## 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.)
Full Text:

### References:

 [1] Aida, S., Kusuocka, S. and Stroock, D. (1993). On the support of Wiener functionals. In Asymptotic Problems in Probability Theory : Wiener Functionals and Asymptotics (K. D. Elworthy and N. Ikeda, eds.). Pitman Res. Notes Math. Ser. 284 3–34. Longman Sci. Tech., Harlow. · Zbl 0790.60047 [2] Bally, V. (2003). Lower bounds for the density of the law of locally elliptic Itô processes. Preprint 4887, INRIA. [3] Bally, V. and Pardoux, E. (1998). Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9 27–64. · Zbl 0928.60040 [4] Bass, R. F. (1997). Diffusions and Elliptic Operators. Probability and Its Applications . Springer, New York. · Zbl 0914.60009 [5] Ben-Arous, G. and Leandre, R. (1991). Decroissance exponentille du noyau de la chaleur sur la diagonale. II. Probab. Theory Related Fields 90 377–402. · Zbl 0734.60027 [6] Dalang, R. and Nualart, E. (2004). Potential theory for hyperbolic SPDEs. Ann. Probab. 32 2099–2148. · Zbl 1054.60066 [7] Fefferman, C. L. and Sanchez-Calle, A. (1986). Fundamental solutions of second order subelliptic operators. Ann. of Math. ( 2 ) 124 247–272. JSTOR: · Zbl 0613.35002 [8] Friedman, A. (1975). Stochastic Differential Equations and Applications . Academic Press, New York. · Zbl 0323.60056 [9] Guérin, H., Méléard, S. and Nulalart, E. (2006). Estimate for the density of a nonlinear Landau proccess. J. Funct. Anal. 238 649–677. · Zbl 1108.60047 [10] Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators . I, 2nd ed. Springer, Berlin. · Zbl 0712.35001 [11] Kusuoka, S. and Stroock, D. (1985). Applications of the Malliavin calculus, part II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 1–76. · Zbl 0568.60059 [12] Kusuoka, S. and Stroock, D. (1987). Applications of the Malliavin calculus, part III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 391–442. · Zbl 0633.60078 [13] Kohatsu-Higa, A. (2003). Lower bounds for densities of uniform elliptic random variables on Wiener space. Probab. Theory Related Fields 126 421–457. · Zbl 1022.60056 [14] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes , 2nd ed. North-Holland, Amsterdam. · Zbl 0684.60040 [15] Millet, A. and Sanz, M. (1997). Points of positive density for the solutions to a hyperbolic SPDE. Potential Anal. 7 623–659. · Zbl 0892.60059 [16] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050 [17] Sanchez-Calle, A. (1986). Fundamental solutions and geometry of the sum of square of vector fields. Invent. Math. 78 143–160. · Zbl 0582.58004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.