Pitman's and Lévy's theorems for Brownian bridges
A synthetic study of Pitman's and Lévy's theorems for one-dimensional Brownian bridges with arbitrary endpoints is provided.
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arXiv subjects
Yuu Hariya
Publications and source records attributed to Yuu Hariya.
Stochastic analysis of Beckner's and related functional inequalities
Beckner's inequality is a family of inequalities that interpolates the two fundamental functional inequalities, the logarithmic Sobolev and Poincaré's inequalities. It is parametrized by exponent $p\in (1,2]$ and it implies the logarithmic Sobolev inequality as $p\to 1$ and agrees with Poincaré's inequality when $p=2$. In this paper, employing a stochastic method, we prove an improvement of Beckner's inequality under the Gaussian measure when $4/3\le p<2$; in particular, when $p=3/2$, the error bound is expressed in terms of the entropy functional. A similar reasoning to the derivation of the improvement also enables us to obtain a Hölder-type inequality that holds among the entropy, variance and related functionals.
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Invariance of three-dimensional Bessel bridges in terms of time reversal
Given $a,b\ge 0$ and $t>0$, let $ρ=\{ ρ_{s}\} _{0\le s\le t}$ be a three-dimensional Bessel bridge from $a$ to $b$ over $[0,t]$. In this paper, based on a conditional identity in law between Brownian bridges stemming from Pitman's theorem, we show in particular that the process given by \begin{align*} ρ_{s}+\Bigl| b-a+ \min _{0\le u\le s}ρ_{u}-\min _{s\le u\le t}ρ_{u} \Bigr| -\Bigl| \min _{0\le u\le s}ρ_{u}-\min _{s\le u\le t}ρ_{u} \Bigr| ,\quad 0\le s\le t, \end{align*} has the same law as the time reversal $\{ ρ_{t-s}\} _{0\le s\le t}$ of $ρ$. As an immediate application, letting $R=\{ R_{s}\} _{s\ge 0}$ be a three-dimensional Bessel process starting from $a$, we obtain the following time-reversal and time-inversion results on $R$: $\{ R_{t-s}\} _{0\le s\le t}$ is identical in law with the process given by \begin{align*} R_{s}+R_{t}-2\min _{s\le u\le t}R_{u},\quad 0\le s\le t, \end{align*} when $a=0$, and $\{ sR_{1/s}\} _{s>0}$ is identical in law with the process given by \begin{align*} R_{s}-2(1+s)\min _{0\le u\le s}\frac{R_{u}}{1+u}+a(1+s),\quad s>0, \end{align*} for every $a\ge 0$.
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Invariance of Brownian motion associated with past and future maxima
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion. As an application of a recent result of ours on exponential functionals of Brownian motion, we show in this paper that, for every fixed $t>0$, the process given by \begin{align*} B_{s}-B_{t}-\Bigl| B_{t}+\max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| +\Bigl| \max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u} \Bigr| ,\quad 0\le s\le t, \end{align*} is a Brownian motion. The path transformation that describes the above process is proven to be an involution, commute with time reversal, and preserve Pitman's transformation. A connection with Pitman's $2M-X$ theorem is also discussed.
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A Girsanov-type formula for a class of anticipative transforms of Brownian motion associated with exponential functionals
In this paper, with the help of a result by Matsumoto--Yor (2000), we prove a Girsanov-type formula for a class of anticipative transforms of Brownian motion which possesses exponential functionals as anticipating factors. Our result unifies existing formulas in earlier works. As an application, we also consider the law of Brownian motion perturbed by a positive weight of a fairly wide class, and prove its invariance under an anticipative transformation associated with the perturbation. In the course of our exploration, a disintegration formula for the Wiener measure related to exponential functionals plays a key role.
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Invariance of Brownian motion associated with exponential functionals
It is well known that Brownian motion enjoys several distributional invariances such as the scaling property and the time reversal. In this paper, we prove another invariance of Brownian motion that is compatible with the time reversal. The invariance, which seems to be new to our best knowledge, is described in terms of an anticipative path transformation involving exponential functionals as anticipating factors. Some related results are also provided.
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On two-dimensional extensions of Bougerol's identity in law
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion and denote by $A_{t},\,t\ge 0$, the quadratic variation of $e^{B_{t}},\,t\ge 0$. The celebrated Bougerol's identity in law (1983) asserts that, if $β=\{ β_{t}\} _{t\ge 0}$ is another Brownian motion independent of $B$, then $β_{A_{t}}$ has the same law as $\sinh B_{t}$ for every fixed $t>0$. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving as the second coordinates the local times of $B$ and $β$ at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.
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The Boué--Dupuis formula and the exponential hypercontractivity in the Gaussian space
This paper concerns a variational representation formula for Wiener functionals. Let $B=\{ B_{t}\} _{t\ge 0}$ be a standard $d$-dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional $F(B)$ of $B$ up to time $1$, the expectation $\mathbb{E}\!\left[ e^{F(B)}\right] $ admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also $F(B)$ to be a functional of $B$ over the whole time interval, we prove that the Boué--Dupuis formula holds true provided that both $e^{F(B)}$ and $F(B)$ are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein--Uhlenbeck semigroup in $\mathbb{R}^{d}$, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the $d$-dimensional Gaussian space.
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Extensions of Bougerol's identity in law and the associated anticipative path transformations
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion and denote by $A_{t},\,t\ge 0$, the quadratic variation of the geometric Brownian motion $e^{B_{t}},\,t\ge 0$. Bougerol's celebrated identity (1983) asserts that, if $β=\{ β(t)\} _{t\ge 0}$ is another Brownian motion independent of $B$, then $β(A_{t})$ is identical in law with $\sinh B_{t}$ for every fixed $t>0$. In this paper, we extend Bougerol's identity to an identity in law for processes up to time $t$, which exhibits a certain invariance of the law of Brownian motion. The extension is described in terms of anticipative transforms of $B$ involving $A_{t}$ as an anticipating factor. A Girsanov-type formula for those transforms is shown. An extension of a variant of Bougerol's identity is also presented.
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Integral representations for the Hartman--Watson density
This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion.
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On some identities in law involving exponential functionals of Brownian motion and Cauchy variable
Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_{t}=\int _{0}^{t}e^{2B_{s}}ds,\,t\ge 0$. Starting from a simple observation of generalized inverse Gaussian distributions with particular sets of parameters, we show, with the help of a result by Matsumoto--Yor (2000), that for every $x\in \mathbb{R}$ and for every finite stopping time $τ$ of the process $\{ e^{-B_{t}}A_{t}\} _{t\ge 0}$, there holds the identity in law \begin{align*} \left( e^{B_τ}\!\sinh x+β(A_{τ}), \, Ce^{B_τ}\!\cosh x+\hatβ(A_{τ}), \, e^{-B_{τ}}\!A_{τ} \right) \stackrel{(d)}{=} \left( \sinh (x+B_{τ}), \, C\cosh (x+B_{τ}), \, e^{-B_{τ}}\!A_{τ} \right) , \end{align*} which extends an identity due to Bougerol (1983) in several aspects. Here $β=\{ β(t)\} _{t\ge 0}$ and $\hatβ=\{ \hatβ(t)\} _{t\ge 0}$ are one-dimensional standard Brownian motions, $C$ is a standard Cauchy variable, and $B$, $β$, $\hatβ$ and $C$ are independent. Using an argument relevant to derivation of the above identity, we also present some invariance formulae for Cauchy variable involving an independent Rademacher variable.
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A unification of the hypercontractivity and its exponential variant of the Ornstein-Uhlenbeck semigroup
Let $γ_{d}$ be the $d$-dimensional standard Gaussian measure and $\{Q_{t}\}_{t\ge 0}$ the Ornstein-Uhlenbeck semigroup acting on $L^{1}(γ_{d})$. We show that the hypercontractivity of $\{Q_{t}\}_{t\ge 0}$ is equivalent to the property that \begin{align*} \left\{ \int_{\mathbb{R}^{d}}\exp \left(e^{2t}Q_{t}f\right) dγ_{d} \right\} ^{1/e^{2t}} \le \int_{\mathbb{R}^{d}}e^{f}\,dγ_{d}, \end{align*} which holds for any $f\in L^{1}(γ_{d})$ with $e^{f}\in L^{1}(γ_{d})$ and for any $t\ge 0$. We then derive a family of inequalities that unifies this exponential variant and the original hypercontractivity, a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. A unification of the reverse hypercontractivity and the exponential variant is also provided.
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On divergence of expectations of the Feynman-Kac type with singular potentials
Motivated by the work of Baras-Goldstein (1984), we discuss when expectations of the Feynman-Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $α$-stable process. In connection with the work of Ishige-Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion.
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A pathwise interpretation of the Gorin-Shkolnikov identity
In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this note, we give a pathwise interpretation to their identity; Jeulin's identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.
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Some asymptotic formulae for Bessel process
We recover in part a recent result of Hamana-Matsumoto (2014) on the asymptotic behaviors for tail probabilities of first hitting times of Bessel process. Our proof is based on a weak convergence argument. The same reasoning enables us to derive the asymptotic behaviors for the tail probability of the time at which the global infimum of Bessel process is attained, and for expected values relative to local infima. In addition, we give another proof of the result of Hamana-Matsumoto with improvement of error estimates, which complements in the case of noninteger dimensions the asymptotic formulae by van den Berg (2007) for first hitting times of multidimensional Brownian motion.
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A variational representation and Prékopa's theorem for Wiener functionals
In 1998, Boué and Dupuis proved a variational representation for exponentials of bounded Wiener functionals. Since their proof involves arguments related to the weak convergence of probability measures, the boundedness of functionals seems inevitable. In this paper, we extend the representation to unbounded functionals under a mild assumption on their integrability. As an immediate application of the extension, we prove an analogue of Prékopa's theorem for Wiener functionals, which is then applied to formulate the Brascamp-Lieb inequality in the framework of Wiener spaces.
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A connection of the Brascamp-Lieb inequality with Skorokhod embedding
We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of variance are also provided.
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Stochastic ranking process with time dependent intensities
We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the infinite particle limit. We give an explicit formula for the limit distribution and show that the limit distribution function is a unique global classical solution to an initial value problem for a system of a first order non-linear partial differential equations with time dependent coefficients.
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