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We consider a class of stochastic heat equations driven by truncated $α$-stable white noises for $1

We study the ergodic property of a continuous-state branching process with immigration and competition, which is an extension of the models studied by Lambert (2005, Ann. Appl. Probab.), Pardoux (2016, Springer) and Berestycki et al. (2018, Probab. Theory Related Fields) with an additional immigration structure. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to branching mechanisms in the full range of criticality, including all stable types. The proof is based on a Markov coupling process and a nonsymmetric control function for the distance, which are designed to identify and to take the advantage of the dominating factor from the branching, immigration and competition mechanisms in different parts of the state space. The approach provides a way of finding explicitly the ergodicity rate.

For a class of non-linear stochastic heat equations driven by $α$-stable white noises for $α\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued càdlàg solutions to such a equation for $p\in(α,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $α$-stable white noises obtained by removing the big jumps from the original $α$-stable white noises. If the $α$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with L��vy mutation and resampling mechanism associated to a general $��$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(��* Z_t) \subseteq S(Z_t)$, where $��$ is the L��vy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process., 23 pages

For a class of $Λ$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $Λ$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $Λ$-Fleming-Viot support processes. In particular, if the $Λ$-coalescent is the Beta$(2-β,β)$ coalescent for $β\in(1,2]$ with $β=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the support process with modulus function $\sqrt{2β/(β-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(β-1)}h(t)$., 27 pages

We construct extensions of the pure-jump $��$-Wright-Fisher processes with frequency-dependent selection ($��$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coalescence processes (EFC). One-to-one correspondences between the nature of the boundary $1$ of the $��$-WF process with selection and the boundary $\infty$ of the block counting process are established. New properties for the $��$-WF processes with selection and the block counting processes of the simple EFC processes are deduced from these correspondences. Some conditions are provided for the selection to be either weak enough for boundary $1$ to be an exit boundary or strong enough for $1$ to be an entrance boundary. When the measure $��$ and the selection mechanism satisfy some regular variation properties, conditions are found in order that the extended $��$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In the latter process, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele which can spread into the population in a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

A generalized stepping stone model with $��$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $��$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $��$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type., 19 pages

In this paper, we consider a class of generalized continuous-state branching processes obtained by Lamperti type time changes of spectrally positive L\'evy processes using different rate functions. When explosion occurs to such a process, we show that the process converges to infinity in finite time asymptotically along a deterministic curve, and identify the speed of explosion for rate function in different regimes. To prove the main theorems, we also establish a new asymptotic result for scale function of spectrally positive L\'evy process., Comment: 29 pages

Consider a spectrally positive L��vy process $Z$ with log-Laplace exponent $��$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $��(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the L��vy process has a negative drift $��:=-��0$ and $R$ is regularly varying at $\infty$ with index $��>��$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.

In this paper we study the draw-down related Parisian ruin problem for spectrally negative L��vy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative L��vy risk process with dividend barrier and Parisian ruin.

In this paper we study a spectrally negative L��vy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for spectrally negative L��vy processes.

For spectrally negative L��vy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative L��vy processes.

For spectrally negative L��vy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure., 23 pages

For refracted spectrally negative L��vy processes, we identify expressions of several quantities related to Laplace transforms on their weighted occupation times until first exit times. Such quantities are expressed in terms of unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative L��vy processes. Previous results on refracted L��vy processes are recovered.

Using a new approach, for spectrally negative L��vy processes we find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last exit time and the associated occupation time, which generalizes some previous results., 16pages

For a spectrally one-sided L��vy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas., Published at http://dx.doi.org/10.3150/15-BEJ695 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H��lder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ��X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $��$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $��$-H��lder continuous for $10$.

Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the $��$-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding $��$-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support., 21 pages

The (\Xi, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the \Xi-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative L��vy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented., 27 pages

In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative L��vy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative L��vy process and its Laplace exponent. Applications to insurance risk models are also presented., corrections in the proof of Theorem 1

Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial., Comment: 19 pages

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration., Comment: 13 pages

Analogues of stepping--stone models are considered where the site--space is continuous, the migration process is a general Markov process, and the type--space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum--sites stepping--stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle--1/2 Cantor set (and hence also to the Brownian zero set)., 41 pages