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Lower bounds for the density of locally elliptic Itô processes. (English) Zbl 1123.60037

The main interest of the author of this paper is to give an explicit lower bound for the density \(p_T(x_0,y)\) of the law of \(X_T\), solution of the SDE on \({\mathbb R}^q\) \[ dX_t=\sigma(X_t) dB_t + b(X_t)dt,\quad X_0=x_0. \] A subject which was intensively discussed during the 1980s. The novelty is that the author assumes only the ellipticity along a differentiable curve joining \(x_0\) and \(y\) (e.g., a tubular neighborhood), apart from the non-boundedness hypothesis on the diffusion coefficient. More precisely, let \(\lambda^*(x)\geq \sigma\sigma^*(x)\geq \lambda_*(x)\), consider the class \(\Phi(x_0,y)\) of functions \(\varphi\in L^2([0,T], {\mathbb R}^d)\) satisfying the following conditions \[ x_0^\varphi=x_0, x_T^\varphi=y,\quad {\lambda_*\over\lambda^*}(x_t^\varphi)\geq {1\over\mu},\;\lambda_*(x_t^\varphi)\geq 1/\chi, \] where \(\mu, \chi\) are given and
\[ dx_t^\varphi=\sum_{j=1}^d \sigma_j(x_t^\varphi)\varphi_t^j\, dt,\quad x_0^\varphi=x_0, \] and define the control distance \[ d(x_0,y)=\inf\biggl\{\sqrt{\int_0^T | \varphi_t| ^2\,dt}; \varphi\in\Phi(x_0,y)\biggr\}. \] Then a Gaussian type lower bound for \(p_T(x_0,y)\) is obtained using \(d(x_0,y), \mu\) and \(\chi\).
Finally, the author develops the conditional version of the Malliavin calculus on one hand and uses the methodology put forward on Wiener functionals by Kohatsu-Higa on other hand. As byproducts, the author obtains general results about the so-called elliptic Itô processes.

MSC:

60H07 Stochastic calculus of variations and the Malliavin calculus
60J60 Diffusion processes
60J35 Transition functions, generators and resolvents
60H30 Applications of stochastic analysis (to PDEs, etc.)
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References:

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