Band
53
Ausgabe
2
Seite(n)
194-215
ISSN
0040-585X
Page URI
https://pub.uni-bielefeld.de/record/1591777

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Bogachev VI, Röckner M, Shaposhnikov SV. Positive Densities of Transition Probabilities of Diffusion Processes. Theory of Probability & Its Applications. 2009;53(2):194-215.
Bogachev, V. I., Röckner, M., & Shaposhnikov, S. V. (2009). Positive Densities of Transition Probabilities of Diffusion Processes. Theory of Probability & Its Applications, 53(2), 194-215. https://doi.org/10.1137/S0040585X97983523
Bogachev, V. I., Röckner, M., and Shaposhnikov, S. V. (2009). Positive Densities of Transition Probabilities of Diffusion Processes. Theory of Probability & Its Applications 53, 194-215.
Bogachev, V.I., Röckner, M., & Shaposhnikov, S.V., 2009. Positive Densities of Transition Probabilities of Diffusion Processes. Theory of Probability & Its Applications, 53(2), p 194-215.
V.I. Bogachev, M. Röckner, and S.V. Shaposhnikov, “Positive Densities of Transition Probabilities of Diffusion Processes”, Theory of Probability & Its Applications, vol. 53, 2009, pp. 194-215.
Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Positive Densities of Transition Probabilities of Diffusion Processes. Theory of Probability & Its Applications. 53, 194-215 (2009).
Bogachev, Vladimir I., Röckner, Michael, and Shaposhnikov, S. V. “Positive Densities of Transition Probabilities of Diffusion Processes”. Theory of Probability & Its Applications 53.2 (2009): 194-215.

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