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Additional file 3: Figure S2. Poteotranscriptomics identify enrichment terms in OABLs. (A) The top 15 CO-UP and CO-DOWN gene sets identified by GSEA analyses are represented in a heatmap of median GSVA score across subgroups. (B) Bar plot of enrichment terms identified by no-coherent dysregulated protein-mRNA pairs. (C) Bubble plot of top 20 discordantly dysregulated gene sets identified by GSEA.

We consider a class of stochastic heat equations driven by truncated $α$-stable white noises for $1

We study the ergodic property of a continuous-state branching process with immigration and competition, which is an extension of the models studied by Lambert (2005, Ann. Appl. Probab.), Pardoux (2016, Springer) and Berestycki et al. (2018, Probab. Theory Related Fields) with an additional immigration structure. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to branching mechanisms in the full range of criticality, including all stable types. The proof is based on a Markov coupling process and a nonsymmetric control function for the distance, which are designed to identify and to take the advantage of the dominating factor from the branching, immigration and competition mechanisms in different parts of the state space. The approach provides a way of finding explicitly the ergodicity rate.

For a class of non-linear stochastic heat equations driven by $α$-stable white noises for $α\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued càdlàg solutions to such a equation for $p\in(α,2]$ by considering a sequence of approximating stochastic heat equations driven by truncated $α$-stable white noises obtained by removing the big jumps from the original $α$-stable white noises. If the $α$-stable white noise is spectrally one-sided, under additional monotonicity assumption on noise coefficients, we prove a comparison theorem on the $L^2$-valued càdlàg solutions of such a equation. As a consequence, the non-negativity of the $L^2$-valued càdlàg solution is established for the above stochastic heat equation with non-negative initial function.

Additional file 6: Figure S5. Splicing factors play an important role in NHL. (A) The mRNA splicing gene set is significantly higher in all subtypes of OABL compared with the control group. (B) Bar plot or enriched domain of dysregulated proteins. (C) Genomic event of top 10 AASE correlated Splicing factors and regulators in DLBCL cohorts.

Additional file 7: Figure S6. Characteristics of ADAR knockdown NHL cell lines. (A) Representative western blots of ADAR expression in SU-DHL-4 and Raji cells with ADAR knockdown (shADAR) from three independent experiments. (B) Complete picture of the western blot. (C) Relative cell viability of SU-DHL-4 and Raji cells with ADAR knockdown (shADAR) and control treated by 100 μM HA-110 HCL for 72 hours. The cell viability is normalized by the corresponding cell treated by DMSO.

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with L��vy mutation and resampling mechanism associated to a general $��$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(��* Z_t) \subseteq S(Z_t)$, where $��$ is the L��vy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process., 23 pages

Additional file 5: Figure S4. Construction of the inflammation-OABL signature. (A) Elbow plot of k-means clustering. (B) Bar plot of top enrichment terms of the inflammation-OABL signature.

Additional file 1. Supplementary methods.

For a class of $Λ$-Fleming-Viot processes with Brownian spatial motion in $\mathbb{R}^d$ whose associated $Λ$-coalescents come down from infinity, we obtain sharp global and local modulus of continuities for the ancestry processes recovered from the lookdown representations. As applications, we prove both global and local modulus of continuities for the $Λ$-Fleming-Viot support processes. In particular, if the $Λ$-coalescent is the Beta$(2-β,β)$ coalescent for $β\in(1,2]$ with $β=2$ corresponding to Kingman's coalescent, then for $h(t)=\sqrt{t\log (1/t)}$, the global modulus of continuity holds for the support process with modulus function $\sqrt{2β/(β-1)}h(t)$, and both the left and right local modulus of continuities hold for the support process with modulus function $\sqrt{2/(β-1)}h(t)$., 27 pages

Additional file 1: Supplemental materials. Figure S1. IL-32 expression levels in human ESCC tumor and peritumor tissues from GEO cohorts, (A) GSE23400, (B) GSE20347, and (C) GSE45670. Data are represented as mean ± SEM. Student’s t-test, *P < 0.05, ***P < 0.001. Figure S2. IL-32 mRNA expression was measured with qRT-PCR in ESCC cell lines and immortalized normal esophageal cell line Het-1A. Data are represented as means ± SEM. Figure S3. IL-32β is the main isoform in ESCC tissues and EC109 cell line. (A-B) IL-32 isoforms were confirmed by DNA sequencing in ESCC tissues and EC109 cell line. (C) Summary diagram described the primers distinguish β, γ and θ isoform of IL-32. (D-E) qRT-PCR assay proved IL-32β was the main isoform in ESCC tissues (n = 9) and EC109 cell line. Three independent experiments were carried out. Data are represented as means ± SEM. Figure S4. MTT assay was performed to measure the growth of EC109 shNC, EC109 shIL-32, KYSE150 vector, and KYSE150 IL-32β cell lines. The data are representative of three independent experiments and represented as means ± SEM. Figure S5. The IL-32 expression and migratory ability of ESCC cell lines. The expression level of IL-32 in ESCC cell lines were detected by ELISA (A) and Western blot (B) assays. (C) Transwell assay was performed to detect the migratory potential of ESCC cells, including KYSE150, KYSE450, EC109 and EC1 cell lines. Five fields were included per assay, and the experiment was repeated three times. Scale bar, 100 μm. Data are represented as mean ± SEM. Student’s t-test, ***P < 0.001. Figure S6. IL-32 and immunocytes in Pearson rank correlation analysis. RNA microarray data was used for analysis. The population of immunocytes were analyzed by xCELL. The expression values of IL-32 were normalized. Pearson’s rank correlation test, *P < 0.05. Figure S7. IHC images showed that IL-32 was highly expressed in the cytoplasm of ESCC cells and was significantly higher in the tumor nest compared with the non-cancerous tissue. Scale bar, 50 μm. Figure S8. The phenotype of induced MDMs. The subset of MDMs were identified by CD14 and CD11b. Figure S9. Malvern spray analyzer showed the size distribution of staining and unstaining EV derived from EC109. Figure S10. MDMs cocultured with EV (50 μg/mL) derived from EC109 shNC, EC109 shIL-32, KYSE150 vector and KYSE150 IL-32β for 72 hours. FACS analysis showed the proportion of M1 macrophage determined by CD14 and CD80. The data are representative of three independent experiments and represented as means ± SEM. Figure S11. Expression of IL-32,TGFβ and IL-10 in ESCC. The expression profile GEO ESCC dataset (GSE75241) was used for analysis (n = 15). Data are represented as means ± SEM. Student’s t-test, ***P < 0.001. Figure S12. MDMs were treated with recombination IL-32β and M-CSF at 50 ng/mL for 48 hours. qRT-PCR was performed to detect the expression of M2 macrophage-related genes (IL-10, Arg1) and M1 macrophage-related genes (IL-1β, iNOS). All data are representative of three independent experiments and represented as means ± SEM. Student’s t-test, *P < 0.05, **P < 0.01, ***P < 0.001. Figure S13. Human IL-32 promotes mouse bone marrow-derived macrophages polarized to M2 macrophage. (A) Representative flow plots showing M2 macrophages (CD45+F4/80+CD11b+CD206+). (B) BMDMs were induced by 20 ng/mL M-CSF (Peprotech, USA) for 7 days, and were treated with PBS, 50 ng/mL recombinant IL-32β or 50 μg/mL EV-IL-32 for 72 hours. The proportion of M2 macrophage was detected by flow cytometry, and the statistical (C) was shown. (D) Western blot assays for phosphorylated FAK and STAT3 expression using β-actin as control in mouse BMDMs treated with rhIL-32β (50 ng/mL) or EV-IL-32 (50 μg/mL) for 3 hours. (E) FAK inhibitor (Y15, 5 μM) decreased the phosphorylation of STAT3 in BMDMs treated rhIL-32β or EV-IL-32. The figures are representatives of three replicates. Scale bar, 100 μm. Student’s t-test, **P < 0.01, ***P < 0.001. Figure S14. Representative dot plots of intratumoral M2 macrophages were defined as CD45+F4/80+CD11b+cells and the proportion was showed in mouse model of KYSE150 vector, KYSE150 IL-32β (A), EC109 shNC and EC109 shIL-32 (B). Figure S15. EV-IL-32 educated M2 macrophage promotes metastasis of EC109 and KYSE150. (A) The coculture system proved that EV-IL-32 induced M2 macrophages promoted migration of ESCC cells in vitro. (B-C) EC109 and KYSE150 were seeded into the upper chamber with 1×105/well and EV-IL-32 educated M2 macrophages into the lower chamber with the same density. Five fields were included per assay, and the experiment was repeated three times. Scale bar, 100 μm. Data are represented as means ± SEM. Student’s t-test, *P < 0.05, ***P < 0.001. Figure S16. Depletion of macrophages attenuating EC109 cells lung metastasis in vivo. (A) The percentage of F4/80+CD11b+ macrophages from blood were analyzed the day after administration of Control liposome or Clodronate liposome using flow cytometry. (B) Lung metastasis was confirmed by H&E (n = 4). Scale bars, 400 or 200 μm. Data are represented as means ± SEM. Student’s t-test, *P < 0.05, **P < 0.01. Figure S17. The global transcriptional profile of EC109 shNC and EC109 shIL-32 was assessed by RNA-Seq. (A) The GO enrichment revealed that IL-32 knockdown in EC109 mainly affected vascular system development and extracellular matrix binding. (B) The data from RNA-seq was used to analyze the up-regulation and down-regulation genes in extracellular matrix binding. (C-D) qRT-RCR was performed to determine the fold change of genes about extracellular matrix binding in EC109 shNC, EC109 shIL-32, KYSE150 vector and KYSE150 IL-32β. Data are represented as means ± SEM. Student’s t-test, *P < 0.05, **P< 0.01, ***P < 0.001. Table S1. Primers for qRT-PCR to distinguish each isoform of IL-32. Table S2. Primers for qRT-PCR to amplify genes. Table S3. Primary antibodies for FACS. Table S4. Primary antibodies for western blot.

Additional file 4: Figure S3. Association between the global protein-mRNA concordance and disease characteristics. (A) Global protein-mRNA concordance is relatively associated with the high Ann Arbor Stage. (B) Kaplan-Meier survival analyses of progression-free survival, overall survival, recurrence-free survival, and local recurrence-free survival. (C) Boxplots show the association between global concordance and different recurrence patterns in OABL, SBL, EMZL, DLBCL, and other subtypes.

Additional file 2: Figure S1. Unsupervised hierarchical clustering heatmap of proteomic and transcriptome results. Highly variant transcripts/proteins (median absolute deviation top 1000) are included in the analyses.

In this work, the BMS-12 pre-sintered material was used as the basic material, and the optimal secondary additive addition amount was found by the wet pressing process. Then, the anisotropic ferrite material was prepared by dry-pressing method, and the effects of camphor adhesives, calcium stearate lubricant and powder particle size on the magnetic properties were studied. The experimental results show that the appropriate addition amount of camphor adhesive and calcium stearate lubricant is 0.6 wt% and 0.8 wt%, respectively. High remanence and coercivity can be obtained for particle size of 0.85–1.00 μm. Superior dry-pressed magnet ferrite was prepared with typical properties of magnetic induction intensity of Br = 421 mT, induction coercivity of Hcb = 296 kA·m−1, intrinsic coercivity of Hcj = 369 kA·m−1 and maximum energy product of (BH)max = 33.3 kJ·m−3, which have reached the level of TDK corporation product of FB5D. This shows that the dry-pressed magnet of La–Ca–Sr–Co system has higher remanence and intrinsic coercivity than the traditional wet-pressed Sr-ferrite.

In this study of resistive random-access memory using CH3NH3PbI3 thin film as the switching layer, the perovskite films are prepared on FTO substrates by a two-step solution spin coating method in air. We study the effect of Zr-doped CH3NH3PbI3 film on the performance of devices. The devices with Zr doping show better bipolar resistive switching behavior including larger storage window, better endurance, better repeatability and stability. In addition, the switching performance is improved. Moreover, the conductive mechanism of the devices is analyzed and the data fitting results indicate that the conduction mechanism transforms from space-charge limited conduction to Schottky emission at high resistance state after Zr doping. This study provides an excellent route to optimize resistive switching-based memory devices through chemical modifications and suggests that the perovskite RRAM with Zr-doping has a great potential in high performance non-volatile memory devices.

Antitumor transcription activator NFE2L1, with the functions to regulate redox homeostasis, protein turnover, and material metabolism, plays an important role in embryonic development and specialization of tissue and organ functions. Deficiency of NFE2L1 gene in different regions yields distinct phenotypes, suggesting that NFE2L1 may have a transcription factor-independent function. Here we originally discovered the non-transcription factor activity of NFE2L1 by constructing a truncated protein-NFE2L1ΔC without 152 aa at the C-terminus which lost the transcription factor activity. The regulation of NFE2L1 on redox homeostasis, proteasome function, and immune response mainly depends on its transcription activator function in nucleus, while the regulation on metabolism, ribosome function, and canceration is germanely to its non-transcription factor activity in cytoplasm. Surprisingly, the results indicated the tumor suppressive effect of NFE2L1 by repression of Wnt/β-catenin signaling in a non-transcription factor manner, indicating the potential value of NFE2L1 as a therapeutic target in clinical cancer treatment independent of its transcription factor activity. Our observations reveal the non-transcription factor activity of NFE2L1 for the first time, and lay foundation for the basic and applied research of NFE2L1.

The customer's review plays an important role in deciding the purchase behaviour for online shopping. This paper intends to extract useful information hidden in customer's reviews using the opinion mining technology. In the reviews, the polarity of the evaluation words reflects the consumer's attitude towards the product, so the discrimination of the polarity of the words plays a crucial role in the sentiment analysis. In this paper, we first make a polarity discrimination based on the sentiment lexicons. For the evaluation words that do not exist in the sentiment lexicon, the similarity between evaluation words and emotional seed words is calculated using the linear superposition of Word2Vec and HowNet (LSWH), and then the polarity of the evaluation word is determined. To evaluate the performance of the proposed method, we comparing with the previous two polarity discriminate methods, the result show that the proposed method has the high accuracy.

This paper discusses Parisian ruin problem with capital injection for Levy insurance risk process. Capital injection takes place at the draw-down time of the surplus process when it drops below a pre-specified function of its last record maximum. The capital is continuously paid to keep the surplus above the draw-down level until either the surplus process goes above the record high or a Parisian type ruin occurs, which is announced at the first instance the surplus process has undergone an excursion below the record for an independent exponential period of time consecutively since the time the capital was first injected. Some distributional identities concerning the excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level of the surplus process) of the ruin time, surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the $q$-potential measure of the surplus process killed at Parisian ruin. Finally, we give expected present value of the total discounted capital payments up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Levy process and are presented semi explicitly in terms of the scale function of the Levy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation period to the ruin probability and to the expected total capital injection., Comment: 16 figures

We construct extensions of the pure-jump $��$-Wright-Fisher processes with frequency-dependent selection ($��$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some duality relationships with the block counting process of simple exchangeable fragmentation-coalescence processes (EFC). One-to-one correspondences between the nature of the boundary $1$ of the $��$-WF process with selection and the boundary $\infty$ of the block counting process are established. New properties for the $��$-WF processes with selection and the block counting processes of the simple EFC processes are deduced from these correspondences. Some conditions are provided for the selection to be either weak enough for boundary $1$ to be an exit boundary or strong enough for $1$ to be an entrance boundary. When the measure $��$ and the selection mechanism satisfy some regular variation properties, conditions are found in order that the extended $��$-WF process with selection makes excursions out from the boundary $1$ before getting absorbed at $0$. In the latter process, $1$ is a transient regular reflecting boundary. This corresponds to a new phenomenon for the deleterious allele which can spread into the population in a set of times of zero Lebesgue measure, before vanishing in finite time almost surely.

A generalized stepping stone model with $��$-resampling mechanism is a two dimensional probability-measure-valued stochastic process whose moment dual is similar to that of the classical stepping stone model except that Kingman's coalescent is replaced by $��$-coalescent. We prove the existence of such a process by specifying its moments using the dual function-valued $��$-coalescent process with geographical labels and migration, and then verifying a multidimensional Hausdorff moment problem. We also characterize the stationary distribution of the generalized stepping stone model and show that it is not reversible if the mutation operator is of uniform jump-type., 19 pages

Consider a spectrally positive L��vy process $Z$ with log-Laplace exponent $��$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $��(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the L��vy process has a negative drift $��:=-��0$ and $R$ is regularly varying at $\infty$ with index $��>��$, the process comes down from infinity and we find a renormalisation in law of its running infimum at small times.

In this paper we study the draw-down related Parisian ruin problem for spectrally negative L��vy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative L��vy risk process with dividend barrier and Parisian ruin.

In this paper we study a spectrally negative L��vy process that is reflected at its draw-down level whenever a draw-down time from the running supremum arrives. Using an excursion-theoretical approach, for such a reflected process we find the Laplace transform of the upper exiting time and an expression of the associated potential measure. When the reflected process is identified as a risk process with capital injections, the expected total amount of discounted capital injections prior to the exiting time and the Laplace transform of the accumulated capital injections until the exiting time are also obtained. The results are expressed in terms of scale functions for spectrally negative L��vy processes.

For spectrally negative L��vy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative L��vy processes.

For spectrally negative L��vy processes, adapting an approach from \cite{BoLi:sub1} we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the permanental process and the Markovian loop soup measure., 23 pages

For refracted spectrally negative L��vy processes, we identify expressions of several quantities related to Laplace transforms on their weighted occupation times until first exit times. Such quantities are expressed in terms of unique solutions to integral equations involving weight functions and scale functions for the associated spectrally negative L��vy processes. Previous results on refracted L��vy processes are recovered.

In this paper we consider a general continuous-state nonlinear branching process which can be identified as a nonnegative solution to a nonlinear version of the stochastic differential equation driven by Brownian motion and Poisson random measure. Intuitively, this process is a branching process with population-size-dependent branching rates and with competition. We construct a sequence of discrete-state nonlinear branching processes and prove that it converges weakly to the continuous-state nonlinear branching process by using tightness arguments and convergence criteria on infinite-dimensional space.

Using a new approach, for spectrally negative L��vy processes we find joint Laplace transforms involving the last exit time (from a semi-infinite interval), the value of the process at the last exit time and the associated occupation time, which generalizes some previous results., 16pages

For a spectrally one-sided L��vy process, we extend various two-sided exit identities to the situation when the process is only observed at arrival epochs of an independent Poisson process. In addition, we consider exit problems of this type for processes reflected either from above or from below. The resulting Laplace transforms of the main quantities of interest are in terms of scale functions and turn out to be simple analogues of the classical formulas., Published at http://dx.doi.org/10.3150/15-BEJ695 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

The original and postnatal scale patterns of guard hairs were compared. The comparisons illustrate that acquired morphological changes take place in the scales on the coarse section of guard hairs. Scales on this part changes from regular smooth to irregular wave. The primary reason would be friction. Scales on the lower part of guard hairs are thick and strong to bear friction. Additionally, they are berried in the bottom layer of pelage where friction is avoided. Scales on the coarse section are thin, broad, dense and overlap, and exposed in the environment as a cover of pelage. So friction always happen on them. Factors which enhance coefficient of friction and weaken keratin bonds are combined to damage hair scales. The results suggest that regular smooth and irregular wave are actually the same type exhibiting the same origin morphological characters, so they should be counted together in the species identification.

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H��lder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ��X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $��$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $��$-H��lder continuous for $10$.

Using the lookdown construction of Donnelly and Kurtz we prove that, at any fixed positive time, the $��$-Fleming-Viot process with underlying Brownian motion has a compact support provided that the corresponding $��$-coalescent comes down from infinity not too slowly. We also find both upper bound and lower bound on the Hausdorff dimension for the support., 21 pages

The (\Xi, A)-Fleming-Viot process with mutation is a probability-measure-valued process whose moment dual is similar to that of the classical Fleming-Viot process except that the Kingman's coalescent is replaced by the \Xi-coalescent, the coalescent with simultaneous multiple collisions. We first prove the existence of such a process for general mutation generator A. We then investigate its reversibility. We also study both the weak and strong uniqueness of solution to the associated stochastic partial differential equation.

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative L��vy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented., 27 pages

In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative L��vy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative L��vy process and its Laplace exponent. Applications to insurance risk models are also presented., corrections in the proof of Theorem 1

Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial., Comment: 19 pages

The hemispherical emittance of the selective coating in all-glass evacuated tubes should be no larger than 0.080 at 80°C± 5°C in “All-glass evacuated collector tubes” of China national standard. A novel equipment to determine the hemispherical emittance of the selective coating in the normal evacuated tubes was designed and set up. Some of hemispherical emittances of Al-N/Al absorbing coatings in all-glass evacuated tubes were measured.

The hemispherical emittance of the selective absorbing coating on the outside of the inner glass tube of an all-glass evacuated collector tube has been determined, using calorimetry at steady state in the temperature range 50–300°C and gas pressure range 1.0×10−3–1.6 Pa at the jacket between cover and inner glass tubes. Calculated gas heat flux qc and equivalent emittance eg based on the theory of gas conduction at medium and low pressures have been determined. The calculations agree well with experiments. The experimental results indicate that heat losses of all-glass evacuated collector tubes due to gas convection and conduction are negligible when the gas pressure in the tube is less than 5×10−2 Pa.

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration., Comment: 13 pages