Probability Theory and Related Fields最新文献

We explore the relationship between recursive distributional equations and convergence results for finite difference schemes of parabolic partial differential equations (PDEs). We focus on a family of random processes called symmetric cooperative motions, which generalize the symmetric simple random walk and the symmetric hipster random walk introduced in Addario-Berry et al. (Probab Theory Related fields 178(1–2):437–473, 2020). We obtain a distributional convergence result for symmetric cooperative motions and, along the way, obtain a novel proof of the Bernoulli central limit theorem. In addition, we prove a PDE result relating distributional solutions and viscosity solutions of the porous medium equation and the parabolic p-Laplace equation, respectively, in one dimension.

We prove a non-extinction result for Fleming–Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.

In this paper, we provide relations among the following properties:

1. (a)

   the tail triviality of a probability measure (mu ) on the configuration space ({varvec{Upsilon }});

2. (b)

   the finiteness of a suitable (L^2)-transportation-type distance (bar{textsf {d} }_{varvec{Upsilon }});

3. (c)

   the irreducibility of local ({mu })-symmetric Dirichlet forms on ({varvec{Upsilon }}).

As an application, we obtain the ergodicity (i.e., the convergence to the equilibrium) of interacting infinite diffusions having logarithmic interaction and arising from determinantal/permanental point processes including (text {sine}_{2}), (text {Airy}_{2}), (text {Bessel}_{alpha , 2}) ((alpha ge 1)), and (text {Ginibre}) point processes. In particular, the case of the unlabelled Dyson Brownian motion is covered. For the proof, the number rigidity of point processes in the sense of Ghosh–Peres plays a key role.

The focal point of thyroidectomy surgery has always been to reduce the incidence of inadvertent damage to the recurrent laryngeal nerve(RLN). The intracapsular thyroidectomy is one such technique with minimum chance of injuring the nerve. To compare retrospectively the surgical outcomes between the two methods of thyroidectomy-coventional thyroidectomy Vs intracapsular thyroidectomy. Materials and methods-55 cases of benign thyroid disease for whom thyroidectomy was performed in our hospital between the period of 2019-2022 were compared retrospectively. Out of these 34 cases had undergone intracapsular thyroidectomy and 21 cases underwent routine extracapsular thyroidectomy. The surgical outcomes including operation time, pain, postoperative infection, postoperative hypocalcemia, postoperative recurrent laryngeal nerve paralysis and mean hospital stay were analyzed. The mean operating time were very low in the intracapsular limb as compared to the other group. The pain and the mean hospital stay was also far lesser for the intracapsular limb. Both cohorts had no incidence of hypocalcemia. The incidence of recurrent laryngeal nerve palsy was very low in the intracapsular cohort (only 1 case of temporary unilateral RLN palsy), whereas it was higher in the routine extracapsular cohort (5 cases of permanent palsy). The risk of having vocal cord palsy (left/right) is 1.172 times more with conventional/standard thyroidectomy as compared to intracapsular thyroidectomy. Intracapsular technique is a much more rewarding method to perform thyroidectomy, without the risk of the recurrent laryngeal nerve palsy as compared to routine thyroidectomy.

Given a probability measure (mu ) on ({{mathbb {R}}}^n), Tukey’s half-space depth is defined for any (xin {{mathbb {R}}}^n) by (varphi _{mu }(x)=inf {mu (H):Hin {{{mathcal {H}}}}(x)}), where (mathcal{H}(x)) is the set of all half-spaces H of ({{mathbb {R}}}^n) containing x. We show that if (mu ) is a non-degenerate log-concave probability measure on ({{mathbb {R}}}^n) then

where (L_{mu }) is the isotropic constant of (mu ) and (c_1,c_2>0) are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of (L_q)-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.