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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
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
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
Linking options:
https://www.mathnet.ru/eng/tvp1725https://doi.org/10.4213/tvp1725 https://www.mathnet.ru/eng/tvp/v53/i2/p213
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