Leonid Mytnik

Optimal local Hölder index for density states of superprocesses with (1+β)-branching mechanism

For 0 0. If d 1 + β but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Holder index.

MUTUALLY CATALYTIC BRANCHING IN THE PLANE: UNIQUENESS

Abstract We study a pair of populations in R 2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

Stochastic integral representation and regularity of the density for the exit measure of super-Brownian motion

This paper studies the regularity properties of the density of the exit measure for super-Brownian motion with (1 + β)-stable branching mechanism. It establishes the continuity of the density in dimension d = 2 and the unboundedness of the density in all other dimensions where the density exists. An alternative description of the exit measure and its density is also given via a stochastic integral representation. Results are applied to the probabilistic representation of nonnegative solutions of the partial differential equation Δu = u 1+β .

Multifractal analysis of superprocesses with stable branching in dimension one

We show that density functions of a (α,1,β)-superprocesses are almost sure multifractal for α>β+1, β∈(0,1) and calculate the corresponding spectrum of singularities.

Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior

Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479–497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.

Longtime Behavior for Mutually Catalytic Branching with Negative Correlations

In several examples, dualities for interacting diffusion and particle systems permit the study of the longtime behavior of solutions. A particularly difficult model in which many techniques collapse is a two-type model with mutually catalytic interaction introduced by Dawson/Perkins for which they proved under some assumptions a dichotomy between extinction and coexistence directly from the defining equations.In the present chapter we show how to prove a precise dichotomy for a related model with negatively correlated noises. The proof uses moment bounds on exit times of correlated Brownian motions from the first quadrant and explicit second moment calculations. Since the uniform integrability bound is independent of the branching rate our proof can be extended to infinite branching rate processes.

Infinite rate mutually catalytic branching.

Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in terms of a planar Brownian motion from which we infer a path property of the process. This is the first paper in a series or three, wherein we also construct an interacting version of this process and study its long-time behavior.

Nonuniqueness for a parabolic SPDE with

Motivated by Girsanovs nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) partial derivative u/partial derivative t = Delta/2 u(t, x) + vertical bar u(t, x)vertical bar(gamma) W(t, x), u(0, x) = 0 Here W is a space time white noise on R x R. More precisely, we show the above stochastic PDE has a nonzero solution for 0 partial derivative u/partial derivative t = Delta/2 u(t, x) + sigma (u(t, x))W(t, x) if a is Holder continuous of index gamma > 3/4. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDEs is therefore similar to their finite dimensional counterparts, but with the index 3/4 in place of 1/2. The case gamma = 1/2 of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

\frac{3}{4}-\varepsilon

We study the solutions of the stochastic heat equation with multiplicative space–time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Holder continuous of index γ>3/4, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Holder continuous.

-Hölder diffusion coefficients

We establish Tanaka like formulae for the local time of (α, d, β)-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L1+θ convergence argument (with 0<θ<β≤1) rather than L2 convergence, for the derivation of the Tanaka formulae.