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Register a new userOn uniqueness of solutions to martingale problems --- counterexamples and sufficient criteria
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The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process is not necessarily determined by its generator. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.
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A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Levy process, and its law is characterized by a triplet $(sigma^2,a,Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(sigma^2,a,Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Levy processes conditioned to stay positive.
We consider a general class of metric measure spaces equipped with a regular Dirichlet form and then provide a lower bound on the hitting time probabilities of the associated Hunt process. Using these estimates we establish (i) a generalization of the classical Liebs inequality on metric measure spaces and (ii) uniqueness of nonnegative super-solutions on metric measure spaces. Finally, using heat-kernel estimates we generalize the local Faber-Krahn inequality recently obtained in [LS18].
This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables $y$ and $z$) in a super-linear way like $|y||ln |y||^{(lambda+1/2)wedge 1}+|z||ln |z||^{lambda}$ for some $lambdageq 0$. For the following four different ranges of the growth power parameter $lambda$: $lambda=0$, $lambdain (0,1/2)$, $lambda=1/2$ and $lambda>1/2$, we give reasonably weakest possible different integrability conditions of the terminal value for the existence of an unbounded solution to the BSDE. In the first two cases, they are stronger than the $Lln L$-integrability and weaker than any $L^p$-integrability with $p>1$; in the third case, the integrability condition is just some $L^p$-integrability for $p>1$; and in the last case, the integrability condition is stronger than any $L^p$-integrability with $p>1$ and weaker than any $exp(L^epsilon)$-integrability with $epsilonin (0,1)$. We also establish the comparison theorem, which yields naturally the uniqueness, when either generator of both BSDEs is convex (concave) in both unknown variables $(y,z)$, or satisfies a one-sided Osgood condition in the first unknown variable $y$ and a uniform continuity condition in the second unknown variable $z$.
We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hu007folder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present.
Several counterexample models to the Nelson-Seiberg theorem have been discovered in previous literature, with generic superpotentials respecting the R-symmetry and non-generic R-charge assignments for chiral fields. This work present a sufficient condition for such counterexample models: The number of R-charge 2 fields, which is greater than the number of R-charge 0 fields, must be less than or equal to the number of R-charge 0 fields plus the number of independent field pairs with opposite R-charges and satisfying some extra requirements. We give a correct count of such field pairs when there are multiple field pairs with degenerated R-charges. These models give supersymmetric vacua with spontaneous R-symmetry breaking, thus are counterexamples to both the Nelson-Seiberg theorem and its extensions.
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