In this paper, we establish the integration by parts formula for the solution of fractional noise driven stochastic heat equations using the method of coupling. As an application, we also obtain the shift Harnack inequalities.
Xiuwei YIN
. AN INTEGRATION BY PARTS FORMULA FOR STOCHASTIC HEAT EQUATIONS WITH FRACTIONAL NOISE*[J]. Acta mathematica scientia, Series B, 2023
, 43(1)
: 349
-362
.
DOI: 10.1007/s10473-023-0119-2
[1] Biagini F, Hu Y, Øksendal B, Zhang T.Stochastic Calculus for Fractional Brownian Motions and Applications. Springer, 2008
[2] Boufoussi B, Hajji S.Stochastic delay differential equations in a Hilbert space driven by fractional Brownian motions. Statist Probab Lett, 2017, 129: 222-229
[3] Cui J, Yan L.Controllability of neutral stochastic evolution equation driving by fractional Brownian motions. Acta Math Sci, 2017, 37B(1): 108-118
[4] Decreusefond L, Üstünel A S.Stochastic analysis of the fractional Brownian motions. Potential Anal, 1998, 10: 177-214
[5] Driver B.Integration by parts for heat kernel measures revisited. J Math Pures Appl, 1997, 76: 703-737
[6] Fan X L.Integration by parts formula and applications for SDEs driven by fractional Brownian motions. Stoch Anal Appl, 2015, 33: 199-212
[7] Fan X L.Stochastic Volterra equations driven by fractional Brownian motion. Front Math China, 2015, 10(3): 595-620
[8] Hu Y.Some recent progress on stochastic heat equations. Acta Math Sci, 2019, 39B: 874-914
[9] Hu Y, Nualart D.Stochastic heat equation driven by fractional noise and local time. Prob Theory Related Fields, 2009, 143: 285-328
[10] Huang X, Liu L X, Zhang S Q.Integration by parts formula for SPDEs with multiplicative noise and its applications. Stoch Dyn, 2019, 18(6): 1950024
[11] Mishura Y S.Stochastic calculus for fractional Brownian motions and related processes. Springer, 2008
[12] Nualart D, Ouknine Y.Regularization of quasilinear heat equations by a fractional noise. Stoch Dyn, 2004, 4: 201-221
[13] Wang F Y.Integration by parts formula and shift Harnack inequality for stochastic equations. Ann Probab, 2014, 42: 994-1019
[14] Wang F Y.Integration by parts formula and applications for SPDEs with jumps. Stochastic, 2016, 88: 737-750
[15] Wang F Y.Harnack Inequality for Stochastic Partial Differential Equations. Springer, 2013
[16] Walsh J.An Introduction to SPDEs, École d’été de Probabilités St Flour XIV, Lect Notes in Math, 1180. Springer, 1986
[17] Zhang S Q.Shift Harnack inequality and integration by part formula for semilinear SPDE. Front Math China, 2016, 11: 461-496
