×

zbMATH — the first resource for mathematics

Stochastic evolution equations and related measure processes. (English) Zbl 0299.60050

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H20 Stochastic integral equations
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bailey, N.T.J, ()
[2] Bharucha-Reid, A.T, ()
[3] Blumenthal, R.M; Getoor, R.K, ()
[4] Breiman, L, ()
[5] Chung, K.L, ()
[6] Doob, J.L, ()
[7] Dynkin, E.B, ()
[8] Feller, W, ()
[9] Itô, K, ()
[10] Itô, K; McKean, H.P, ()
[11] Loève, M, ()
[12] Meyer, P.A, ()
[13] Meyer, P.A, ()
[14] Dawson, D.A, Generalized stochastic integrals and equations, Trans. amer. math. soc., 147, 473-506, (1970) · Zbl 0199.52402
[15] Gihman, I.I; Skorohod, A.V, ()
[16] Itô, K, On a formula concerning stochastic differentials, Nagoya math. J., 3, 55-65, (1951) · Zbl 0045.07603
[17] Itô, K, On stochastic differential equations, Mem. amer. math. soc., 4, (1961)
[18] Lipcer, R.S; Sirjaev, A.N, On the absolute continuity of measures corresponding to processes of diffusion type relative to a Wiener measure, Math. U.S.S.R. izvestija, 6, 839-882, (1972)
[19] McKean, H.P, ()
[20] Skorohod, A.V, ()
[21] Stroock, D.W; Varadhan, S.R.S; Stroock, D.W; Varadhan, S.R.S, Diffusion processes with continuous coefficients II, Comm. pure and appl. math., Comm. pure and appl. math., 22, 479-530, (1969) · Zbl 0175.44802
[22] Baklan, V.V, On the existence of solutions of stochastic equations in Hilbert space, Dopevidi akad. nauk. ukrain. RSR, 1299-1303, (1963) · Zbl 0129.10605
[23] Baklan, V.V, Equations in variational derivatives and Markov processes in Hilbert spaces, Dokl. akad. nauk. SSSR, 159, 707-710, (1964) · Zbl 0295.60040
[24] Bensoussan, A; Temam, R, Équations aux dérivées partielles stochastiques non linéaires (1), Israel J. math., 11, 95-129, (1972) · Zbl 0241.35009
[25] Bensoussan, A; Temam, R, Équations stochastiques du type Navier Stokes, J. functional analysis, 13, 195-222, (1973) · Zbl 0265.60094
[26] Bharucha-Reid, A.T, ()
[27] {\scBose}, A. and {\scDawson}, D. A. Measure Diffusion Processes, in preparation.
[28] Cabaña, E, Stochastic integration in separable Hilbert spaces, Publ. inst. de math. y estadistica, Uruguay, 4, 49-79, (1966) · Zbl 0154.18702
[29] Cabaña, E, On stochastic differential in Hilbert space, (), 259-265 · Zbl 0167.16903
[30] Cabaña, E, The vibrating string forced by white noise, Z. wahr., 15, 111-130, (1970) · Zbl 0193.45101
[31] Curtain, R.F; Falb, P.L, Itô’s lemma in infinite dimensions, J. math. anal. appl., 31, 434-448, (1970) · Zbl 0233.60051
[32] Curtain, R.F; Falb, P.L, Stochastic differential equations in Hilbert space, J. differential equations, 10, 412-430, (1971) · Zbl 0225.60028
[33] Daletskii, Yu.L, Infinite dimensional elliptic operators and parabolic equations connected with them, Uspekhi mat. nauk., 22, 1-53, (1967), Translation, Russ. Math. Surveys · Zbl 0164.41304
[34] Daletskii, Yu.L, ()
[35] Daletskii, Yu.L; Teterina, N.I, Multiplicative stochastic integrals, Uspekhi math. nauk., 27, 167-168, (1972)
[36] Daletskii, Yu.L; Fomin, S.V, Generalized measures in Hilbert space and Kolmogorov’s forward equation, Sov. math. dokl., 13, 993-997, (1972) · Zbl 0266.46037
[37] Daletskii, Yu.L; Paramova, S.N, Stochastic integrals with respect to a normally distributed additive set function, Sov. math., 14, 96-99, (1973)
[38] Dawson, D.A, Stochastic evolution equations, Math. biosciences, 15, 287-316, (1972) · Zbl 0251.60040
[39] Kannan, D; Bharucha-Reid, A.T, An operator-valued stochastic integral, (), 472-476 · Zbl 0256.60029
[40] Kannan, D, An operator-valued stochastic integral (II), Ann. inst. poincare, sect. B, 8, 9-32, (1972) · Zbl 0256.60030
[41] Kunita, H, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya math. J., 38, 41-52, (1970) · Zbl 0234.60071
[42] Kuo, H.H, Stochastic integrals in abstract Wiener space, Pac. J. math., 41, 469-483, (1972) · Zbl 0226.60076
[43] Kuo, H.H, An operator-valued stochastic integral, Bull. amer. math. soc., 79, 478-482, (1973)
[44] Kuo, H.H; Piech, M.A, Stochastic integrals and parabolic equations in abstract Wiener space, Bull. amer. math. soc., 79, 478-482, (1973) · Zbl 0256.60028
[45] Kuo, H.H, Stochastic integrals in abstract Wiener space (II); regularity properties, Nagoya math. J., 50, 89-116, (1973) · Zbl 0317.60026
[46] Kuo, H.H, Absolute continuity of measures corresponding to diffusion processes in Banach space, Ann. of prob., 1, 513-518, (1973) · Zbl 0288.28012
[47] Kuo, H.H, Differential and stochastic equations in abstract Wiener space, J. functional analysis, 12, 246-256, (1973) · Zbl 0243.35017
[48] Mandrekar, V; Salehi, H, Operator-valued wide sense Markov processes and solution of infinite dimensional linear differential systems driven by white noise, Math. sys. theor., 4, 340-356, (1970) · Zbl 0208.43704
[49] Mortensen, R.E, Existence and uniqueness of measure-valued solutions of a stochastic integral equation, (), 441-449 · Zbl 0214.40501
[50] Pardoux, E, Équations aux dérivées partielles stochastiques monotones, C. R. acad. sci., Paris, ser. A, 275, 101-103, (1972) · Zbl 0236.60039
[51] Pardoux, E, Thesis, (1974)
[52] Piech, M.A, Some regularity properties of diffusion processes on abstract Wiener space, J. functional analysis, 8, 153-172, (1971) · Zbl 0215.49102
[53] Piech, M.A, Diffusion semigroups on abstract Wiener space, Trans. amer. math. soc., 166, 411-430, (1972) · Zbl 0247.47034
[54] {\scViot, M.} (to appear). Solution en loi d’une équation aux dérivées partielles stochastique non linéaire: Méthode de compacité. C. R. Acad. Sci., Paris.
[55] {\scViot, M.} (to appear). Solution en loi d’une équation aux dérivées partielles stochastique non linéaire: méthode de monotonie. C. R. Acad. Sci., Paris.
[56] Yor, M, Existence et unicité de diffusions à valeurs dans un espace de Hilbert, Ann. inst. Poincaré, 10, 55-88, (1974) · Zbl 0281.60094
[57] Yor, M, Sur LES intégrales stochastiques à valeurs dans un espace de Banach, Ann. inst. Poincaré, 10, 31-36, (1974) · Zbl 0295.60041
[58] Bochner, I.S, ()
[59] Choquet, G, ()
[60] Conley, C.C; Smoller, J.A, Topological methods in the theory of shock waves, () · Zbl 0368.35040
[61] Dudley, R.M, Random linear functionals, Trans. amer. math. soc., 136, 1-24, (1969) · Zbl 0186.21301
[62] Fernique, X, Processes linéaires, processus généralisés, Ann. inst. Fourier, 17, 1-92, (1967) · Zbl 0167.16702
[63] Gel’fand, I.M; Vilenkin, N.Ja, ()
[64] Gross, L, Abstract Wiener space, (), 31-42
[65] Hida, T, (), and Univ. of Tokyo Press
[66] Kahane, J.P, Suite de Fourier aléatoire, Univ. de montréal, montréal, Canada, (1966)
[67] Kahane, J.P, ()
[68] Kato, T, Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya math. J., 19, 93-125, (1961) · Zbl 0114.06102
[69] Kato, T; Tanabe, H, On the abstract evolution equation, Osaka math. J., 14, 107-133, (1962) · Zbl 0106.09302
[70] Kato, T, ()
[71] Kurtz, T.G, Extensions of Trotter’s operator semigroup approximation theorems, J. functional analysis, 3, 354-375, (1969) · Zbl 0174.18401
[72] Kurtz, T.G, Semigroups of conditioned shifts and approximation of Markov processes, ms., (1974), Univ. of Wisconsin
[73] Lions, J.L, ()
[74] Lions, J.L, LES semi groupes distributions, Portugal math., 19, 141-165, (1960) · Zbl 0103.09001
[75] Marsden, J.E, On product formulas for nonlinear semigroups, J. functional analysis, 13, 51-74, (1973) · Zbl 0258.47042
[76] Mourier, E, Random elements in linear space, (), 31-42 · Zbl 0187.41001
[77] Newell, G.F, Asymptotic extreme value distribution for one dimensional diffusion processes, J. math. mech., 11, 481-496, (1963) · Zbl 0115.13602
[78] Prohorov, Yu.V, Random measures on a compactum, Sov. math. dokl., 2, 539-541, (1961) · Zbl 0111.32202
[79] Rao, M.M, Local fields and generalized random fields, Bull. amer. math soc., 74, 288-293, (1968) · Zbl 0204.50208
[80] Rao, M.M, Local functionals and generalized random fields with independent values, Th. prob. appl., 16, 466-483, (1971) · Zbl 0248.60037
[81] Riesz, F; Nagy, B.Sz, ()
[82] Sato, H, Gaussian measure on a Banach space and abstract Wiener measure, Nagoya math. J., 36, 65-81, (1969) · Zbl 0185.44303
[83] Segal, I.E, Non-linear semi-groups, Ann. of math., 78, 339-364, (1963) · Zbl 0204.16004
[84] Schwartz, L, ()
[85] {\scSchwartz, L.} (to appear). Randon measures on arbitrary topological spaces. Tata Inst. Fund. Res., Bombay, India.
[86] Schwartz, J.T, ()
[87] Trotter, H, On the product of semi-groups of operators, (), 545-551 · Zbl 0099.10401
[88] Zygmund, A, ()
[89] Bailey, N.T.J, Stochastic birth, death and migration processes for spatially distributed populations, Biometrika, 55, 189-198, (1963) · Zbl 0164.51002
[90] Bartlett, M.S, Deterministic and stochastic models for recurrent epidemics, (), 81-109 · Zbl 0070.15004
[91] Bensoussan, A, ()
[92] Daley, D.J; Verre-Jones, D, A summary of the theory of point processes, () · Zbl 0278.60035
[93] Davis, A.W, On the theory of birth, death and diffusion processes, J. appl. prob., 2, 293-322, (1965) · Zbl 0161.15101
[94] Davis, A.W, Some generalizations of Bailey’s birth, death and migration model, Adv. appl. prob., 2, 83-109, (1970) · Zbl 0216.47201
[95] Falb, P.L, Infinite dimensional filtering: the Kalman-bucy filter in Hilbert space, Inf. and control, 11, 102-137, (1967) · Zbl 0178.18902
[96] Feller, W, Diffusion processes in genetics, (), 227-246
[97] Fleming, W.H, Distributed parameter stochastic systems in population biology, m.s., (1974), Brown University Providence, R.I
[98] Frisch, U, Wave propagation in random media, () · Zbl 0727.76064
[99] Harris, T.E, Random measures and motions of point processes, Z. wahr., 18, 85-115, (1971) · Zbl 0194.49204
[100] Harris, T.E, Counting measures, monotone random set functions, Z. wahr., 10, 102-119, (1968) · Zbl 0165.18902
[101] Hinton, F.L, Nonequilibrium theory of fluid fluctuations, Phys. of fluids, 13, 857-866, (1970) · Zbl 0212.30504
[102] Ikeda, N; Nagasawa, M; Watanabe, S; Ikeda, N; Nagasawa, M; Watanabe, S, Branching Markov processes I, II, III, J. math. Kyoto univ., J. math. Kyoto univ., 9, 95-160, (1969) · Zbl 0233.60070
[103] Jagers, P, Aspects of random measures and point processes, () · Zbl 0333.60059
[104] Jirina, M, Branching processes with measure-valued states, (), 333-357
[105] Kallenberg, O, Characterization and convergence of random measures and point processes, Z. wahr., 27, 9-21, (1973) · Zbl 0253.60037
[106] Kingman, J.F.C, Completely random measures, Pacific J. math., 21, 59-78, (1967) · Zbl 0155.23503
[107] Kushner, H.J, On the optimal control of a system governed by a linear parabolic equation with white noise inputs, S.I.A.M. J. control, 6, 596-614, (1968) · Zbl 0186.23404
[108] Kushner, H.J, ()
[109] Lesieur, M; Frisch, U; Brissaud, A, A Markovian random coupling model for turbulence, J. fluid mech., 65, 145-152, (1974) · Zbl 0285.76021
[110] Malécot, G, LES mathématiques de l’hérédité, (1948), Masson et Cie Paris · Zbl 0031.17304
[111] {\scMartin-Lof, A. and Spitzer, F.} (1973). Private communication.
[112] Montroll, E.W; West, B.J, Models of population growth, competition and rearrangement, () · Zbl 0276.92022
[113] Moyal, J.E, The general theory of stochastic population processes, Acta math., Stockholm, 108, 1-31, (1962) · Zbl 0128.40302
[114] Moyal, J.E, Multiplicative population processes, J. appl. prob., 1, 267-283, (1964) · Zbl 0203.17303
[115] Nelson, E, Quantum fields and Markov fields, (), 413-421
[116] Newman, C.M, The construction of stationary two dimensional markoff fields with an application to quantum field theory, J. functional analysis, 13, 44-62, (1973) · Zbl 0268.60045
[117] Prigogine, I; Nicolis, G; Babloyantz, A, Thermodynamics of evolution, (), 23-28
[118] Segel, L.A, On the collective motions of chemotactic cells, () · Zbl 0226.76034
[119] Sevast’yanov, B.A, Branching stochastic processes for particles diffusing in a restricted domain with absorbing boundaries, Theor. prob. appl., 3, 111-126, (1958) · Zbl 0198.22403
[120] Skellam, J.G, The formulation and interpretation of mathematical models of diffusing processes in population biology, () · Zbl 0083.15602
[121] Skorohod, A.V, Branching diffusion processes, Th. prob. appl., 9, 492-497, (1967)
[122] Szafirski, B, Characteristic functionals and turbulent diffusion, Bull. acad. Pol. ser. math., 19, 785-789, (1971) · Zbl 0297.76049
[123] Szafirski, B, Diffusion by turbulent movements, Bull. acad. Pol. ser. math., 19, 791-797, (1971) · Zbl 0229.76049
[124] Uhlenbeck, G.E; Ornstein, L.S, On the theory of Brownian motion I, Phys., 38, 823-841, (1930) · JFM 56.1277.03
[125] Uhlenbeck, G.E; Wang, M.C, On the theory of Brownian motion II, Rev. mod. phys., 17, 323-342, (1945) · Zbl 0063.08172
[126] van Vliet, K.M, Markov approach to density fluctuations and scattering I, mathematical formalism; II application, J. math. phys., 12, 1998-2012, (1971) · Zbl 0239.60106
[127] Watanabe, S, On the branching process for Brownian particles with an absorbing boundary, J. math. Kyoto univ., 4, 385-398, (1965) · Zbl 0134.34401
[128] Whittle, P, Stochastic processes in several dimensions, Bull. inst. int. statist., 40, 974-994, (1963) · Zbl 0129.10603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.