Professor Leonid Mytnik

General Information

Dr. Leonid Mytnik has an M.Sc. in Operation Research from the Technion. His Ph.D. concentrated on stochastic processes is also from the Technion. He held postdoctoral positions at the Department of Mathematics of The University of British Columbia, Canada (1996-1998), and then at the Laboratoire de Probabilites, University of Paris VI (1998-1999).

He returned to Israel in 1999 to assume a position at the Faculty of Industrial Engineering and Management, at the Technion.

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Research Summary

Dr. Leonid Mytnik has been doing research in various branches of the theory of Stochastic Processes. His main contribution over the past several years has been in the area of measure-valued processes or superprocesses. This class of processes describes the motion of mass in a scenario where some reproductive mechanism is also at play. Superprocesses can be constructed as a limit of branching migrating particles with small masses. The martingale problem approach, introduced by Stroock Varadhan for the study of finite dimensional diffusions, appeared to be a powerful tool to handle the problems of existence and uniqueness for the measure-valued processes. The success in the mathematical study of superprocesses is largely due to the strong independence assumptions: in the ``particle picture'' distinct particles branch and migrate independently of each other. When dependence on the other particles in the population is introduced into these two random mechanisms the situation becomes much more difficult, especially in the part of uniqueness question.

Over the past several years Dr. Mytnik had significant contribution in the applying ``duality'' technique for establishing uniqueness of the martingale problems corresponding to measure-valued processes with interactions. More recently he has been involved in a number of projects where ``duality'' approach proved to be a strong tool not only in proving uniqueness but also in deriving important properties of superprocesses with interactions.

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Current Research Projects

Superprocesses with hard killing.
Mutually catalytic superprocesses.
Stochastic partial differential equations driven by stable noise.
Computer simulations of spatial stochastic models

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Selected Publications

Adler R., Mytnik L., Bisexual branching diffusions,
Advances in Applied Probability, 27, 980-1018, 1995.
Mytnik L., Superprocesses in random environments,
The Annals of Probability, 24, 1953-1978, 1996.
Mytnik L., Collision measure and collision local time for (alpha,d,beta) superprocesses,
Journal of Theoretical Probability, 11, 733-763, 1998.
Mytnik L., Weak uniqueness for the heat equation with noise,
The Annals of Probability, 26, 968-984, 1998.
Mytnik L., Uniqueness for a mutually catalytic branching model,
Probability Theory and Related Fields, 112(2), 245-253, 1998.
Mytnik L., Uniqueness for a competing species model,
Canadian Journal of Mathematics, 51(2), 372-448, 1999.
Dawson D.A., Etheridge A.M., Fleischmann K., Mytnik L., Perkins E.A., Xiong J.,
Mutually catalytic branching in the plane: finite measure states.
Mytnik L.,
Stochastic partial differential equation driven by stable noise.
Mytnik L., Perkins E.A.,
Regularity and irregularity of (1 + beta)-stable super-Brownian motion.
Fleischmann, K., Mytnik L.,
Competing species superprocesses with infinite variance.
Burdzy, K., Mytnik L.,
Super-Brownian motion with reflecting historical paths. II. Convergence of approximations.
Mytnik L., Xiang K.-N.,
Tanaka formulae for (alpha ,d, beta)-superprocesses.
Le Gall J.-F., Mytnik L.,
Regularity and irregularity of the exit measure density for $(1+beta)$-stable super-Brownian motion.
L. Mytnik, E. Perkins, A. Sturm.,
On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficient
Mytnik L., Villa J.,
Self-Intersection local time of $(alpha ,d,beta )$-superprocess.

Annales de l'Institut Henry Poincare, 43(4), 481-507, 2007.
Fleischmann K., Mytnik L., Wachtel V., Optimal local Hölder index for density states of superprocesses with (1+β)-branching mechanism, The Annals of Probability, Vol 38(3), 1180-1220, 2010.
Fleischmann K., Mytnik L., Wachtel V., Hölder index at a given point for density states of super-$alpha$-stable motion of index $1+beta$. J. Theoret. Probab. 24 (2011), no. 1, 66-92.
Mueller C., Mytnik L., Quastel J., Effect of noise on front propagation in reaction-diffusion equations of KPP type. Invent. Math. 184 (2011), no. 2, 405�453.
Mytnik L., Perkins E.A., Pathwise uniqueness for stochastic heat equations with Holder continuous coefficients: the white noise case. Prob. Theory and Rel. Fields, 149 (2011), no. 1-2, 1�96.
Li Z., Mytnik L., Strong solutions for stochastic differential equations with jumps. Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 4 (2011), 1055-1067.
Mytnik L., Klenke A., Infinite Rate Mutually Catalytic Branching,
The Annals of Probability Vol 38(4), 1690-1716, 2010.
Mytnik L., Klenke A., Infinite Rate Mutually Catalytic Branching in Infinitely Many Colonies: The Longtime Behaviour, to appear in The Annals of Probability, 2011.
Mytnik L., Klenke A., Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence, to appear in Prob. Theory and Rel. Fields, 2011.
Mytnik L., Xiong J., Zeitouni O., Snake representation of a superprocess in random environment, to appear in ALEA, 2011.
Doering L., Mytnik L., Longtime Behavior for Mutually Catalytic Branching with Negative Correlations, to appear in Springer proceedings in mathematics.
Doering L., Mytnik L., Mutually Catalytic Branching Processes and Voter Processes with Strength of Opinion. Preprint (2011).

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