On some properties of space inverses of stochastic flows

Abstract

We derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of continuous SDEs, we derive the existence and uniqueness of classical solutions of linear parabolic second order SPDEs by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case.

Appendix

Let V be an arbitrary Banach space. The following lemma and its corollaries are indispensable in this paper.

Lemma 5.1

Let $Q\subseteq {\mathbf {R}}^{d}$ be an open bounded cube, $p\ge 1$, $\delta \in (0,1]$, and f be a V-valued integrable function on Q such that

$$\begin{aligned} \left[ f\right] _{\delta ;p;Q;V}:=\left( \int _{Q}\int _{Q}\frac{ |f(x)-f(y)|_{V}^{p}}{|x-y|^{2d+\delta p}}dxdy\right) ^{1/p}<\infty . \end{aligned}$$

Then f has a ${\mathcal {C}}^{\delta }(Q;V)$-modification and there is a constant $N=N(d,\delta ,p )$ independent of f and Q such that

$$\begin{aligned}{}[f]_{\delta ;Q;V}\le N\left[ f\right] _{\delta ,p;Q;V} \end{aligned}$$

and

$$\begin{aligned} \sup _{x\in Q}|f(x)| _{V}\le N| Q| ^{\delta /d}[f] _{\delta ;p;Q;V}+|Q| ^{-1/p}\left( \int _{Q}|f(x)| _{V}^{p}dx\right) ^{1/p}, \end{aligned}$$

where |Q| is the volume of the cube.

Proof

If $V={\mathbf {R}}$, then the existence of a continuous modification of f and the estimate of $\left[ f\right] _{\delta ;Q}$ follows from Lemma 2 and Exercise 5 in Chap. 10, Sect. 1, in [7]. The proof for a general Banach space is the same. For all $x\in Q$, we have

$$\begin{aligned} |f(x)|_{V}&\le \frac{1}{|Q|}\int _{Q}| f(x)-f(y)|_{V}dy+\frac{1}{|Q|} \int _{Q}| f(y)|_{V}dy \\&\le N\frac{1}{|Q|}\left[ f\right] _{\delta ,p;Q}\int _{Q}|x-y|^{\delta }dy+\frac{1}{| Q|}\int _{Q}| f(y)|_V dy \\&\le N|Q|^{\delta /d}[f] _{\delta ,p;Q}+| Q|^{-1/p}\left( \int _{Q}| f(y)| _{V}^{p}dy\right) ^{1/p}, \end{aligned}$$

which proves the second estimate. $\square$

The following is a direct consequence of Lemma 1.

Corollary 5.2

Let $p\ge 1$, $\delta \in (0,1]$, and f be a V-valued function on ${\mathbf {R}}^d$ such that

$$\begin{aligned} |f| _{\delta ;p;V}:=\left( \int _{{\mathbf {R}}^d}|f(x)|^p_Vdx+ \int _{|x-y|<1}\frac{ |f(x)-f(y)|_{V}^{p}}{|x-y|^{2d+\delta p}}dxdy\right) ^{1/p}<\infty . \end{aligned}$$

Then f has a ${\mathcal {C}}^{\delta }({\mathbf {R}}^d;V)$-modification and there is a constant $N=N(d,\delta , p)$ independent of f such that

$$\begin{aligned} | f| _{\delta ;V}\le N| f| _{\delta ;p;V}. \end{aligned}$$

Corollary 5.3

Let X be a V-valued random field defined on ${\mathbf {R}}^{d}$. Assume that for some $p\ge 1$, $l\ge 0,$ and $\beta \in (0,1]$ with $\beta p>d$ there is a constant $\bar{N}>0$ such that for all $x,y\in {\mathbf {R}}^{d}$,

$$\begin{aligned} {\mathbf {E}}\left[ |X(x)|_{V}^{p}\right] \le \bar{N}r_{1}(x)^{lp} \end{aligned}$$

(5.14)

and

$$\begin{aligned} {\mathbf {E}}\left[ |X(x)-X(y)|_{V}^{p}\right] \le \bar{N}\left[ r_{1}(x)^{lp}+r_1(y)^{lp}\right] |x-y|^{\beta p}. \end{aligned}$$

(5.15)

Then for all $\delta \in (0,\beta -\frac{d}{p})$ and $\epsilon >\frac{d}{p}$ , there exists a ${\mathcal {C}}^{\delta }({\mathbf {R}}^{d};V)$-modification of $r_{1}^{-(l+\epsilon )}X$ and a constant $N=N(d,p,\delta ,\epsilon )$ such that

$$\begin{aligned} {\mathbf {E}}\left[ |r_{1}^{-(l+\epsilon )}X|_{\delta }^{p}\right] \le N\bar{N}. \end{aligned}$$

Proof

Fix $\delta \in (0,\beta -\frac{d}{p})$ and $\epsilon >\frac{d}{p}$. Owing to (5.14), there is a constant $N=N(d,p,\bar{N},\delta ,\epsilon )$ such that

$$\begin{aligned} \int _{{\mathbf {R}}^{d}}{\mathbf {E}}\left[ |r_{1}(x)^{-(l+\epsilon )}X(x)|_{V}^{p}\right] dx\le \bar{N}\int _{{\mathbf {R}}^{d}}r_{1}(x)^{-p\epsilon }dx\le N\bar{N}. \end{aligned}$$

Appealing to (5.15) and (3.19), we find that there is a constant $N=N(d,p,\delta ,\epsilon )$ such that

$$\begin{aligned}&\int _{|x-y|<1}\frac{{\mathbf {E}}\left[ |r_{1}(x)^{-(l+\epsilon )}X(x)-r_{1}(y)^{-(l+\epsilon )}X(y)|_{V}^{p}\right] }{|x-y|^{2d+\delta p}}dxdy\\&\quad \le \bar{N}\int _{|x-y|<1}\frac{r_{1}(x)^{-p\epsilon }+r_{1}(y)^{-p\epsilon }}{ |x-y|^{2d-(\beta -\delta )p}}dxdy\\&\quad \quad +\,\bar{N}\int _{|x-y|<1}\frac{ r_{1}(y)^{pl}|r_{1}(x)^{-(l+\epsilon )}-r_{1}(y)^{-(l+\epsilon )}|^{p}}{ |x-y|^{2d+\delta p}}dxdy\\&\quad \le N\bar{N}+N\bar{N}\int _{|x-y|<1}\frac{r_{1}(x)^{-p(1+\epsilon )}+r_{1}(y)^{-p(1+\epsilon )}}{|x-y|^{2d-(1-\delta )p}}dxdy\le N\bar{N}. \end{aligned}$$

Therefore, ${\mathbf {E}}[r_{1}^{-(l+\epsilon )}X]_{\delta ,p}^{p}\le N\bar{N}$, and hence, by Corollary 5.3, $r_{1}^{-(l+\epsilon )}X$ has a ${\mathcal {C}}^{\delta }({\mathbf {R}}^{d};V)$-modification and the estimate follows immediately. $\square$

Corollary 5.4

Let $(X^{(n)})_{n\in {\mathbf {N}}}$ be a sequence of V-valued random fields defined on ${\mathbf {R}}^{d}$. Assume that for some $p\ge 1$, $l\ge 0$ and $\beta \in (0,1],$ with $\beta p>d$ there is a constant $\bar{N}>0$ such that for all $x,y\in {\mathbf {R}}^{d}$ and $n\in {\mathbf {N}}$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\zeta <|x-y|<1 }\frac{{\mathbf {E}}\left[ |r_{1}(x)^{- (l+\epsilon )}X_{n}(x)-r_{1}(y)^{-(l+\epsilon )}X_{n}(y)|_{V}^{p}\right] }{{|x-y|^{2d+\delta p}}}dxdy=0. \end{aligned}$$

and

$$\begin{aligned} {\mathbf {E}}\left[ |X^{(n)}(x)-X^{(n)}(y)|_{V}^{p}\right] \le \bar{N}(r_{1}(x)^{lp}+r_1(y)^{lp})|x-y|^{\beta p}. \end{aligned}$$

Moreover, assume that for all $x\in {\mathbf {R}}^{d}$, $\lim _{n\rightarrow \infty }{\mathbf {E}}\left[ |X^{(n)}(x)|^{p}\right] = 0.$ Then for all $\delta \in (0,\beta -\frac{d}{p})$ and $\epsilon >\frac{d}{p}$,

$$\begin{aligned} \lim _{n\rightarrow \infty }{\mathbf {E}}\left[ |r_{1}^{-(l+\epsilon )}X^{(n)}|_{\delta }^{p}\right] =0. \end{aligned}$$

Proof

Fix $\delta \in (0,\beta -\frac{d}{p})$ and $\epsilon >\frac{d}{p}$. Using the Lebesgue dominated convergence theorem, we attain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{{\mathbf {R}}^{d}}{\mathbf {E}}\left[ |r_{1}(x)^{-(l+\epsilon )}X^{(n)}(x)|_{V}^{p}\right] dx=0, \end{aligned}$$

and therefore for all $\zeta \in (0,1)$,

$$\begin{aligned} \lim _{n}\int _{\zeta <|x-y|<1 }\frac{{\mathbf {E}}\left[ |r_{1}(x)^{-(l+\epsilon )}X_{n}(x)-r_{1}(y)^{-(l+\epsilon )}X_{n}(y)|_{V}^{p}\right] }{|x-y|^{2d+\delta p}} dxdy=0. \end{aligned}$$

By repeating the proof of Corollary 5.3, we obtain that there is a constant N such that

$$\begin{aligned}&\int _{|x-y|\le \zeta }\frac{{\mathbf {E}}\left[ |r_{1}(x)^{-(l+\epsilon )}X^{(n)}(x)-r_{1}(y)^{-(l+\epsilon )}X^{(n)}(y)|_{V}^{p}\right] }{|x-y|^{2d+\delta p}}dxdy\\&\quad \le \bar{N}\int _{|x-y|\le \zeta }\frac{r_{1}(x)^{-p\epsilon }+r_{1}(y)^{-p\epsilon }}{ |x-y|^{2d+(\delta -\beta )p}}dxdy\\&\quad \quad +\,\bar{N}\int _{|x-y|\le \zeta }\frac{ r_{1}(x)^{-p(1+\epsilon )}+r_{1}(y)^{-p(1+\epsilon )}}{|x-y|^{2d+(\delta -1)p}}dxdy\\&\quad \le \bar{N}\zeta ^{\beta p-\delta p-d}. \end{aligned}$$

Therefore, $\lim _{n\rightarrow \infty }{\mathbf {E}}\left[ [r_{1}^{-(l+\epsilon )}X]_{\delta ,p}^{p}\right] =0$, and the statement is confirmed. $\square$