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Russian Mathematical Surveys, 2022, Volume 77, Issue 6, Pages 1155–1157
DOI: https://doi.org/10.4213/rm10088e
(Mi rm10088)
 

Brief Communications

On the Davis–Monroe problem

P. A. Yaskov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
References:
Funding agency Grant number
Russian Science Foundation 18-71-10097
This work was supported by the Russian Science Foundation under grant no. 18-71-10097, https://rscf.ru/project/18-71-10097/.
Received: 10.11.2022
Russian version:
Uspekhi Matematicheskikh Nauk, 2022, Volume 77, Issue 6(468), Pages 207–208
DOI: https://doi.org/10.4213/rm10088
Bibliographic databases:
Document Type: Article
MSC: Primary 60J65; Secondary 60G30
Language: English
Original paper language: Russian

In this paper we solve the Davis and Monroe problem [2] about the values of $\varepsilon>0$ for which $\mu_\varepsilon$, the distribution of the Brownian motion with nonlinear drift $B_\varepsilon=B+\varepsilon F$ considered as a random element of $C[0,1]$, is equivalent to (is singular with respect to) the Wiener measure $\mu$ on $C[0,1]$, where $B=(B(t))_{t\in[0,1]}$ is the standard Brownian motion, $F(t)=\sqrt{(t-\tau)^+}$, $t\in[0,1]$, and $\tau$ is a random moment of time that is uniformly distributed on $[0,1]$ and independent of $B$. This problem is closely related to the theory of Gaussian multiplicative chaos.

Note that by the Cameron–Martin theorem (see [4], Theorem 1.38) this kind of power drift is critical: if $F(t)\equiv t^\alpha$ and $\alpha>0$, then for all $\varepsilon>0$ the above measures $\mu_\varepsilon$ are equivalent to $\mu$ ($\mu_\varepsilon\sim \mu$) when $\alpha>1/2$ and singular with respect to $\mu$ ($\mu_\varepsilon\perp \mu$) when $\alpha\leqslant 1/2$. The case $\alpha=1/2$ becomes more delicate when the drift occurs at a random moment of time $\tau$. As shown in [2], if $F(t)\equiv\sqrt{(t-\tau)^+}$ and $\tau$ has the above properties, then $\mu_\varepsilon\ll\mu$ (that is, $\mu_\varepsilon$ is absolutely continuous with respect to $\mu$) for $\varepsilon<2$ and $\mu_\varepsilon\perp\mu$ for $\varepsilon>\sqrt{8}$ . Our main result, Theorem 1 below, covers the unexplored case of $\varepsilon\in [2,\sqrt{8}\,]$.

Theorem 1. The following relations hold: $\mu_\varepsilon\sim\mu$ for $0<\varepsilon<\sqrt{8}$ and $\mu_\varepsilon\perp\mu$ for $\varepsilon\geqslant\sqrt{8}$.

To prove Theorem 1 in the case when $\varepsilon\geqslant\sqrt{8}$ we use the following observation: $\varlimsup M_n(Z)\leqslant 0$ almost surely at $Z=B$ and $\varlimsup M_n(Z)>0$ almost surely at $Z=B_\varepsilon$ for $\varepsilon\geqslant\sqrt{8}$ , where the upper limit is taken as $n\to\infty$,

$$ \begin{equation*} M_n(Z)=\max_{k=0,\dots,4^n-2^n-1}\biggl\{\frac1{\sqrt{\ln(4^n-k)}} \int_{(k+1)4^{-n}}^1\frac{dZ(t)}{\sqrt{t-k\,4^{-n}}}- \sqrt{2\ln(4^n-k)}\,\biggr\}, \end{equation*} \notag $$
and the integral is understood in the Riemann–Stieltjes sense.

To prove Theorem 1 in the case when $\varepsilon<\sqrt{8}$ we consider the process $B_{\varepsilon,\delta}=B+\varepsilon F_\delta$ with values in the measurable space $(C[0,1],\mathcal C)$ and its distribution $\mu_{\varepsilon,\delta}$, where $\delta>0$, $F_\delta(t)=\sqrt{(t-\tau)^++\delta}-\sqrt{\delta}$ for $t\in[0,1]$, and $\mathcal C$ is the Borel $\sigma$-algebra in the space $C[0,1]$ with the uniform metric. Then it is obvious that $B_{\varepsilon,\delta}\to B_{\varepsilon}$ almost surely in the uniform metric as $\delta\to0$. The rest of the proof that $\mu_\varepsilon\ll \mu$ comes down to verifying that $\mu_{\varepsilon,\delta}\ll\mu$ for all $\delta>0$ and

$$ \begin{equation} \frac{d\mu_{\varepsilon,\delta}}{d\mu} \text{ converges in }L_1(C[0,1],\mathcal C,\mu)\text{ as }\delta\to0. \end{equation} \tag{1} $$

Further, let $(\Omega,\mathcal F,\mathsf P)=(C[0,1],\mathcal C,\mu)$ be a probability space with the Brownian motion $B$ defined by $B(\omega)\equiv \omega$, $\omega\in\Omega$. According to the Cameron–Martin–Girsanov formula, we have $\mu_{\varepsilon,\delta}\ll\mu$ and $\mu_{\varepsilon,\delta}/d\mu=I(\varepsilon X_\delta/2)$, where $X_\delta(t):=\displaystyle\int_t^1 \frac{dB(s)}{\sqrt{s-t+\delta}}$ , $t\in [0,1]$, and $I(X):=\displaystyle\int_0^1\mathcal E(X(t))\,dt$ for the Gaussian process $X=(X(t))_{t\in[0,1]}$ on $(\Omega,\mathcal F,\mathsf P)$ with values in $C[0,1]$ and $\mathcal E(\xi):=e^{\xi-\mathsf E\xi^2/2}$, for the Gaussian variable $\xi$ (here $\mathsf E$ is the expectation with respect to the measure $\mathsf P$).

The model of our interest corresponds to $\delta=0$. By setting $\delta=0$, under the integral sign in $I(\varepsilon X_0/2)$ we obtain formally a ‘Gaussian’ process $X_0(t):=\displaystyle\int_t^1 \frac{dB(s)}{\sqrt{s-t}}$ , whose covariance function is non-negative and has the form $K(s,t)=-\ln|s-t|+g(s,t)$ for some continuous function $g$ on $[0,1]^2\setminus (1,1)$. There is no such process in the usual sense since $K(t,t)=\infty$ for all $t\in[0,1)$. However, $X_0$ can be defined as the family of Gaussian variables $\displaystyle\int_0^1X_0(t)\,\rho(dt)$ indexed by non-negative measures $\rho$ on $(0,1)$ with the standard Borel $\sigma$-algebra such that $\displaystyle\iint_{(0,1)^2}K(s,t)\,\rho(ds)\,\rho(dt)<\infty$.

It is well known in the theory of Gaussian multiplicative chaos (see, for instance, [1]) that if $g$ in the representation of $K$ is continuous on $[0,1]^2$, then for $\varepsilon<\sqrt{8}$ the integral $I(\varepsilon X_0/2)$ can be defined as the limit of $I(\varepsilon X_{\delta,G}/2)$ in $L_1$ as $\delta\to0$, where $X_{\delta,G}(t)=\displaystyle\int_0^1 X_0(s)\,d\biggl(1- G\biggl(\dfrac{t-s}{\delta}\biggr)\biggr)$ and $G\colon\mathbb R\to[0,1]$ is a probability distribution function with support in $(0,1)$ and such that $\displaystyle\sup_{0\leqslant t\leqslant 1/2}\displaystyle\int_0^1 \ln\dfrac1{|t-s|}\,dG(s)<\infty$. The above limit does not depend on the particular choice of $G$. Formally, our case is not covered by the known results. Nevertheless, it can be analysed by modifying the method from [1] (see the proof of Theorem 4.1.1 in [3]). In this way we can establish (1), from which it follows that $\mu_{\varepsilon}\ll\mu$ and $d\mu_\varepsilon/d\mu=I(\varepsilon X_0/2)$ is the limit of $I(\varepsilon X_{\delta}/2)$ in $L_1$ for $\varepsilon<\sqrt{8}$.

To prove that $\mu\ll \mu_{\varepsilon}$ we only need to check that $\mu(I=0)=0$ for $I=I(\varepsilon X_0/2)$. This follows from the estimate below:

$$ \begin{equation} I\geqslant \frac{I_1(\gamma)}2\inf_{t\in [0,1/2-\gamma]} \mathcal E\biggl(\frac{\varepsilon Z_0(t)}{2}\biggr)+ \frac{I_2}{2} \quad\text{for all } \gamma\in\biggl(0,\frac{1}{2}\biggr), \end{equation} \tag{2} $$
where $I_1(\gamma)=2\displaystyle\int_{0}^{1/2-\gamma} \mathcal E\biggl(\dfrac{\varepsilon Y_0(t)}{2}\biggr)\,dt$, $\gamma\in\biggl(0,\dfrac{1}{2}\biggr)$, and $I_2=2\displaystyle\int_{1/2}^1 \mathcal E\biggl(\frac{\varepsilon X_0(t)}{2}\biggr)\,dt$ are independent (the integrals $I_1(\gamma)$ and $I_2$ are found by taking limits in $L_1$ similarly to $I$), and we have
$$ \begin{equation*} Y_0(t):=\int_t^{1/2}\frac{dB(s)}{\sqrt{s-t}} \end{equation*} \notag $$
and
$$ \begin{equation*} Z_0(t):=\int_{1/2}^1\frac{dB(s)}{\sqrt{s-t}}= \frac{B(1)}{\sqrt{1-t}}-\frac{B(1/2)}{\sqrt{1/2-t}}+ \frac{1}{2}\int_{1/2}^1\frac{B(s)\,ds}{(s-t)^{3/2}} \end{equation*} \notag $$
for $t\in[0,1/2)$. In this case, as follows from a suitable change of variables under the integral sign and the properties of the Brownian motion, $I_1(0)$ and $I_2$ have the same distribution as $I$. In addition, since trajectories of the Gaussian process $Z_0$ are continuous almost surely, the infimum in (2) is positive almost surely for all $\gamma\in(0,1/2)$. In particular, all this allows us to show that
$$ \begin{equation*} \begin{aligned} \, \mu(I=0)&\leqslant \mu\bigl(I_1(\gamma)=0 \ \forall\,\gamma\in(0,1/2);I_2=0\bigr) \\ &=\mu\bigl(I_1(0)=0;I_2=0\bigr)=[\mu(I=0)]^2, \end{aligned} \end{equation*} \notag $$
that is, $\mu(I=0)=0$ or $1$. Since $\mathsf E I=\lim_{\delta\to 0}\mathsf E I(\varepsilon X_\delta/2)=1$, we have $\mu(I=0)=0$; hence $\mu\ll \mu_{\varepsilon}$.


Bibliography

1. N. Berestycki, Electron. Commun. Probab., 22 (2017), 27, 12 pp.  crossref  mathscinet  zmath
2. B. Davis and I. Monroe, Ann. Probab., 12:3 (1984), 922–925  crossref  mathscinet  zmath
3. H. Lacoin, Real and complex Gaussian multiplicative chaos, 32$^{\rm o}$ Colóquio Brasileiro de Matemática, Reprint of the 2019 original, IMPA, Rio de Janeiro, 2021, 68 pp.  mathscinet
4. P. Mörters and Y. Peres, Brownian motion, Camb. Ser. Stat. Probab. Math., 30, Cambridge Univ. Press, Cambridge, 2010, xii+403 pp.  crossref  mathscinet  zmath

Citation: P. A. Yaskov, “On the Davis–Monroe problem”, Uspekhi Mat. Nauk, 77:6(468) (2022), 207–208; Russian Math. Surveys, 77:6 (2022), 1155–1157
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