Support Theorem for Stochastic Differential Equations with Sobolev Coefficients

Abstract

In this paper we prove a support theorem for stochastic differential equations with Sobolev coefficients in the framework of DiPerna-Lions theory.

Appendix

Proof of Eqs. 45–48

We prove only Eq. 46 for one-dimensional Brownian motion W. The others are analogous. For $t>2^{-n}$, write

$$\begin{array}{@{}rcl@{}} \mathcal{L} &:=& \mathbb{E} {{\int}_{t_{n}}^{{t}_{n}^{+}}} {{\int}_{t_{n}}^{s}} {\dot{W}}_{r}^{m} {\dot{W}}_{s}^{n} \mathrm{d} r \mathrm{d} s = 2^{n} \mathbb{E} (W_{t_{n}} - W_{t^{-}_{n}}) {{\int}_{t_{n}}^{{t}_{n}^{+}}} ({W}_{s}^{m} - {W}_{s_{m}}^{m})\mathrm{d} s\\ && +\, 2^{n} \mathbb{E} (W_{t_{n}} - W_{{t}_{n}^{-}}) {{\int}_{t_{n}}^{{t}_{n}^{+}}} ({W}_{s_{m}}^{m} - {W}_{t_{n}}^{m}) \mathrm{d} s =: \mathcal{L}^{1} + \mathcal{L}^{2}. \end{array} $$

By Eq. 10 and Brownian motion properties, we have (see Figure)

$$\begin{array}{@{}rcl@{}} \mathcal{L}^{1} &=& 2^{n+m} \mathbb{E} \left[(W_{t_{n}} - W_{t^{-}_{n}}) {{\int}_{t_{n}}^{t^{+}_{n}}} (s-s_{m}) (W_{s_{m}} - W_{s^{-}_{m}}) \mathrm{d} s\right]\\ &=& 2^{n+m} {{\int}_{t_{n}}^{t_{n}+ 2^{-m}}} (s-s_{m}) ({t_{n}} - {s}_{m}^{-}) \mathrm{d} s = 2^{n-2m-1}, \end{array} $$

and

$$\begin{array}{@{}rcl@{}} \mathcal{L}^{2} &=& 2^{n} \mathbb{E} \left[(W_{t_{n}} - W_{t^{-}_{n}}) {{\int}_{t_{n}}^{t^{+}_{n}}} (W_{s^{-}_{m}} - W_{t_{n}-2^{-m}}) \mathrm{d} s\right]\\ &=& 2^{n} {{\int}_{t_{n}}^{t^{+}_{n}}} [{s}_{m}^{-} \wedge t_{n} - (t_{n} - 2^{-m})] \mathrm{d} s\\ &=& 2^{n} ({t}_{n}^{+} - t_{n} - 2^{-m}) 2^{-m} = 2^{-m} - 2^{n-2m}. \end{array} $$

Summing them up, we find that

$$\mathcal{L} = 2^{-m} - 2^{n-2m-1}. $$

Thus, Eq. 46 is obtained. □