On the harmonic measure of stable processes

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real L{\'e}vy process. This gives a simple and unified proof of several results in the literature, old and recent. We also provide a full description of the corresponding Green functions. As a by-product, we compute the hitting probabilities of points and describe the non-negative harmonic functions for the stable process killed outside a finite interval

History of depression and survival after acute myocardial infarction

Objective: To compare survival in post-myocardial (MI) participants from the Enhancing Recovery In Coronary Heart Disease (ENRICHD) clinical trial with a first episode of major depression (MD) and those with recurrent MID, which is a risk factor for mortality after acute MI. Recent reports suggest that the level of risk may depend on whether the comorbid MD is a first or a recurrent episode. Methods: Survival was compared over a median of 29 months in 370 patients with an initial episode of MD, 550 with recurrent MD, and 408 who were free of depression. Results: After adjusting for an all-cause mortality risk score, initial Beck Depression Inventory score, and the use of selective serotonin reuptake inhibitor antidepressants, patients with a first episode of MD had poorer survival (18.4% all-cause mortality) than those with recurrent MD (11.8%) (hazard ratio (HR)=1.4; 95% Confidence Interval (CI)=1.0-2.0; p=.05). Both first depression (HR=3.1; 95% CI=1.6-6.1; p=.001) and recurrent MD (HR=2.2; 95% CI=1.1-4.4; p=.03) had significantly poorer survival than did the nondepressed patients (3.4%). A secondary analysis of deaths classified as probably due to a cardiovascular cause resulted in similar HRs, but the difference between depression groups was not significant. Conclusions: Both initial and recurrent episodes of MD predict shorter survival after acute MI, but initial MD episodes are more strongly predictive than recurrent episodes. Exploratory analyses suggest that this cannot be explained by more severe heart disease at index, poorer response to depression treatment, or a higher risk of cerebrovascular disease in patients with initial MD episodes

On h h -transforms of one-dimensional diffusions stopped upon hitting zero

For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are $h$-transforms of the process stopped upon hitting zero, where $h$'s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the $h$-transforms are investigated

Construction of markov processes from hitting distributions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47648/1/440_2004_Article_BF00538487.pd

Excision of a strong Markov process

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47651/1/440_2004_Article_BF00532853.pd

Potentials of stable processes

For a stable process, we give an explicit formula for the potential measure of the process killed outside a bounded interval and the joint law of the overshoot, undershoot and undershoot from the maximum at exit from a bounded interval. We obtain the equivalent quantities for a stable process reflected in its infimum. The results are obtained by exploiting a simple connection with the Lamperti representation and exit problems of stable processes.Comment: 10 page

Uniqueness and Nondegeneracy of Ground States for (−Δ)sQ+Q−Qα+1=0(-\Delta)^s Q + Q - Q^{\alpha+1} = 0 in R\mathbb{R}

We prove uniqueness of ground state solutions $Q = Q(|x|) \geq 0$ for the nonlinear equation $(-\Delta)^s Q + Q - Q^{\alpha+1}= 0$ in $\mathbb{R}$, where $0 < s < 1$ and $0 < \alpha < \frac{4s}{1-2s}$ for $s < 1/2$ and $0 < \alpha < \infty$ for $s \geq 1/2$. Here $(-\Delta)^s$ denotes the fractional Laplacian in one dimension. In particular, we generalize (by completely different techniques) the specific uniqueness result obtained by Amick and Toland for $s=1/2$ and $\alpha=1$ in [Acta Math., \textbf{167} (1991), 107--126]. As a technical key result in this paper, we show that the associated linearized operator $L_+ = (-\Delta)^s + 1 - (\alpha+1) Q^\alpha$ is nondegenerate; i.\,e., its kernel satisfies $\mathrm{ker}\, L_+ = \mathrm{span}\, \{Q'\}$. This result about $L_+$ proves a spectral assumption, which plays a central role for the stability of solitary waves and blowup analysis for nonlinear dispersive PDEs with fractional Laplacians, such as the generalized Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page

Construction of Markov processes and associated multiplicative functionals from given harmonic measures

Let E be a noncompact locally compact second countable Hausdorff space. We consider the question when, given a family of finite nonzero measures on E that behave like harmonic measures associated with all relatively compact open sets in E (i.e. that satisfy a certain consistency condition), one can construct a Markov process on E and a multiplicative functional with values in [0, ∞) such that the hitting distributions of the process “inflated” by the multiplicative functional yield the given harmonic measures. We achieve this construction under weak continuity and local transience conditions on these measures that are natural in the theory of Markov processes, and a mild growth restriction on them. In particular, if the space E equipped with the measures satisfies the conditions of a harmonic space, such a Markov process and associated multiplicative functional exist. The result extends in a new direction the work of many authors, in probability and in axiomatic potential theory, on constructing Markov processes from given hitting distributions (i.e. from harmonic measures that have total mass no more than 1).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47657/1/440_2005_Article_BF01192513.pd

Two refreshing views of Fluctuation Theorems through Kinematics Elements and Exponential Martingale

In the context of Markov evolution, we present two original approaches to obtain Generalized Fluctuation-Dissipation Theorems (GFDT), by using the language of stochastic derivatives and by using a family of exponential martingales functionals. We show that GFDT are perturbative versions of relations verified by these exponential martingales. Along the way, we prove GFDT and Fluctuation Relations (FR) for general Markov processes, beyond the usual proof for diffusion and pure jump processes. Finally, we relate the FR to a family of backward and forward exponential martingales.Comment: 41 pages, 7 figures; version2: 45 pages, 7 figures, minor revisions, new results in Section

Hausdorff dimension of operator semistable L\'evy processes

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process in \rd with exponent $E$, where $E$ is an invertible linear operator on \rd and $X$ is semi-selfsimilar with respect to $E$. By refining arguments given in Meerschaert and Xiao \cite{MX} for the special case of an operator stable (selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we determine the Hausdorff dimension of the partial range $X(B)$ in terms of the real parts of the eigenvalues of $E$ and the Hausdorff dimension of $B$.Comment: 23 page