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Teoriya Veroyatnostei i ee Primeneniya, 2008, Volume 53, Issue 2, Pages 213–239
DOI: https://doi.org/10.4213/tvp1725
(Mi tvp1725)
 

This article is cited in 16 scientific papers (total in 16 papers)

Positive Densities of Transition Probabilities of Diffusion Processes

V. I. Bogacheva, M. Röcknerb, S. V. Shaposhnikovc

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Bielefeld University
c M. V. Lomonosov Moscow State University
References:
Abstract: For diffusion processes in $\mathbf R^d$ 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 $\mathcal L^*\mu=0$ with respect to measures on $\mathbf R^d\times (0,1)$ with the operator $\mathcal L u:=\partial_t u+\partial_{x_i}(a^{ij}\partial_{x_j}u)+b^i\partial_{x_i}u$. It is shown that if the diffusion coefficient $A=(a^{ij})$ is sufficiently regular and the drift coefficient $b=(b^i)$ satisfies the condition $\exp(\kappa |b|^2)\in L_{\mathrm{loc}}^1(\mu)$, where the measure $\mu$ is nonnegative, then $\mu$ has a continuous density $\varrho(x,t)$ which is strictly positive for $t>\tau$ provided that it is not identically zero for $t\le\tau$. Applications are obtained to finite-dimensional projections of stationary distributions and transition probabilities of infinite-dimensional diffusions.
Keywords: density of transition probability, stationary distribution, parabolic equation, infinite-dimensional diffusion.
Received: 26.11.2007
English version:
Theory of Probability and its Applications, 2009, Volume 53, Issue 2, Pages 194–215
DOI: https://doi.org/10.1137/S0040585X97983523
Bibliographic databases:
Language: Russian
Citation: V. I. Bogachev, M. Röckner, S. V. Shaposhnikov, “Positive Densities of Transition Probabilities of Diffusion Processes”, Teor. Veroyatnost. i Primenen., 53:2 (2008), 213–239; Theory Probab. Appl., 53:2 (2009), 194–215
Citation in format AMSBIB
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\paper Positive Densities of Transition Probabilities of Diffusion Processes
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\pages 213--239
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\jour Theory Probab. Appl.
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  • https://www.mathnet.ru/eng/tvp/v53/i2/p213
  • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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