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Register a new userThe quasi-stationary distribution of the subcritical contact process
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We show that the quasi-stationary distribution of the subcritical contact process on $mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.
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The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
Suppose that $X$ is a subcritical superprocess. Under some asymptotic conditions on the mean semigroup of $X$, we prove the Yaglom limit of $X$ exists and identify all quasi-stationary distributions of $X$.
The stationary asymptotic properties of the diffusion limit of a multi-type branching process with neutral mutations are studied. For the critical and subcritical processes the interesting limits are those of quasi-stationary distributions conditioned on non-extinction. Limiting distributions for supercritical and critical processes are found to collapse onto rays aligned with stationary eigenvectors of the mutation rate matrix, in agreement with known results for discrete multi-type branching processes. For the sub-critical process the quasi-stationary distribution is obtained to first order in the overall mutation rate, which is assumed to be small. The sampling distribution over allele types for a sample of given finite size is found to agree to first order in mutation rates with the analogous sampling distribution for a Wright-Fisher diffusion with constant population size.
We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $mu le 1$ for each particle at that site. We show the critical birth rate $lambda_c(mu)$ for weak survival is of order $sqrt{mu}$ as $mu to 0$. Mean-field calculations predict that when $mu < 1/2$ there is a discontinuous transition as $lambda$ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to physicists the paper that introduced the model, the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued c`adl`ag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Levy driven SDEs under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.
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