Quantitative Heat-Kernel Estimates for Diffusions with Distributional Drift

Nicolas Perkowski,Willem Van Zuijlen

doi:10.1007/s11118-021-09984-3

Nicolas Perkowski, Willem Van Zuijlen

Open Access

https://doi.org/10.1007/s11118-021-09984-3

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Abstract

We consider the stochastic differential equation on mathbb {R}^{d} given bydXt=b(t,Xt)dt+dBt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{array}{@{}rcl@{}} \\mathrm{d} X_{t} = b(t,X_{t}) \\mathrm{d} t + \\mathrm{d} B_{t}, \\end{array} $$\\end{document}where B is a Brownian motion and b is considered to be a distribution of regularity > -frac 12. We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat-kernel estimates for Γt with explicit dependence on t and the norm of b.

Highlights

Introduction and Main ResultsIn this paper we consider the stochastic differential equation on Rd given by dXt = b(t, Xt ) dt + dBt, (1)where distribution of regularitySuch singular diffusions appear as models for stochastic processes in random media ( b would be random, but independent of B), for example in [4–6]
We show that the martingale solution of the SDE has a transition kernel Γt and prove upper and lower heat-kernel estimates for Γt with explicit dependence on t and the norm of b
In non-singular SPDEs, the stochastic characteristics would be formulated in terms of the Brownian motion, and they may be useful tools to infer information about the long-time behavior of the SPDE

Summary

Introduction and Main Results

In this paper we consider the stochastic differential equation on Rd given by dXt = b(t, Xt ) dt + dBt , Such singular diffusions (diffusions with distributional drift) appear as models for stochastic processes in random media ( b would be random, but independent of B), for example in [4–. Since we are interested in the long-time behavior, we need quantitative control of the transition probabilities on arbitrarily long time intervals.

Literature

Notation and Conventions

Diffusions with Distributional Drift and Their Heat-Kernel Estimates

Heat-Kernel Upper Bounds

Heat-Kernel Lower Bounds

A Appendix