Kohei Uchiyama
The distribution of the first hitting time of a disc for the standard two dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairy detailed asymptotic estimates of its density valid uniformly with respect to the point where the Brownian motion starts from.
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In this short article, we will focus on the different links between some stochastic processes resulting from Brownian motion and two notions of probability theory (proportional increments and last hitting times).
In this paper we consider a (reflected) Brownian motion with broken drift hitting a random boundary. Some dedicated calculations allow us to obtain the formula on the joint Laplace transform of the hitting time and hitting position. These develop the research on first rendezvous times of (reflected) Brownian motion and compound Poisson-type processes by Perry et al. (2004).
Explicit formulae for the densities of the first hitting times to the sphere of Brownian motions with drifts are given. We need to consider the joint distributions of the first hitting times to the sphere and the hitting positions of the standard Brownian motion and explicit expression for their Laplace transforms are given, which are different from the known formulae in the literature and are of independnt interest.
The purpose of the paper is to find the joint distribution of the hitting time and place of two-dimensional Brownian motion hitting the negative horizontal axis. We provide various formulas for Green functions as well as for the conditional distributions. In the end we treat an analogous problem for the case when vertical component has a negative drift.
Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.
(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
We present functional versions of recent results on the univariate distributions of the process $V_{x,u} = x + W_{uτ(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $τ(x) =\inf\{t>0 : W_{t}=-x\}$.
In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $ρ$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $ρ$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.
Let $S^{d-1}_r$ be the sphere in $\bR^d$ whose center is the origin and the radius is $r$, and $σ_r$ be the first hitting time to it of the standard Brownian motion $\{B_t\}_{t\geqq0}$, possibly with constant drift. The aim of this article is to show explicit formulae by means of spherical harmonics for the density of the joint distribution of $(σ_r,B_{σ_r})$ and to study the asymptotic behavior of the distribution function.
The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joini…
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burd…
Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change …
Hitting probability for Reflected Brownian Motion at Small Target
本文研究含反射、吸收边界及小目标的光滑区域中反射布朗运动的小目标击中概率，通过目标附近局部化与显式计算得到其关于 $O(ε)$ 的渐近形式。
Let $B^{H}$ be a $d$-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f:[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the fractional Brownian motion $B^{H}$ with the deterministic drift $f$. It aims to highlight the role of the regularity properties of the drift $f$ as well as that of the dimension of $E$ in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$ for $F$ a subset of $\mathbb{R}^{d}$ and also fo…
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary $t\mapsto a+bt,\ a\geq 0,\,b\in \R,$ by a reflecting Brownian motion. The main tool hereby is Doob's formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter $δ>…
We consider a stochastic flow $ϕ_t(x,ω)$ in $\mathbb{R}^n$ with initial point $ϕ_0(x,ω)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|ϕ_t(x)\|)}{\|ϕ_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonemp…
Let $τ$ be the first hitting time of the point 1 by the geometric Brownian motion $X(t)= x \exp(B(t)-2μt)$ with drift $μ\geq 0$ starting from $x>1$. Here $B(t)$ is the Brownian motion starting from 0 with $E^0 B^2(t) = 2t$. We provide an integral formula for the density function of the stopped exponential functional $A(τ)=\int_0^τX^2(t) dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in \cite{BGS}, the present paper also covers the case of arbitrary drifts $μ\geq 0$ and provides a significant unification and extension of results of the ab…
We consider two depending Wiener processes which have membranes at zero with different permeability coefficients. Starting from different points, the processes almost surely do not meet at any fixed point except that where membranes are situated. The necessary and sufficient conditions for the meeting of the processes are found. It is shown that the probability of meeting is equal to zero or one.
We compute the joint distribution of the site and the time at which a $d$-dimensional standard Brownian motion $B_t$ hits the surface of the ball $ U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site $B_0$ becomes large. Our results entail that if Brownian motion is started at ${\bf x}$ and conditioned to hit $U(a)$ at time $t$ for the first time, the distribution of the hitting site approaches the uniform distribution or the point mass at $a{\bf x}/|{\bf x}|$ according as $|{\bf x}|/t$ tends to zero or inf…
The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the invol…
The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in $\R^{n+1}$, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space …
Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes $(Z_{t}=X_{t}+iY_{t},t\geq0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$(β_{t},t\geq0)$: for any fixed $u>0$: \sinh(β_{u}) \stackrel{(law)}{=} \hatβ_{(\int^{u}_{0}ds\exp(2β_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^θ_{c}\equiv\inf\{t:θ_{t} =c \}, (c>0)$, $T^θ_{-d,c}\equiv\inf\{t:θ_{t}\notin(-d,c) \}, (c,d>0)$ and more specifically of $T^θ_{-c,…
Hitting Probabilities for Hypoelliptic Differential Equations Driven by Fractional Brownian Motion
本文研究了由分数布朗运动驱动的双曲椭圆微分方程的击中概率估计，通过控制距离定义的牛顿型容量，推导出双侧估计并推广了经典椭圆结果。
We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index $H\le1/2$. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. B…
This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbolαu$ with $\boldsymbolα\in \mathbb{R}^d$, $\boldsymbolμ \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some c…
Branching capacity and Brownian snake capacity
本文在 $\mathbb R^d$ 中定义布朗蛇容量，并证明重新缩放后的分支容量淡收敛于该布朗蛇容量，证明依赖于击中概率逼近分支容量的精确收敛速率。
For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$, and the first hitting time $τ=\inf\{t\ge 0:B_t=0\}$, we find the probability density of $B_{uτ}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.
Hitting probabilities, thermal capacity, and Hausdorff dimension results for the Brownian sheet
该论文研究布朗片击中紧集的概率与交集维数，给出正概率相交的充要条件，并以热容量刻画交集豪斯多夫维数的本质确界，推广了布朗运动及区间情形下的已有结果。
We obtain sharp asymptotic estimates for hitting probabilities of a critical branching Brownian motion in one dimension with killing at 0 We also obtain sharp asymptotic formulas for the tail probabilities of the number of particles killed at 0. In the special case of double-or-nothing branching, we give exact formulas for both the hitting probabilities, in terms of elliptic functions, and the distribution of the number of killed particles.
Let $T_1^{(μ)}$ be the first hitting time of the point 1 by the Bessel process with index $μ\in \R$ starting from $x>1$. Using an integral formula for the density $q_x^{(μ)}(t)$ of $T_1^{(μ)}$, obtained in Byczkowski, Ryznar (Studia Math., 173(1):19-38, 2006), we prove sharp estimates of the density of $T_1^{(μ)}$ which exibit the dependence both on time and space variables. Our result provides optimal estimates for the density of the hitting time of the unit ball by the Brownian motion in $\mathbb{R}^n$, which improve existing bounds. Another application is to provide sharp estimates for the …
The purpose of the paper is to provide a general method for computing hitting distributions of some regular subsets D for Ornstein-Uhlenbeck type operators of the form 1/2Δ+ F\cdot\nabla, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method developed in the paper is based on stochastic calculus and on skew product representation of multidimensional Brownian motion and yields more complete resul…
Applying the technique of characteristic functions developped for one-dimensional regular surfaces (curves) with compact support, we obtain the distribution of hitting probabilities for a wide class of finite membranes on square lattice. Then we generalize it to multi-dimensional finite membranes on hypercubic lattice. Basing on these distributions, we explicitly construct the Brownian self-transport operator which governs the Laplacian transfer. In order to verify the accuracy of the distribution of hitting probabilities, numerical analysis is carried out for some particular membranes.
By the Cameron--Martin theorem, if a function $f$ is in the Dirichlet space $D$, then $B+f$ has the same a.s. properties as standard Brownian motion, $B$. In this paper we examine properties of $B+f$ when $f \notin D$. We start by establishing a general 0-1 law, which in particular implies that for any fixed $f$, the Hausdorff dimension of the image and the graph of $B+f$ are constants a.s. (This 0-1 law applies to any Lévy process.) Then we show that if the function $f$ is Hölder$(1/2)$, then $B+f$ is intersection equivalent to $B$. Moreover, $B+f$ has double points a.s. in dimensions $d\le 3…
In this article, we consider the properties of hitting times for $G$-martingale and the stopped processes. We prove that the stopped processes for $G$-martingales are still $G$-martingales and that the hitting times for a class of $G$-martingales including $G$-Brownian motion are quasi-continuous. As an application, we improve the $G$-martingale representation theorems in [Song10].
For a hyperbolic Brownian motion on the Poincaré half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(η, α)$ inside a hyperbolic disc $U$ of radius $\barη$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar η,\bar α)$. For $\barη \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $η$ and $\bar η$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{η_1}<T_{η_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $η_1$ and $η_2$ ar…
The joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient betwee…
We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.
Fractional Brownian motion with deterministic drift: How critical is drift regularity in hitting probabilities
本文研究带确定性漂移的分数布朗运动的击中概率，揭示漂移正则性、集合维数及容量对上下界的关键影响，并应用于点的极性判定。
In this paper we introduce a new method for the simulation of the exit time and position of a $δ$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $δ$-dimensional Bessel process and the $δ$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with exis…
We establish a general formula for the Laplace transform of the hitting times of a Gaussian process. Some consequences are derived, and particular cases like the fractional Brownian motion are discussed.
Given a Brownian path $β(t)$ on $\mathbb{R}$, starting at $1$, a.s. there is a singular time set $T_β$, such that the first hitting time of $β$ by an independent Brownian motion, starting at $0$, is in $T_β$ with probability one. A couple of problems regarding hitting measure for random processes are presented.
Asymptotic estimates of the hitting distribution of a long segment on the real axis for two dimensional random walks on ${\bf Z}^2$ of zero mean and finite variances are obtained: some are general and exhibit its apparent similarity to the corresponding Brownian density, while others are so detailed as to involve certain characteristics of the random walk.
We study a one-dimensional diffusion process in a drifted Brownian potential. We characterize the upper functions of its hitting times in the sense of Paul Lévy, and determine the lower limits in terms of an iterated logarithm law.
In Hernández-del-Valle (2010) the author studies the connection between Schrödinger's equation and first hitting densities of Brownian motion. Although the author is able to find solutions of a Schrödinger type pde he fails---except in some special cases---to construct a solution which satisfies the boundary on the space variable at $x=0$. In this paper we use an approach used in Bluman and Shtelen (1996) to find solutions which satisfy the pde and boundary condition when $t=0$.
Last Hitting Time Distributions for Solvable Diffusions
本文针对一维时齐可解扩散过程，建立了计算末次击中时间分布及其与有限时间范围内过程值联合分布的完整分析框架，推导了相关公式并给出应用实例。
A classic result on the 1-dimensional Brownian motion shows that conditionally on its first hitting time of 0, it has the distribution of a 3-dimensional Bessel bridge. By applying a certain time-change to this result, Matsumoto and Yor showed a theorem giving a relation between Brownian motions with opposite drifts. The relevant time change is the one appearing in Lamperti's relation. Sabot and Zeng showed that a family of Brownian motions with interacting drifts, conditioned on the vector of hitting times of 0, also has the distribution of independent 3-dimensional Bessel bridges. Moreover, …
Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable function. We are interested in finding conditions on $X$ and $f$ which imply that the perpetual integral functional $$ I^X_\infty(f):=\int_0^\infty f(X_t) dt $$ is identical in law with the first hitting time of a point for some other diffusion. This phenomenon may often be explained using random time change. Because of some potential applications in mathematical finance, we are considering mainly the case when $X$ is a Brownian motion with drift $μ>0,$ denoted $\{B^{(μ)}_t: t\geq 0\},$ but it is obvious that the method pre…
We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for $B$ a Brownian motion and $T_1$ its first hitting time of the level one, this remarkable law allows us to understand some properties of the process $(B_{uT_1}/\sqrt{T_1}, u\leq 1)$ under uniform random sampling.
In this paper we derive weak limits for the discretization errors of sampling barrier-hitting and extreme events of Brownian motion by using the Euler discretization simulation method. Specifically, we consider the Euler discretization approximation of Brownian motion to sample barrier-hitting events, i.e. hitting for the first time a deterministic "barrier" function; and to sample extreme events, i.e. attaining a minimum on a given compact time interval or unbounded closed time interval. For each case we study the discretization error between the actual time the event occurs versus the time t…
We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{é}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.
Small Brownian Loops Hitting SLE$_2$: An Exact Natural-Content Limit
该研究在有界解析若尔当域中证明了击中弦形$\text{SLE}_2$的小布朗环的根强度测度经$ε^{5/4}$缩放后依$L^1$收敛到自然含量测度的常数倍，给出精确极限常数，并得到淡收敛结果与布朗环汤的大数定律应用。
An ergodic theorem for the maximum of branching Brownian motion with absorption
本文研究具有吸收的分支布朗运动，证明其最大值的经验分布函数几乎处处收敛于一个随机位移的Gumbel分布。
\def\R{\mathbb R} We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on $E=(0,\infty)\times \R^d$, which is not invariant by translation. We then prove that the hitting probability of a Borel set $A\subset E$ for the graph of the Brownian snake starting at $(0,0)$ is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass $δ_0$ hits immediately $A$ (that is $(0,0)$ is G-regular for $A^c$) if and only if its capacity is infinite. As a di…
A new stochastic process is introduced and considered - squared Bessel process with special stochastic time. The analogues of fundamental properties for Brownian motion are deduced for squared Bessel process. In particular an analogue of the celebrated strong Markov construction of Brownian motion independent of a given sigma field is presented and proved. This result has strong consequences. It allows for deeper understanding the nature of first hitting time of squared Bessel process. The joint distribution of two correlated squared Bessel processes is presented. For squared Bessel process it…
The first passage time problem for Brownian motions hitting a barrier has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the barrier itself have appeared. Most interestingly, Peskir(2002b) demonstrates that a master integral equation can be used to generate a countable number of new equations via differentiation or integration by parts. In this article, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new cla…
The goal of this work is to find the asymptotics of the hitting probability of a distant point for the voter model on the integer lattice started from a single 1 at the origin. In dimensions 2 or 3, we obtain the precise asymptotic behavior of this probability. We use the scaling limit of the voter model started from a single 1 at the origin in terms of super-Brownian motion under its excursion measure. This invariance principle was stated by Bramson, Cox and Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less precise estimates are derived in dimensions greater than 4.
Let be $(X_t, t\geq 0)$ be a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time $τ_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{τ_x}-x$ the overshoot and $L_x:= x-X_{τ_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(τ_x, K_x, L_x).$
Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $C_c^\infty(\mathbf{R}^3\setminus \{0\})$. We will prove that this energy form is a regular Dirichlet form with core $C_c^\infty(\mathbf{R}^3)$. The associated diffusion $X$ behaves like a $3$-dimensional Brownian motion with a mild radial drift when far from $0$, subject to an ever-stronger push toward $0$ near that point. In particular $\{0\}$ is not a polar set with respect to $X$. The diffusion $X$ is rotation invariant, and admits a skew-produc…
A Moving Boundary Problem for Brownian Particles with Singular Forward-Backward Interactions
该研究提出一类带奇异正反向相互作用的布朗粒子系统，将其转化为新型分层运动边界偏微分方程问题，证明了该问题的经典适定性，并据此求解原正反向系统且证明其唯一性。
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