Introduction .................................................... 7
Chapter 1. Deterministic Volterra equations .................... 13
1.1. Notations and preliminaries ......................... 13
1.2. Resolvents and well-posedness ....................... 14
1.3. Kernel functions .................................... 16
1.4. Parabolic equations and regular kernels ............. 17
1.5. Approximation theorems .............................. 20
Chapter 2. Probabilistic background ............................ 25
2.1. Notations and conventions ........................... 25
2.2. Classical infinite dimensional Wiener process ....... 26
2.3. Stochastic integral with respect to cylindrical
Wiener process ...................................... 29
2.3.1. Properties of the stochastic integral ........ 35
2.4. The stochastic Fubini theorem and the Itô formula ... 35
Chapter 3. Stochastic Volterra equations in Hilbert space ...... 37
3.1. Notions of solutions to stochastic Volterra
equations ........................................... 38
3.1.1. Introductory results ......................... 39
3.1.2. Results in general case ...................... 42
3.2. Existence of strong solution ........................ 47
3.3. Fractional Volterra equations ....................... 52
3.3.1. Convergence of α-times resolvent families .... 53
3.3.2. Strong solution .............................. 57
3.4. Examples ............................................ 62
Chapter 4. Stochastic Volterra equations in spaces of
distributions ....................................... 65
4.1. Generalized and classical homogeneous Gaussian
random fields ....................................... 65
4.2. Regularity of solutions to stochastic Volterra
equations ........................................... 69
4.2.1. Stochastic integration ....................... 69
4.2.2. Stochastic Volterra equation ................. 73
4.2.3. Continuity in terms of Γ ..................... 74
4.2.4. Some special cases ........................... 76
4.3. Limit measure to stochastic Volterra equations ...... 78
4.3.1. The main results ............................. 79
4.3.2. Proofs of theorems ........................... 82
4.3.3. Some special case ............................ 84
4.4. Regularity of solutions to equations with infinite
delay ............................................... 85
4.4.1. Introduction and setting the problem ......... 85
4.4.2. Main results ................................. 88
References ..................................................... 95
Index .......................................................... 99
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