Is a diffusion process determined by its intrinsic metric? (English) Zbl 0939.58032
Summary: J. R. Norris [Acta Math. 179, 79-103 (1997; Zbl 0912.58041)] proved that the small time asymptotic \(\lim_{t\to 0}2t \cdot\log p(t,x,y)\) of a symmetric elliptic diffusion on \(\mathbb{R}^n\) (or, more general, on a Lipschitz manifold) is determined by the intrinsic metric defined in terms of the associated Dirichlet form. Here we ask the question: Is the Dirichlet form (or the diffusion process) determined uniquely by its intrinsic metric (i.e., by its small time asymptotic)?
The answer is NO. For any symmetric elliptic diffusion there exists another one with the same small time asymptotic but with strictly smaller diffusion coefficients.
However, the answer is YES if a priori we know that the diffusion coefficients are continuous.
The answer is NO. For any symmetric elliptic diffusion there exists another one with the same small time asymptotic but with strictly smaller diffusion coefficients.
However, the answer is YES if a priori we know that the diffusion coefficients are continuous.
MSC:
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 58J37 | Perturbations of PDEs on manifolds; asymptotics |
| 31C25 | Dirichlet forms |
| 60J60 | Diffusion processes |
Citations:
Zbl 0912.58041References:
| [1] | Norris, J. R., Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. In press.; Norris, J. R., Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. In press. · Zbl 0912.58041 |
| [2] | Fukushima, M.; Oshima, Y.; Takeda, M., (Dirichlet Forms and Symmetric Markov Processes (1994), De Gruyter: De Gruyter Berlin) · Zbl 0838.31001 |
| [3] | Sturm, K. T., Analysis on local Dirichlet spaces—I. Recurrence, conservativeness and \(L^P\)-Liouville properties, J. Reine Angew. Math., 456, 173-196 (1994) · Zbl 0806.53041 |
| [4] | Sturm, K. T., On the geometry defined by Dirichlet forms, (Bolthausen, E.; etal., Seminar on Stochastic Analysis, Random Fields and Applications. Seminar on Stochastic Analysis, Random Fields and Applications, Ascona, 1993 (1995), Birkhäuser: Birkhäuser Berlin), 231-242 · Zbl 0834.58039 |
| [5] | Norris, J. R., Small time asymptotics for heat kernels with measurable coefficients, C.R. Acad. Sci. Paris, Serie I, 322, 339-344 (1996) · Zbl 0846.35054 |
| [6] | De Cecco, G.; Palmieri, G., Integral distance on a Lipschitz Riemannian manifold, Math. Z., 207, 223-243 (1991) · Zbl 0722.58006 |
| [7] | De Cecco, G.; Palmieri, G., LIP manifolds—from metric to Finslerian structures, Math. Z., 218, 223-237 (1995) · Zbl 0819.53014 |
| [8] | Varadhan, S. R.S., On the behaviour of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20, 431-455 (1967) · Zbl 0155.16503 |
| [9] | Aronson, D. G., Non-negative solutions of linear parabolic equations, Ann. Sci. Norm. Sup. Pisa, 22, 607-694 (1967) · Zbl 0182.13802 |
| [10] | Davies, E. B., Explicit constants for gaussian upper bounds on heat kernels, Am. J. Math., 109, 319-334 (1987) · Zbl 0659.35009 |
| [11] | Léandre, R., Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal., 74, 399-414 (1987) · Zbl 0637.58034 |
| [12] | Norris, J. R.; Stroock, D. W., Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients, (Proc. Lond. Math. Soc., 62 (1991)), 373-402 · Zbl 0694.35075 |
| [13] | Zheng, W., A large deviation result for a class of Dirichlet processes, Prob. Theory Related Fields, 101, 237-249 (1995) · Zbl 0813.60031 |
| [14] | Zheng, W., Diffusion processes on a Lipschitz Riemannian manifold and their applications, (Proc. of Symposia in Pure Mathematics, 57 (1995)), 373-381 · Zbl 0824.60083 |
| [15] | Sturm, K. T., Analysis on local Dirichlet spaces—II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32, 275-312 (1995) · Zbl 0854.35015 |
| [16] | Sturm, K. T., Sharp estimates for capacities and applications to symmetric diffusions, Prob. Theory Related Fields, 102, 73-89 (1995) · Zbl 0828.60062 |
| [17] | Sturm, K. T., Analysis on local Dirichlet spaces—III. The parabolic Harnack inequality, J. Math. Pure Appl., 75, 273-297 (1996) · Zbl 0854.35016 |
| [18] | Sturm, K. T., How to construct diffusion processes on metric spaces. Potential Analysis; Sturm, K. T., How to construct diffusion processes on metric spaces. Potential Analysis · Zbl 0929.60060 |
| [19] | Sturm K. T., Diffusion processes and heat kernels on length spaces. Ann. Prob; Sturm K. T., Diffusion processes and heat kernels on length spaces. Ann. Prob · Zbl 0936.60074 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
