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1. Exact sampling of the infinite horizon maximum of a random walk over a nonlinear boundary

   https://doi.org/10.1017/jpr.2019.9  

   Blanchet, Jose; Dong, Jing; Liu, Zhipeng (March 2019, Journal of Applied Probability)

   Abstract We present the first algorithm that samples max n ≥0 { S n − n α }, where S n is a mean zero random walk, and n α with $$\alpha \in ({1 \over 2},1)$$ defines a nonlinear boundary. We show that our algorithm has finite expected running time. We also apply this algorithm to construct the first exact simulation method for the steady-state departure process of a GI/GI/∞ queue where the service time distribution has infinite mean. 

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2. On the three ball theorem for solutions of the Helmholtz equation

   https://doi.org/10.1007/s40627-021-00070-3  

   Berge, Stine Marie; Malinnikova, Eugenia (June 2021, Complex Analysis and its Synergies)

   null (Ed.)

   Abstract Let $$u_{k}$$ u k be a solution of the Helmholtz equation with the wave number k , $$\varDelta u_{k}+k^{2} u_{k}=0$$ Δ u k + k 2 u k = 0 , on (a small ball in) either $${\mathbb {R}}^{n}$$ R n , $${\mathbb {S}}^{n}$$ S n , or $${\mathbb {H}}^{n}$$ H n . For a fixed point p , we define $$M_{u_{k}}(r)=\max _{d(x,p)\le r}|u_{k}(x)|.$$ M u k ( r ) = max d ( x , p ) ≤ r | u k ( x ) | . The following three ball inequality $$M_{u_{k}}(2r)\le C(k,r,\alpha )M_{u_{k}}(r)^{\alpha }M_{u_{k}}(4r)^{1-\alpha }$$ M u k ( 2 r ) ≤ C ( k , r , α ) M u k ( r ) α M u k ( 4 r ) 1 - α is well known, it holds for some $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and $$C(k,r,\alpha )>0$$ C ( k , r , α ) > 0 independent of $$u_{k}$$ u k . We show that the constant $$C(k,r,\alpha )$$ C ( k , r , α ) grows exponentially in k (when r is fixed and small). We also compare our result with the increased stability for solutions of the Cauchy problem for the Helmholtz equation on Riemannian manifolds. 

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3. Random Interpolating Sequences in Dirichlet Spaces

   https://doi.org/10.1093/imrn/rnab110  

   Chalmoukis, Nikolaos; Hartmann, Andreas; Kellay, Karim; Wick, Brett Duane (May 2021, International Mathematics Research Notices)

   Abstract We discuss random interpolating sequences in weighted Dirichlet spaces $${{\mathcal{D}}}_\alpha $$, $$0\leq \alpha \leq 1$$, when the radii of the sequence points are fixed a priori and the arguments are uniformly distributed. Although conditions for deterministic interpolation in these spaces depend on capacities, which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $$\alpha =1/2$$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $${{\mathcal{D}}}_\alpha $$ are exactly the almost sure separated sequences when $$0\le \alpha <1/2$$ (which includes the Hardy space $$H^2={{\mathcal{D}}}_0$$), and they are exactly the almost sure zero sequences for $${{\mathcal{D}}}_\alpha $$ when $$1/2 \leq \alpha \le 1$$ (which includes the classical Dirichlet space $${{\mathcal{D}}}={{\mathcal{D}}}_1$$). 

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4. Asymptotics of the eigenmodes and stability of an elastic structure with general feedback matrix

   https://doi.org/10.1093/imamat/hxz019  

   Shubov, Marianna A; Kindrat, Laszlo P (October 2019, IMA Journal of Applied Mathematics)

   Abstract The distribution of natural frequencies of the Euler–Bernoulli beam subject to fully non-dissipative boundary conditions is investigated. The beam is clamped at the left end and equipped with a 4-parameter ($$\alpha ,\beta ,k_1,k_2$$) linear boundary feedback law at the right end. The $$2 \times 2$$ boundary feedback matrix relates the control input (a vector of velocity and its spatial derivative at the right end), to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space equipped with the energy norm. The dynamics generator has a purely discrete spectrum (the vibrational modes) denoted by $$\{\nu _n\}_{n\in \mathbb {Z}^{\prime}}$$. The role of the control parameters is examined and the following results have been proven: (i) when $$\beta \neq 0$$, the set of vibrational modes is asymptotically close to the vertical line on the complex $$\nu$$-plane given by the equation $$\Re \nu = \alpha + (1-k_1k_2)/\beta$$; (ii) when $$\beta = 0$$ and the parameter $$K = (1-k_1 k_2)/(k_1+k_2)$$ is such that $$\left |K\right |\neq 1$$ then the following relations are valid: $$\Re (\nu _n/n) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iii) when $$\beta =0$$, $|K| = 1$, and $$\alpha = 0$$, then the following relations are valid: $$\Re (\nu _n/n^2) = O\left (1\right )$$ and $$\Im (\nu _n/n) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$; (iv) when $$\beta =0$$, $|K| = 1$, and $$\alpha>0$$, then the following relations are valid: $$\Re (\nu _n/\ln \left |n\right |) = O\left (1\right )$$ and $$\Im (\nu _n/n^2) = O\left (1\right )$$ as $$\left |n\right |\to \infty$$. 

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5. Heterogeneous diffusion in an harmonic potential: the role of the interpretation

   https://doi.org/10.1088/1367-2630/addf72  

   Pacheco-Pozo, Adrian; Sokolov, Igor_M; Metzler, Ralf; Krapf, Diego (June 2025, New Journal of Physics)

   Abstract Diffusion in heterogeneous energy and diffusivity landscapes is widespread in biological systems. However, solving the Langevin equation in such environments introduces ambiguity due to the interpretation parameter $$\alpha$$, which depends on the underlying physics and can take values in the range $$0<\alpha<1$$. The typical interpretations are It\^o ($$\alpha=0$$), Stratonovich ($$\alpha=1/2$$), and Hanggi-Klimontovich ($$\alpha=1$$). Here, we analyse the motion of a particle in an harmonic potential---modelled as an Ornstein-Uhlenbeck process---with diffusivity that varies in space. Our focus is on two-phase systems with a discontinuity in environmental properties at $x=0$. We derive the probability density of the particle position for the process, and consider two paradigmatic situations. In the first one, the damping coefficient remains constant, and fluctuation-dissipation relations are not satisfied. In the second one, these relations are enforced, leading to a position-dependent damping coefficient. In both cases, we provide solutions as a function of the interpretation parameter $$\alpha$$, with particular attention to the It\^o, Stratonovich, and Hanggi-Klimontovich interpretations, revealing fundamentally different behaviours, in particular with respect to an interface located at the potential minimum. 

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