In this note we provide a stochastic method for approximating solutions of ordinary differential equations. To this end, a stochastic variant of the Euler scheme is given by means of Markov chains. For an ordinary differential equation, these approximations are shown to satisfy a Large Number Law, and a Central Limit Theorem for the corresponding fluctuations about the solution of the differential equation is proven.

Via Phi-Sobolev inequalities, we give some sharp integrability conditions on $F$ for the large deviation principle of the empirical mean $\frac{1}{T}{\int_{0}^{T}{F(X_{s})}ds}$ for large time $T$, where $F$ is unbounded with values in some separable Banach space. Several examples are provided.

A non-commutative version of the sewing lemma is proved, with some applications

We revisit the work of Bourgain et al. (1992) - referred to as "BKKKL" in the title - about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\{1,...,r\}^N$, generalizing one of the main results of Talagrand (1994).

We establish in this paper an exact formula which links the dimension of the harmonic measure, the asymptotic entropy and the rate of escape for a random walk on a discrete subgroup of the isometry group of a Gromov hyperbolic space. This completes a result obtained by the author in a previous paper, where only an upper bound for the dimension was proved.

In this note we extend a classical equivalence result for Gaussian stationary processes to the more general setting of Gaussian processes with stationary increments. This will allow us to apply it in the setting of aggregated independent fractional Brownian motions.

Abstract. We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on $\mathbb{R}^n$. The proof is based on arguments introduced in Bakry and al, but for the sake of completeness, all details are provided.

We extend to matrix-valued stochastic processes, some well-known relations between real-valued diffusions and classical orthogonal polynomials, along with some recent results about Lévy processes and martingale polynomials. In particular, joint semigroup densities of the eigenvalue processes of the generalized matrix-valued Ornstein-Uhlenbeck and squared Ornstein-Uhlenbeck processes are respectively expressed by means of the Hermite and Laguerre polynomials of matrix arguments. These polynomials also define martingales for the Brownian matrix and the generalized Gamma process. As an application, we derive a chaotic representation property for the eigenvalue process of the Brownian matrix.

Central limit theorems for functionals of general state space Markov chains are of crucial importance in sensible implementation of Markov chain Monte Carlo algorithms as well as of vital theoretical interest. Different approaches to proving this type of results under diverse assumptions led to a large variety of CLT versions. However due to the recent development of the regeneration theory of Markov chains, many classical CLTs can be reproved using this intuitive probabilistic approach, avoiding technicalities of original proofs. In this paper we provide a characterization of CLTs for ergodic Markov chains via regeneration and then use the result to solve the open problem posed in [Roberts & Rosenthal 2005]. We then discuss the difference between one-step and multiple-step small set condition.

Benford's law states that for many random variables $X > 0$ its leading digit $D = D(X)$ satisfies approximately the equation $\mathbb{P}(D = d) = \log_{10}(1 + 1/d)$ for $d = 1,2,\ldots,9$. This phenomenon follows from another, maybe more intuitive fact, applied to $Y := \log_{10}X$: For many real random variables $Y$, the remainder $U := Y - \lfloor Y\rfloor$ is approximately uniformly distributed on $[0,1)$. The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of $Y$ or some derivative of it. These bounds are an interesting and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford's law.

We provide a characterisation of Gaussian time series which optimise the one-step prediction error subject to the covariance sequence being completely monotone with the first $m$ covariances specified.

Metric entropy of the class of probability distribution functions on $[0,1]$ with a $k$-monotone density is studied through its connection with the small ball probability of $k$-times integrated Brownian motions.

In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniquenes of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed.

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus $({\mathbb Z} / N{\mathbb Z})^d$, $d \geq 3$, until $uN^d$ time steps, $u > 0$, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large $N$, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time $uN^d$ converges to independent copies of the random interlacement at level $u$.

Let $H$ be a separable real Hilbert space and let $\mathbb{F}=(\mathscr{F}_t)_{t\in [0,T]}$ be the augmented filtration generated by an $H$-cylindrical Brownian motion $(W_H(t))_{t\in [0,T]}$ on a probability space $(\Omega,\mathscr{F},\mathbb{P})$. We prove that if $E$ is a UMD Banach space, $1\le p<\infty$, and $F\in \mathbb{D}^{1,p}(\Omega;E)$ is $\mathscr{F}_T$-measurable, then $$ F = \mathbb{E} (F) + \int_0^T P_{\mathbb{F}} (DF)\,dW_H,$$ where $D$ is the Malliavin derivative of $F$ and $P_{\mathbb{F}}$ is the projection onto the ${\mathbb{F}}$-adapted elements in a suitable Banach space of $L^p$-stochastically integrable $\mathscr{L}(H,E)$-valued processes.

Consider a long-range, one-dimensional voter model started with all zeroes on the negative integers and all ones on the positive integers. If the process obtained by identifying states that are translations of each other is positively recurrent, then it is said that the voter model exhibits interface tightness. In 1995, Cox and Durrett proved that one-dimensional voter models exhibit interface tightness if their infection rates have a finite third moment. Recently, Belhaouari, Mountford, and Valle have improved this by showing that a finite second moment suffices. The present paper gives a new short proof of this fact. We also prove interface tightness for a long range swapping voter model, which has a mixture of long range voter model and exclusion process dynamics.

We show that the Bernoulli conjecture holds for sets with small one-dimensional projections, i.e. any bounded Bernoulli process indexed by such set may be represented as a sum of a uniformly bounded process and a process dominated by a bounded Gaussian process.

The scale-invariant spacings lemma due to Arratia, Barbour and Tavaré establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant

We show that an asymmetric Bernoulli random variable is symmetry resistant in the sense that any independent random variable, which when added to it produces a symmetric sum, must have a variance at least as much as itself. The main instrument is to use Skorokhod embedding to transfer the discrete problem to the realm of stochastic calculus.

The purpose of this note is twofold. Firstly to complete a recent accumulation of results concerning extended version of Ito's formula for any one dimensional Lévy processes, $X$. Secondly, we use the latter to characterise the parabolic generator of $X$, \[ {\bf A}:= \left\{ (f,g) : f(X_\cdot,\cdot) - \int_0^\cdot g(X_s, s)ds \text{ is a local martingale} \right\}. \] We also establish a necessary condition for a pair of functions to be in the domain of the parabolic generator when $X$ has a Gaussian component.

A compound Poisson process is considered. We estimate the current position of the stochastic process based on past discrete-time observations (non-linear discrete filtering problem) in Bayesian setting. We obtain bounds for the asymptotic rate of the expected square error of the filter when observations become frequent. The bounds depend linearly on jump intensity. Also, estimation of process' parameters is addressed.

We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\Lambda(\cdot)_{\mid \mathcal{A}}$.

It was proved by Michel Talagrand in [10] that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application of the differentiability to prove non self-averaging of the overlap outside of the replica symmetric region.

We give a new proof of a theorem by Le Gall and Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2-sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces.

The edges of the complete graph on $n$ vertices are assigned independent exponentially distributed costs. A $k$-matching is a set of $k$ edges of which no two have a vertex in common. We obtain explicit bounds on the expected value of the minimum total cost $C_{k,n}$ of a $k$-matching. In particular we prove that if $n = 2k$ then $\pi^2/12 < EC_{k,n} < \pi^2/12 + \log n/n$.

We give a characterization of the family of all probability measures on the extended line $(-\infty,+\infty]$, which may be obtained as the distribution of the maximum of some linear Bernoulli process.

The stochastic growth model Internal Diffusion Limited Aggregation was defined in 1991 by Diaconis and Fulton. Several shape results are known when the underlying state space is the d-dimensional lattice, or a discrete group with exponential growth. We prove an extension of the shape result of Blachere and Brofferio for Internal Diffusion Limited Aggregation on a wide class of Markov chains on non-amenable graphs.

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \sigma - o( N^{-6/11+\varepsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\varepsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\varepsilon <0, \ $ one has $ \|A_N\| =2 \sigma + o( N^{-6/11+\varepsilon}) $ with probability going to $ 1 $ as $N \to \infty$.

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions $d < 8$, and it is potentially also applicable to percolation in dimensions $d < 6$.

We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao's theorem (Shao, Q.-M., 1995. On a conjecture of Révész. Proc. Amer. Math. Soc. 123, 575-582) in the gaussian case.

We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree.

This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.

We study the precise relationship between the subordinate killed and killed subordinate processes in the case of an underlying Hunt process, and show that, under minimal conditions, the former is a subprocess of the latter obtained by killing at a terminal time. Moreover, we also show that the killed subordinate process can be obtained by resurrecting the subordinate killed one at most countably many times.

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.

In this work, we approximate a diffusion process by its Euler scheme and we study the convergence of the density of the marginal laws. We improve previous estimates especially for small time.

We give necessary and sufficient conditions for a Gaussian vector with non-zero mean, to have infinitely divisible squares for all scalar multiples of the mean, and show how the this vector is related to the local times of a Markov chain determined by the covariance matrix of the Gaussian vector. Our results add to results of Griffiths, Bapat, Eisenbaum and Kaspi.

We show that for all $p<p_c(\mathbb{Z}^d)$ percolation parameters, the probability that the cluster of the origin is finite but has at least $t$ vertices at distance one from the infinite cluster is exponentially small in $t$. We use this to give a short proof of the strongest version of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. This implies, e.g., that simple random walk on the largest cluster of a finite box $[-n,n]^d$ with high probability has $L^\infty$-mixing time $\Theta(n^2)$, and that the heat kernel (return probability) on the infinite cluster a.s. decays like $p_n(o,o)=O(n^{-d/2})$. Versions of these results have been proven by Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau (2006). For general infinite graphs, we prove that anchored isoperimetric properties survive supercritical percolation, provided that the probability of the cluster of the origin being finite with large boundary decays rapidly; this is the case for a large class of graphs when $p$ is close to 1. As an application (with the help of some entropy inequalities), we give a short conceptual proof of a theorem of Angel, Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedge in $\mathbb{Z}^3$ is a.s. transient whenever the wedge itself is transient.

It has been shown that certain types of random walks in random potentials and Brownian motion in Poissonian potentials undergo a phase transition from sub-ballistic to ballistic behavior when the strength of the underlying drift is increased. The ballistic behavior has been manifested by indicating a limiting area for the normalized motion. In the present article, we provide a refined description of this limiting area with a further development for the case of rotation invariant Poissonian potentials.

It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. What is more, we allow for a certain amount of dependence among the matrix entries, in the spirit of a recent generalization of Wigner's theorem, due to Schenker and Schulz-Baldes. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{+}$ with expectation $1$ and variance $V.$ Let $\mu _{n}$ denote the $n$-time free multiplicative convolution of measure $\mu $ with itself. Then, for large $n$ the length of the support of $\mu _{n}$ is asymptotically equivalent to $eVn$, where $e$ is the base of natural logarithms, $ e=2.71\ldots $.

For real Lévy processes $(X_t)_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $\beta$, the estimate $E \sup_{s \leq t} |X_s - a_p s|^p \leq C_p t$ for every $t \in [0,1]$ and suitable $a_p \in R$ has been established by Millar for $\beta < p \leq 2$ provided $X_1 \in L^p$. We derive extensions of these estimates to the cases $p < 2$ and $p \leq\beta$.

We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami (2001), essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we adapt the approach by Götze and Kösters (2008), who proved the analogous result for the Hermitian case.

We derive a number of estimates for the probability that a chordal SLE$_\kappa$ path in the upper half plane $\mathbb{H}$ intersects a semicircle centred on the real line. We prove that if $0<\kappa <8$ and $\gamma:[0,\infty) \to \overline{\mathbb{H}}$ is a chordal SLE$_\kappa$ in $\mathbb{H}$ from $0$ to $\infty$, then $P\{\gamma[0,\infty) \cap \mathcal{C}(x;rx) \neq \emptyset\} \asymp r^{4a-1}$ where $a=2/\kappa$ and $\mathcal{C}(x;rx)$ denotes the semicircle centred at $x<0$ of radius $rx$, $00$. For $4<\kappa<8$, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE$_\kappa$ path in $\mathbb{H}$ from $0$ to $\infty$.

Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each $n$ a fragmentation process $(\Pi_{n,k}, 1 \leq k \leq n)$ such that $\Pi_{n,k}$ is distributed like the partition generated by cycles of a uniform random permutation of $\{1,2,\ldots,n\}$ conditioned to have $k$ cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions.

We illustrate how a technique from the theory of random iterations of functions can be used within the theory of products of matrices. Using this technique we give a simple proof of a basic theorem about the asymptotic behavior of (deterministic) "backwards products" of row-stochastic matrices and present an algorithm for perfect sampling from the limiting common row-vector (interpreted as a probability-distribution).

Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\in$, $Z$ be an Hermite random variable of index $q$, and $H_q$ denote the $q$th Hermite polynomial. For any $n\geq 1$, set $V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H<1-1/(2q)$, an upper bound for the total variation distance between the laws $\mathscr{L}(Z_n)$ and $\mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case where $H<1-1/(2q)$, corresponding to the case where one has normal approximation.

In this note, we study the asymptotic behaviour of Lévy processes with no positive jumps conditioned to stay positive and some related processes. In particular, we establish an integral test for the lower envelope at $0$ and at $+\infty$ and an analogue of Khintchin's law of the iterated logarithm at 0 and at $+\infty$, for the upper envelope of the reflected process at its future infimum.

We consider the $r$-th power quantization error arising in the optimal approximation of a $d$-dimensional probability measure $P$ by a discrete measure supported by the realization of $n$ i.i.d. random variables $X_1,...,X_n$. For all $d \geq 1$ and $r \in $ we establish mean and variance asymptotics as well as central limit theorems for the $r$-th power quantization error. Limiting means and variances are expressed in terms of the densities of $P$ and $X_1$. Similar convergence results hold for the random point measures arising by placing at each $X_i, 1 \leq i \leq n,$ a mass equal to the local distortion.

Jiang, Dickey, and Kuo [12] gave the multivariate c-characteristic function and showed that it has properties similar to those of the multivariate Fourier transformation. We first give the multivariate c-characteristic function of a random functional of a Ferguson-Dirichlet process over the unit sphere. We then find out its probability density function using properties of the multivariate c-characteristic function. This new result would generalize that given by [11].

The so-called Bercovici-Pata bijection maps the set of classical infinitely divisible laws to the set of free infinitely divisible laws. The purpose of this work is to study the free infinitely divisible laws corresponding to the classical Generalized Gamma Convolutions (GGC). Characterizations of their free cumulant transforms are derived as well as free integral representations with respect to the free Gamma process. A random matrix model for free GGC is built consisting of matrix random integrals with respect to a classical matrix Gamma process. Nested subclasses of free GGC are shown to converge to the free stable class of distributions.

It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.

We consider a two-type oriented competition model on the first quadrant of the two-dimensional integer lattice. Each vertex of the space may contain only one particle of either Red type or Blue type. A vertex flips to the color of a randomly chosen southwest nearest neighbor at exponential rate 2. At time zero there is one Red particle located at $$. The main result is a partial shape theorem: Denote by $R (t)$ and $B (t)$ the red and blue regions at time $t$. Then (i) eventually the upper half of the unit square contains no points of $B (t)/t$, and the lower half no points of $R (t)/t$; and (ii) with positive probability there are angular sectors rooted at $$ that are eventually either red or blue. The second result is contingent on the uniform curvature of the boundary of the corresponding Richardson shape.

The nonlinear filter associated with the discrete time signal-observation model $(X_k,Y_k)$ is known to forget its initial condition as $k\to\infty$ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condition regardless of any properties of the signal. We show that for observations of additive type $Y_k=h(X_k)+\xi_k$ with invertible observation function $h$ (under mild regularity assumptions on $h$ and on the distribution of the noise $\xi_k$), the filter is indeed stable in a weak sense without any assumptions at all on the signal process. If the signal satisfies a uniform continuity assumption, weak stability can be strengthened to stability in total variation.

The question of recurrence and transience of branching Markov chains is more subtle than for ordinary Markov chains; they can be classified in transience, weak recurrence, and strong recurrence. We review criteria for transience and weak recurrence and give several new conditions for weak recurrence and strong recurrence. These conditions make a unified treatment of known and new examples possible and provide enough information to distinguish between weak and strong recurrence. This represents a step towards a general classification of branching Markov chains. In particular, we show that in homogeneous cases weak recurrence and strong recurrence coincide. Furthermore, we discuss the generalization of positive and null recurrence to branching Markov chains and show that branching random walks on $Z$ are either transient or positive recurrent.

We give a simple probabilistic proof of the classical fact from complex analysis that the zeros of a holomorphic function of several variables are never isolated and that they are not contained in any compact set. No facts from complex analysis are assumed other than the Cauchy-Riemann definition. From stochastic analysis only the Ito formula and the standard existence theorem for stochastic differential equations are required.

We consider a continuous time Markov process $(X,L)$, where $X$ jumps between a finite number of states and $L$ is a piecewise linear process with state space $\mathbb{R}^d$. The process $L$ represents an "inert drift" or "reinforcement." We find sufficient and necessary conditions for the process $(X,L)$ to have a stationary distribution of the product form, such that the marginal distribution of $L$ is Gaussian. We present a number of conjectures for processes with a similar structure but with continuous state spaces.

In a non-parametric framework, we establish some non-asymptotic bounds for self-normalized sums and quadratic forms in the multivariate case for symmetric and general random variables. This bounds are entirely explicit and essentially depends in the general case on the kurtosis of the Euclidean norm of the standardized random variables.

For the one-dimensional Brownian motion $B=(B_t)_{t\geq 0}$, started at $x<0$, and the first hitting time $\tau=\inf\{t\geq 0:B_t=0\}$, we find the probability density of $B_{u\tau}$ for a $u\in$, i.e. of the Brownian motion on its way to hitting zero.

We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.

Let $X=(X_t)_{t \ge 0}$ be a stable Lévy process of index $\alpha \in $ with the Lévy measure $\nu(dx) = (c/x^{1+\alpha}) I_{ dx$ for $c<0$, let $x<0$ be given and fixed, and let $\tau_x = \inf\{ t<0 : X_t=x \}$ denote the first hitting time of $X$ to $x$. Then the density function $f_{\tau_x}$ of $\tau_x$ admits the following series representation: $$f_{\tau_x}(t) = \frac{x^{\alpha-1}}{\pi ( \Gamma(-\alpha) t)^{2-1/\alpha}} \sum_{n=1}^\infty \bigg[(-1)^{n-1} \sin(\pi/\alpha) \frac{\Gamma(n-1/\alpha)}{\Gamma(\alpha n-1)} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t} \Big)^{n-1} $$ $$- \sin\Big(\frac{n \pi}{\alpha}\Big) \frac{\Gamma(1+n/\alpha)}{n!} \Big(\frac{x^\alpha}{c \Gamma(-\alpha)t}\Big)^{(n+1)/\alpha-1} \bigg]$$ for $t<0$. In particular, this yields $f_{\tau_x}(0+)=0$ and $$ f_{\tau_x}(t) \sim \frac{x^{\alpha-1}}{\Gamma(\alpha-1), \Gamma(1/\alpha)} (c \Gamma(-\alpha)t)^{-2+1/\alpha} $$ as $t \rightarrow \infty$. The method of proof exploits a simple identity linking the law of $\tau_x$ to the laws of $X_t$ and $\sup_{0 \le s \le t} X_s$ that makes a Laplace inversion amenable. A simpler series representation for $f_{\tau_x}$ is also known to be valid when $x<0$.

Let $\alpha$ be a fixed number from the interval $[0,1]$. We obtain the sharp probability bounds for the maximal function of the process which is $\alpha$-differentially subordinate to a bounded submartingale. This generalizes the previous results of Burkholder and Hammack.